A1 -Homotopy Theory of Schemes

Anwar Alameddin Supervisor: Vladimir Guletski˘ı University of Liverpool, UK

e-mail: [email protected]

April 6, 2016

Contents Introduction

1 3 3 4 18 18

Chapter 2. Model Categories 1. Preliminaries 2. Homotopy in Model Categories 3. Quillen’s Pair and Derived Functors 4. Existence of Model Structures 5. Cofibranty Generated Model Categories 6. Proper Model Categories 7. Simplicial Model Category 8. Localisation of Model Categories

19 19 33 39 42 68 79 81 86

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Chapter 1. Algebraic Topology Preliminary Notation 1. Homotopies 2. CW Complexes 3. Compactly Generated Hausdorff Spaces

109 109 110 110 118 120

Chapter 4. 1. The 2. The 3. The

123 123 132 166

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Chapter 3. The Model Structure on sSet 1. Weak Equivalences 2. Cofibrations 3. Kan Fibrations 4. The Canonical Model Structure 5. Kan Extension Ex∞

Simplicial Pre-sheaves Category of Simplicial Pre-sheaves Model Structures on Simplicial Pre-sheaves Model Structures on Simplicial Sheaves

Appendix A. Simplicial and Cosimplicial Objects 1. Triangulated Spaces 2. Simplicial Category 3. Extended Simplicial Category 4. (Co)Simplicial Objects 5. The Simplicial Homotopy Theory of Simplicial Objects

169 169 170 171 171 172

Appendix B. Simplicial Sets 1. Geometric Realisation and Singular Functors 2. Generators of Simplicial Sets 3. The Category of Simplicial Sets 4. Classical Examples of Simplicial Sets 5. Function Complexes and Cartesian Product 6. Abstract Realisations 7. Examples of Simplicial Objects 8. Complexes of a Simplicial Sets 9. Pointed Simplicial Sets 10. Topological Homotopy of Simplicial Sets 11. Simplicial Homotopy

177 177 179 182 184 191 193 194 194 195 195 198 i

ii

12.

Homotopy Category

201 203

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Bibliography

Introduction These notes aim to develop a comprehensive description of A1 -homotopy theory of schemes. It is based mainly on [MV99], and expands material in each chapter to the level understood by the beginners. It contains the author’s understanding of some of the materials that are being studied in the frame of undertaking the PhD degree at the University of Liverpool under the supervision and direction of Dr Vladimir Guletski˘ı.

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The current version of theses notes studies mainly model categories, their localisation and examples needed for A1 -homotopy theory of schemes. That starts with a recollection of motivational results from algebraic topology. It is far from being complete, and only meant to set the topological motivation for the study of homotopical algebra and model categories. Then, in chapter 2, we provide a more comprehensive survey on model categories and their localisations. In the study of model categories, we mainly follow [Hov99],[Hir03] and [Qui67], and the layout of the material is set having the objective of understanding localisation of model categories and the argument of the existence of Bousfield localisation, as in [Hir03] and the local injective model structure on simplicial (pre)-sheaves, as in [Jar87]. The transfinite small object argument, theorem 4.48, and the reduction of right lifting property to the boundedness conditions, theorem 5.15, are the main arguments needed to understand localisation of model categories.

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In the next two chapters, we move to study examples of model categories, namely the classical model category of simplicial sets due to [Qui67], in chapter 3. This chapter is mainly based on [Hov99] and [GJ09]. We also provide an account on simplicial objects and sets in the appendix. Then, based on the understanding of the classical model structure of simplicial sets we elaborate the local injective model category of simplicial (pre)sheaves on a Grothendieck site, due to [Jar87], in chapter 4. This chapter expands on [ibid.] and [MV99]’s description of the local injective model structure. These note are far from being complete, they are being updated regularly and they will be expanding on other topics related to motive homotopy theory. April 6, 2016

1

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CHAPTER 1

Algebraic Topology Algebraic topology studies algebraic invariance of topological spaces, i.e. invariants up to homeomorphisms. It consists of different directions of study, namely homotopy, (co)homology, CW-complexes. In this chapter we mainly follow [Swi75], [May99] and [Hat02].

Preliminary Notation Throughout the study of algebraic topology n important rule is played by some distinguish Euclidean geometrical objects, and they are recalled here by.

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Definition 0.1 (n−Cube). Let n ≥ 1. Then, the n−Euclidean cube is defined to be the topological sub-space I n ∶= {x = (x1 , x2 , . . . , xn ) ∈ Rn ∣0 ≤ xi ≤ 1, i = 1..n}.

In particular, I 1 = I is the unit interval.

Definition 0.2 (n−Sphere). Let n ≥ 0. Then, the n−sphere is defined to be the topological sub-space n+1

S n ∶= {x = (x1 , x2 , . . . , xn+1 ) ∈ Rn+1 ∣ ∑ x2i = 1}. i=1

In particular, S 0 = {±1}, and S 1 is the unit circle in R2 .

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Definition 0.3 (n−Disk). Let n ≥ 1. Then, the (n)−disk is defined to be the topological sub-space n

Dn ∶= {x = (x1 , x2 , . . . , xn ) ∈ Rn ∣ ∑ x2i ≤ 1}. i=1

Hence, ∂Dn+1 = S n , ∀n ≥ 0.

Definition 0.4 (Standard n−Simplex). Let n ≥ 0. Then, the topological standard n−simplex is defined to be the topological sub-space n

∆ntop = {(t0 , t1 , ..., tn ) ∈ Rn+1 ∣ ∑ ti = 1, ti ≥ 0} i=0

t2 t1

v2 v1 0 v0 t0

Figure 1. ∆2top 3

4

1. Homotopies Although algebraic topology studies algebraic invariant up to homeomorphisms. Most of these invariants are actually up to homotopy type. Hence, we start by recalling the notion of homotopy and its related concepts and properties. Definition 1.1 (Homotopy). Let f0 , f1 ∶ L → X be continuous maps. A topological homotopy (or homotopy) from f0 to f1 is defined to be a continuous map H ∶ L × I → X, such that H(−, 0) = f0 , and H(−, 1) = f1 i.e. that makes the following diagram commutative: i0

L

/ L×I o

i1

L

H

(  v X

f0

f1

where i0 = idL × ∣ ∂11 ∣ and i1 = idL × ∣ ∂10 ∣. We say that f0 is homotopic to f1 if there exists such a homotopy H, and we write f0 ∼H f1 ∶ L → X, or just f0 ∼ f1 if no confusion is caused.

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For any homotopy we can define a family of maps ft ∶= H(−, t) for t ∈ I. Also, notice that for every x ∈ X, one can consider the continuous map ιx ∶ ∗ → X with ιx (∗) = x, then one can talk of homotopy from point x0 to x1 in X as a homotopy ιx0 ∼ ιx1 ∶ ∗ → X. The homotopy given in definition 1.1 should have been called a left homotopy, whereas the right homotopy is defined as follows: Definition 1.2 (Right Homotopy). Let f0 , f1 ∶ L → X be continuous maps. A right homotopy from f0 to f1 is defined to be a continuous map Hr ∶ L → X I such that (Hr (−))(0) = f0 , and (Hr (−))(1) = f1

i.e. that makes the following diagram commutative:

L

f0

∗

D

f1

 Xo

Hr

p1

 XI

p0

 /X

∗

where p0 = ∣ ∂11 ∣ and p1 = ∣ ∂10 ∣ are the evaluation maps at 0 and 1, respectively. We say that f0 is right r homotopic to f1 if there exists such a right homotopy H, and we write f0 ∼H f1 ∶ L → X to emphasise the r considered right homotopy, or just f0 ∼ f1 if no confusion is caused. However, since I is a locally compact Hausdorff space, then we have a natural bijection of sets, natural in L and X: ≅ Top(L × I, X) Ð→ Top(L, X I ) Hence, there is a natural correspondence between left homotopies (live in the left set) and right homotopies (live in the right set), which also explains the terminology! Therefore, we can use both notations interchangeably. Lemma 1.3. Let L, X be topological spaces, such that Y ≠ ∅. Then, the homotopy relation on Top(L, X) is an equivalence relation, that is compatible with composition. Proof. (1) (Reflexivity) For any continuous map f ∶ L → X, we have the constant homotopy H ∶= f ○ P r1 , where P r1 ∶ L × I → L is the projection map. (2) (Symmetry) Let f0 ∼ f1 then there is a homotopy H ∶ L × I → X from f0 to f1 . Let H ′ ∶ L × I → X defined by H ′ = H ○ (idL × InvI ) where InvI ∶ I → I given by InvI (t) = 1 − t. Then, one sees that H ′ is a homotopy from f1 to f0 . Hence, f1 ∼ f0 . (3) (Transitivity) Let f0 ∼ f1 and f1 ∼ f2 , then there are Homotopies H and H ′ from f0 to f1 and from f1 to f2 , respectively. Then, define H′ ⋅ H ∶ L × I (1)

(l, t)

X

Ð→ ↦

{

H(l, 2t) H ′ (l, 2t − 1)

t ≤ 1/2 t > 1/2

One sees easily that H ′ ⋅ H is a homotopy from f0 to f2 , hence f0 ∼ f2 .

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On the other hand, let f0 ∼H f1 ∶ X → Y , and g0 ∼H ′ g1 ∶ Y → Z. Then, define H′ ○ H ∶ X × I → Z given by (H ′ ○ H)(−, −) = H2 (H1 × P r2 )

(2)

Then, (H ′ ○ H)(−, 0) = g0 f0 and (H ′ ○ H)(−, 1) = g1 f1 . Hence, g0 f0 ∼H ′ ○H g1 f1 .



Then, the homotopy category of topological spaces is defined to be the quotient category hTop = Top/ ∼

where ∼ is the homotopy relation. The objects of hTop are topological spaces, wheras its morphisms are homotopy classes of continuous maps. It is common to use the notation [X, Y ] ∶= hTop(X, Y ) Then, there is an evident functor (3)

Ho ∶ Top → hTop

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that is the identity on objects and sends each continuous map to the its homotopy class. for a continuous map f , we denote Ho(f ) by [f ].

DR AF

Definition 1.4 (Homotopy Equivalence). Let f ∶ X → Y be continuous map. Then, f is called a homotopy equivalence from X to Y if [f ] is an isomorphism in hTop, i.e. if there exist a continuous map g ∶ Y → X such that [f ][g] = [idY ] and [g][f ] = [idX ]. X is said to be homotopic equivalence to Y if there exist a homotopy equivalence f ∶ X → Y . Remark 1.5. One sees easily that homotopy equivalence is an equivalence relation on the class of objects of Top, and its equivalence classes are called homotopy types, so if X is homotopy equivalence to Y , we say that they are of the same homotopy type. Most of algebraic topology invariants are invariants under homotopy type. Example 1.6. A space which is of the homotopy type of a point is called contractable space. Hence, the unit disk Dn ’s and Euclidean spaces Rn ’s are contractable. Lemma 1.7. The homotopy category hTop coincide with the localisation of Top with respect to homotopy equivalences. Proof. Let H be the class of homotopy equivalences in Top. Then, define F∶

hTop

X [f ]

Ð→ Top[H −1 ] ↦ X ↦ f /1.

F is well-defined that ∀[f0 ], [f1 ] ∈ hTop such that [f0 ] = [f1 ], then f0 ∼H f1 for some homotopy H. TBC by the definitions of hTop, Top[H −1 ], F is bijection on objects. Let [f0 ], [f1 ] ∈ hTop such that f0 /1 = f1 /1, then h0 f0 h′0 = h1 f1 h′1 for h0 , h′0 , h1 , h′1 being homotopy equivalences. On the one hand, we need to show that the homotopy classes of homotopy equivalences are isomorphisms. Let f ∶ X → Y be a homotopy equivalences, hence there exist g ∶ Y → X such that f g ∼ idY and gf ∼ idX , i.e. [f ][g] = [idY ] and [g][f ] = [idX ] in hTop, i.e. [f ] and [g] are isomorphisms. On the other hand, we need to show that homotopic maps  Definition 1.8 (Constant Homotopy). Let H ∶ L × I → X be a homotopy. Then, H is called a constant homotopy if it factorise as H = f P r1 L×I i.e. it is independent of its second argument.

P r1

/L

f

/X

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Lemma 1.9. Let H ∶ L × I → X be a homotopy, then H is a constant homotopy iff any of the following diagrams commutes, and as a result all, for t ∈ I: L×I

// L × I

id it P r1

H

/ X.

where it ∶ L ≅ L×∗ → L×I is induced by the embedding of the point in the unit interval at t, i.e. it (−, ∗) = (−, t). Proof. Having a constant homotopy H, then the commutativity of the diagrams is straightforward. On the other hand, let the above diagram be commutative for some t ∈ I, then precomposition of the above diagram with it′ , shows that following diagram commutes: L×∗

it′

// L × I

it

H

/ X.

for every t′ ∈ I, hence the homotopy H is constant at H(−, t).



K ×I

T

Definition 1.10 (Relative Homotopy). Let i ∶ K ↪ L be a subspace, and f0 , f1 ∶ L → X be continuous maps. Then, a homotopy from f0 to f1 is called a homotopy relative to K if there exists a homotopy H from f0 to f1 such that the composition H ○ (i × idI ) is a constant homotopy, i.e. if the following diagram commute: /K

P r1

f0 ∣K =f1 ∣K

i×idI

 / X.

DR AF

 L×I

H

One sees that homotopy to relative K, defined above, is an equivalence relation on Top(X, Y ), that is compatible with composition. A case of particular interest, when studying homotopy groups later on, is when K = {l0 }, for l0 ∈ L. That, pointed continuous maps f0 , f1 ∶ (L, l0 ) → (X, x0 ) are called homotopic as continuous pointed maps if they are homotopic relative to {l0 } as continuous maps. Then, similar to the definitions above, the pointed homotopy category of pointed topological spaces is defined to be the quotient category hTop∗ = Top ∗ / ∼

where ∼ is the homotopy relative to the base point. The objects of hTop∗ are topological spaces, wheras its morphisms are homotopy classes of pointed continuous maps. It is common to use the notation [X, Y ]∗ ∶= hTop∗ ((X, x0 ), (Y, y0 )) However, when no confusion can arise the star will be suppressed, and we write [X, Y ]. Also, there is an evident functor (4)

Ho∗ ∶ Top∗ → hTop∗

that is the identity on objects and sends each pointed continuous map to the its homotopy class. If no confusion may arise, then for a continuous map f we denote Ho∗ (f ) by [f ]∗ , and we might shorten the notation to [f ] if no confusion may arise. Definition 1.11 (Fibre-wise Homotopy). Let H ∶ L × I → X be a homotopy, and p ∶ X → Y a continuous map, we say that H is a fibre-wise homotopy with respect to p if the following diagram commutes: L×I P r1

 L

H

/X p

g

 / Y.

for some continuous map g ∶ L → Y . I.e. the induced homotopy pH is a constant homotopy. We say that f is ≅ fibre-wise homotopic to f ′ with respect to p if there exist a fibre-wise homotopy with respect to p H ∶ f → f ′ ∶ L → X, and we write f ∼p f ′ .

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If such g exists, then it is unique that g = pH(−, t), for any t ∈ I, in particular g = pH(−, 0). Hence H is fibre-wise homotopy with respect to p if there exists a continuous map g ∶ L → Y such that H(l, t) ∈ Xg(l) = XpH(l,0) , ∀t ∈ I. Therefore, each two homotpic points of X with respect to p lie in one fibre of p along their homotopy. I.e. this homotopy lives actually in the fibres. Notice that every homotopy H ∈ Hom(L, X)1 is is a fibre-wise homotopy with respect to X → ∗. Moreover, fibre-wise homotopy with respects to p introduce homotopy on the fibres, which justify the name, as we see below: Lemma 1.12. Let p ∶ X → Y be a continuous map, and let f, f ′ ∶ L → X, ∀y0 ∈ Y , fy0 , f ′ y0 the canonical induced morphisms on fibres. Then, f ∼p f ′ iff fy0 ∼ f ′ y0 for ∀y0 ∈ Y . Proof. consider the Cartesian diagram: // XO

f

LO ? Ly0

f

′

? // Xy0

fy0 f

p

′

/Y O ? /∗

y0

TBC

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It might be useful to rephrase the homotopy-related concepts in terms of right homotopy. . 1.1. Homotopy Groups. Fundamental group of topological spaces is an important homotopic invariant of topological spaces. In this section, we recall the construction of homotopy groups and their basic results. Let (S n , s0 ) be the pointed space of the Euclidean n−sphere, where s0 ∈ S n , then the representable functors [(S n , s0 ), −] in hTop∗ gives rise to homotopy groups as follows. Definition 1.13 (Homotopy Groups). Let n ≥ 0. Then, we define the n-homotopy ’group’ functor to be πn (−, ∗) ∶ hTop∗ → Set

(S n ,s0 )

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given by the representable functor πn (−) = h = [(S n , s0 ), −]. In particular, π0 is called the path connected components functor, π1 the fundamental group functor, whereas πn ’s are called higher homotopy groups for n ≥ 2, for reasons to be recalled afterwards. Remark 1.14. Let (X, x0 ) be a pointed topological space. Then, π1 (X, x0 ) is the set of homotopy classes of loops passing through x0 in X. Lemma 1.15. Let n ≥ 0, then the solid diagram Set ; ∗ US

hTop∗

πn

 / Set

admits a canonical dotted lift, where US ∶ Set∗ → Set is the forgetful functor. Proof. Since πn = [(S n , s0 ), −] is a representable functor on hTop∗ to Set, we only need to show that πn (X, x0 ) is a pointed set for any pointed space (X, x0 ) and that πn ([f ]) is a pointed map for every continuous pointed map f ∶ (X, x0 ) → (Y, y0 ). Let (X, x0 ) be a pointed space. Then, the set πn (X, x0 ) is non-empty. That, there exists a canonical constant map (S n , s0 ) → (X, x0 ) that sends all the points of S n to x0 , and it is usually denoted by ex0 , hence [ex0 ] ∈ πn (X, x0 ). Let f ∶ (X, x0 ) → (Y, y0 ) be a continuous pointed map. Then, the map πn ([f ]) ∶ πn (X, x0 ) → πn (Y, y0 ) preserves base points that n πn ([f ])([ex0 ]) = h(S ,s0 ) ([f ])([ex0 ]) = [f ][ex0 ] = [f ex0 ] = [ey0 ] that f is a continuous pointed map.

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Then, one sees readily that hTop∗ (X, x0 ) [f ]

(5)

Ð→ Set∗ ↦ ([(S n , s0 ), (X, x0 )], ex0 ) n ↦ h(S ,s0 ) ([f ])

is a functor, in particular it is the desired lift.



One can abuse notation and uses πn to denote the above lift hTop∗ → Set∗ . Lemma 1.16 (Homotopy Groups). Let n ≥ 1, then the solid diagram (6)

; Grp UG

hTop∗

πn

 / Set∗

admits a canonical dotted lift, where UG ∶ Grp → Set∗ is the forgetful functor. This lift is also denoted by πn . Proof. TBC.



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We notice that for n = 0, S 0 consists of two points {±1} with the base point +1. Then, each continuous pointed map from (S 0 , +1) to (X, x0 ) is determined by a point of X namely the image of −1 that +1 is always mapped to x0 . Hence, homotopy relative to +1 between such continuous pointed maps are in one-to-one correspondence with paths between their images of −1 in X. Hence, Since, there is a path between points of X iff they are in the same path connected component, then the set π0 (X, x0 ) of classes of homotopy relative to +1 between continuous pointed maps from {±1} to (X, x0 ) are in one-to-one corresponding to set of path connected components, which explained the name above. We already saw in lemma 1.15 that π0 (X, x0 ) is a pointed set. Following the proof of 1.15, the base point of π0 (X, x0 ) is the path connected component that includes x0 . The notion of path connected component functor can be generalised for topological spaces, not necessary pointed as follows. Definition 1.17 (Path Connected Components π0 (X)). The path connected components functor is defined to be π0 (−) ∶ hTop → Set

D

(7)

∗

given by the representable functor π0 (−) = h = [∗, −], where ∗ is a singleton topological space. A similar discussion to the previous one, shows that π0 (X) is the set of path connected components of X. Notice that, for pointed spaces (X, x0 ), the above defined π0 (X, x0 ) can be constructed from πn (X) as in the below lemma. Lemma 1.18 (π0 (X, x0 )). The solid diagram 5 Set∗ US

hTop∗

UH

/ hTop

π0

 / Set

admits a canonical doted lift, where UH , US are the forgetful functors. Moreover, this lift coincide with π0 (−, ∗) ∶ hTop∗ → Set∗ , given above in lemma 1.15. Proof. Since π0 = [∗, −] is a representable functor on hTop to Set, we only need to show that π0 (X) is a pointed set for any pointed space (X, x0 ) and that π0 ([f ]) is a pointed map for every continuous pointed map f ∶ (X, x0 ) → (Y, y0 ). Let (X, x0 ) be a pointed space. Then, the set π0 (X) is non-empty. That, [ιx0 ] the pat-connected component that include x0 , denoted by Cx0 , belongs to π0 (X). Let f ∶ (X, x0 ) → (Y, y0 ) be a continuous pointed map. Then, the map π0 ([f ]) ∶ π0 (X) → π0 (Y ) preserves base points, i.e. it sends Cx0 to Cy0 , that 0

π0 ([f ])(Cx0 ) = hS ([f ])([ιx0 ]) = [f ][ιx0 ] = [f ιx0 ] = [ιy0 ] = Cx0

9

that f is a continuous pointed map. Then, one sees readily that hTop∗ (X, x0 ) [f ]

Ð→ Set∗ ↦ ([S n , X], Cx0 ) 0 ↦ hS ([f ])

is the desired functor.



getting back to the homotopy groups of pointed spaces. Let f ∶ (X, x0 ) → (Y, y0 ) be a pointed space, then we denote πn ([f ]) by f∗ if no confusion can arise, that it sends each homotopy class [g] in πn (X, x0 ) to [f g] in πn (Y, y0 ). Lemma 1.19. Let n ≥ 0. Then, there is a unique homotopy class in [(S n , s0 ), (S n+1 , 0)]. Proof. TBC.



We denote the unique homotopy class in [(S n , s0 ), (S n+1 , 0)] by [in ]. Then, there is a canonical natural ⋅ transformation [in ]∗ ∶ πn+1 → πn , given by precomposition by [in ].

T

Lemma 1.20. Homotopy groups commute with filtered colimits. Sketch of proof. Let n ≥ 0, J be a filtered category,

F ∶ J → hTop∗

DR AF

be a functor, then we need to show that

colim πn (F ) ≅ πn (colim F ) J

J

Notice that the universal property of the colimit induces the canonical morphism, obtain by composition with colimit arrow i ∶ colim πn (F ) ≅ πn (colim F ) J

J

First, notice that i is surjective that, ∀ψ ∈ πn (colim F ), there exists a representative f ∶ (S n , s0 ) → colim F J

J

of ψ. Hence, {f −1 (F (j))∣j ∈ J} forms a covering for (S n , s0 ). Since S n is compact, then there is a finite full subcategory if ∶ Jf of J such that {f −1 (F (j))∣j ∈ Jf } is a covering for S n . Since, J is filtered, then there exist jf ∈ J such that jf = colim if . Then, im f ⊂ F (jf ), hence there exist a continuous map gf ∶ (S n , s0 ) → F (jf ) Jf

such that gf (s) = f (s). Then, i([gf ]) = [f ].

Now, let ϕ, ϕ′ ∈ colim πn (F ) such that i(ϕ) = i(ϕ′ ). Then, there exist j, j ′ ∈ J such that ϕ ∈ πn (F (j)), ϕ ∈ J

πn (F (j ′ )). Let, g ∶ (S n , s0 ) → F (j), g ′ ∶ (S n , s0 ) → F (j ′ ) be representatives for ϕ, ϕ′ , respectively. Let ψ ∶ (S n , n) → colim F ....TBC.  J

Same result and proof applies for the path connected components functor. An important role of topological homotopy groups that they distinguish homotopy equivalence, as recalled below in Whitehead theorem. Definition 1.21 (Weak Equivalence). Let f ∶ (X, x0 ) → (Y, y0 ) be a continuous pointed map. Then, f is called a weak equivalence iff it induces isomorphisms f∗ ∶ πn (X, x0 ) → πn (Y, y0 ) for n ≥ 0 The definition of homotopy groups shows implies that every homotopic equivalence is a weak equivalence. The inverse statement is not always true. However, it holds for CW complexes, due to Whitehead. Hereby, we recall Whitehod theorem, followed by a counter example of the general case. Theorem 1.22 (Whitehead Theorem). Let f ∶ (X, x0 ) → (Y, y0 ) be a continuous pointed map between CW complexes. Then, f is a homotopic equivalence iff it is a weak equivalence. Proof. See [Swi75, P89, Th.6.32].



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Counter Example 1.23. For counter example, see [Swi75, P89, Th.6.32]. For topological spaces with finite fundamental groups see [Tei92]. 1.2. Smash Product and Function Spaces. In this following, we recall a very important result in homotopy theory, that allows us to express higher homotopy groups in terms of the fundamental homotopy group. In order to do so, we need to recall wedge sum, smash product, suspension, loop space, cone and paths. Recall that the category Top is a complete and cocomplete category, in particular it has the Cartesian product × as its product and the disjoint union as its coproduct ⊔. Then, a natural question arise whether same notations hold in Top∗ , and if not, does Top∗ possess an equivalent notation. It is easy to see that the disjoint union of pointed space is not a coproduct in Top∗ that there is no canonical way of choosing a base point of the disjoint union that satisfy the coproduct conditions. However, the Cartesian product of pointed spaces has a canonical base point, namely the pair of base points. And, it is easy to see that it satisfies the condition of product in the category Top∗ . Definition 1.24 (Wedge Sum). Let (X, x0 ), (Y, y0 ) be pointed spaces. Then, the wedge sum (X, x0 ) ∨ (Y, y0 ) is defined to be the pointed topological space given by the topological spaces

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(X ⊔ Y )/ ∼ where ∼ is the smallest equivalence relation that identifies x0 and y0 in X ⊔ Y , and the base point the equivalent class of both x0 , y0 . If the base points are understood, notation might be abused and one may write (X ∨ Y, ∗) instead of (X, x0 ) ∨ (Y, y0 ). The wedge sum of pointed spaces is basically gluing these spaces along their base points. Then, one can see easily that ∨ is the coproduct in Top∗ . Hence, the universal property of coproduct defines a bifunctor (8)

− ∨− ∶ Top∗ × Top∗ Ð→ Top∗

Particularly, for pointed continuous maps f ∶ (X, x0 ) → (Y, y0 ), f ′ ∶ (X ′ , x′0 ) → (Y ′ , y0′ ), then the base point of X ∨ X ′ is mapped to the base point of Y ∨ Y ′ through the continuous map f ⊔ f ′ .

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Definition 1.25 (Smash Product). Let (X, x0 ), (Y, y0 ) be pointed spaces. Then, the smash product (X, x0 ) ∧ (Y, y0 ) is defined to be the pointed topological space given by the topological spaces (X × Y )/(X ∨ Y )

and the base point the equivalent class of points of X ∨ Y .Where the above quotient is taken with respect to the evident embeddings of (X, x0 ), (Y, y0 ) in (X × Y, (x0 , y0 )). If the base points are understood, notation might be abused and one may write (X ∧ Y, ∗) instead of (X, x0 ) ∧ (Y, y0 ). We denote the equivalence class of (x, y) ∈ X × Y by < x, y >. Recall that the quotient spaces are topologised by requiring the projection from the original space to the quotient to be continuous. Then, the continuity of projection (X × Y ) Ð→ (X ∧ Y ) = (X × Y )/(X ∨ Y ) shows that X ∧ Y is obtained from X × Y , by contracting X ∨ Y into a point. Notice that if f ∶ (X, x0 ) → (Y, y0 ), f ′ ∶ (X ′ , x′0 ) → (Y ′ , y0′ ) are continuous pointed maps, then the base point of X ∧ X ′ is mapped to the base point of Y ∧ Y ′ through the continuous map f × f ′ . That induces a continuous pointed map f ∧ f ′ ∶ X ∧ X ′ → Y ∧ Y ′ in Top∗ . Moreover, one can see that ∧ defines a bifunctor (9)

− ∧− ∶ Top∗ × Top∗ Ð→ Top∗

Then, for any pointed space (X, x0 ), we have the functor (10)

(X, x0 ) ∧ − ∶ Top∗ Ð→ Top∗

The following example is of a particular interest in understanding homotopy groups. Example 1.26. Let (X, x0 ), (Y, y0 ) be the pointed spaces of the pink and red circles, in the below figure, respectively; with base points being the intersection point. Then, the Cartesian product (X, x0 ) × (Y, y0 ) in the pointed space of the tours and the intersection point as its base point. Whereas, (X, x0 ) ∨ (Y, y0 ) is the gluing of the two spaces through their intersection point, with the intersection point as its base point. Whereas the smash product (X, x0 ) ∧ (Y, y0 ) is obtain by contracting the two circle into a point. That can be done by first

11

Figure 2. S 1 × S 1 , [Kri14]. contracting the pink circle into a point, then contracting the red circle to obtain a figure homomorphism to S 2 . Hence, (S 1 , s0 ) ∧ (S 1 , s0 ) = (S 2 , s0 )

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(11)

Lemma 1.27. Let (X, x0 ) be a pointed space, such that X is Hausdorff and locally compact. Then, the representable functor h(X,x0 ) lifts to Top∗ , i.e. the below solid diagram (12)

<

Top∗

UT

Top∗

h(X,x0

 / Set )

admits the dotted lift, where UT is the forgetful functor.

Proof. We need to show that h(X,x0 ) (Y, y0 ) is a pointed space for any pointed space (Y, y0 ) and that (f ) is a continuous pointed map for every continuous pointed map f ∶ (Y, y0 ) → (Z, z0 ).

(X,x0 )

h

D

(Y, y0 )(X,x0 ) is endued with compact-open topology. There exists a canonical base point of (Y, y0 )(X,x0 ) , namely the constant map fy0 that maps points of X to y0 . Then, ((Y, y0 )(X,x0 ) , fy0 ) is a pointed space. h(X,x0 ) (f ) preserves base points that h(X,x0 ) (f )(fy0 ) = f fy0 = fz0 . Then, one sees readily that −(X,x0 ) ∶

Top∗ (Y, y0 ) f

Ð→ ↦ ↦

T op∗ ((Y, y0 )(X,x0 ) , fy0 ) h(X,x0 ) (f )

is the desired lift.



Definition 1.28 (Function Space). Let (X, x0 ) be a pointed space, such that X is Hausdorff and locally compact. Then, the function space functor is defined to be −(X,x0 ) ∶ Top∗ → Top∗ given in the above lemma 1.27. We have seen that we have functors (X, x0 ) ∧ −, −(X,x0 ) ∶ Top∗ → Top∗ . When, we have functors Top∗ → Top∗ , it is natural to ask if such functrs respects homotopy, i.e. if they induce canonically functors on hTop∗ → hTop∗ . This proves positive for smash product and function spaces, as we see below. Lemma 1.29. Let (X, x0 ) be a pointed space. Then, the smash product functor is extended to the pointed homotopy category of topological spaces hTop∗ (X, x0 ) ∧ − ∶ hTop∗ → hTop∗ .

12

Proof. We need to show that the solid diagram Top∗

(X,x0 )∧−

/ Top ∗

Ho∗

Ho

 hTop∗

 / hTop ∗

admits the doted extension. Then, define F∶

hTop∗ (Y, y0 ) [f ]

Ð→ ↦ ↦

hTop∗ (X ∧ Y, ∗) [(X, x0 ) ∧ f ] .

F is well-defined on objects, and it is well-defined on morphisms that: For [f0 ] = [f1 ] ∶ (Y, y0 ) → (Z, z0 ) in hTop∗ , there exist a homotopy H ∶ (Y × I, (y0 , 0)) → (Z, z0 ) such that f0 ∼H f1 ∶ (Y, y0 ) → (Z, z0 ). Define (X ∧ Y, ∗) × (I, 0) (< x, y >, t)

Ð→ (X ∧ Z, ∗) ↦ < x, H(y, t) >

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H′ ∶

H ′ is well defined that for (< x, y >, t) = (< x′ , y ′ >, t′ ) ∈ (X ∧ Y, ∗) × (I, 0) and t = t′ , then < x, y >=< x′ , y ′ >. We distinguish two cases: ● < x, y >≠ ∗, then the definition of the smash product implies that x = x′ and y = y ′ , hence H ′ (< x, y >, t) = (< x′ , y ′ >, t′ ). ● < x, y >= ∗, i.e. either x = x0 or y = y0 , without loose of generality, let x = x0 . Then, either x′ = x0 or y ′ = y0 . If x′ = x0 , then H ′ (< x, y >, t) =< x0 , H(y, t) >= ∗ =< x0 , H(y ′ , t′ ) >= H ′ (< x′ , y ′ >, t′ ). Otherwise, y ′ = y0 , i.e. H(y ′ , t′ ) = H(y0 , t′ ) = z0 , hence

H ′ (< x, y >, t) =< x0 , H(y, t) >= ∗ =< x′ , z0 >=< x′ , H(y ′ , t′ ) >= H ′ (< x′ , y ′ >, t′ ).

D

Hence, H ′ is well-defined. Also, the above argument shows that H ′ (∗, t) = ∗, i.e. H ′ is a homotopy of continuous pointed maps H ′ (< −, − >, 0) to H ′ (−, 1). Notice that H ′ (< −, − >, 0) =< −, H(−, 0) >=< −, f0 (−) >= (X, x0 )∧f0 , and H ′ (< −, − >, 1) =< −, H(−, 1) >=< −, f1 (−) >= (X, x0 )∧f1 Therefore, [(X, x0 ) ∧ f0 ] = [(X, x0 ) ∧ f1 ]. Hence, F is well-defined on morphisms. Since (X, x0 ) ∧ − is a functor, then one can see readily that F is a functor, and it is the desire extension.



Remark 1.30. It is common to abuse notation and use (X, x0 ) ∧ − to denote F in the proof of lemma 1.29, above. Lemma 1.31. Let (X, x0 ) be a pointed space, such that X is Hausdorff and locally compact. Then, the function space functor is extended to the pointed homotopy category of topological spaces hTop∗ −(X,x0 ) ∶ hTop∗ → hTop∗ . Proof. Similar to the proof of lemma 1.29.



Restricting attention to ’nice’ enough topological spaces, one obtain an adjunction between smash products and function spaces. Theorem 1.32. Let (X, x0 ), (Y, y0 ), (Z, z0 ) be pointed spaces, such that X, Z are Hausdorff, and Z locally compact. Then, there is a natural isomorphism (13)

⋅

[(Z ∧ X, ∗), (Y, y0 )] ≅ [(X, x0 ), ((Y, y0 )(Z,z0 ) , fz0 )].

Hence, (Z, z0 ) ∧ − is a left adjoint of −(Z,z0 ) in locally compact Hausdorff. Proof. See [Swi75, P12].



Among pointed spaces, we distinguish two spaces of particular interest in homotopy theory, namely the pointed unit circle (S 1 , s0 ) and the pointed unit interval (I, 0), both of which are Hausdorff locally compact. Hereby, we study the consequences of the above theorem 1.32, for (Z, z0 ) being either (S 1 , s0 ) or (I, 0).

13

Definition 1.33 (Reduced Suspension). The reduced suspension functor is defined to be the functor Σ ∶ Top∗ → Top∗ given by Σ = (S 1 , s0 ) ∧ −. Let (X, x0 ) be a pointed space. Then, Σn (X, x0 ) is called the n− fold iterated reduced suspension of (X, x0 ), and is usually denoted by (Σn X, ∗). Also, due to lemma 1.29, the reduced suspension functor can be though of to be: Σ ∶ hTop∗ → hTop∗ . Example 1.34. The reduced suspension of the n−sphere is homeomorphic to the (n + 1)−sphere. Definition 1.35 (Loop Space). The loop space functor is defined to be the function space functor Ω ∶ Top∗ → Top∗ given by Ω = −(S ,s0 ) 1 as in lemma 1.27. Let (X, x0 ) be a pointed space. Then, (X, x0 )(S ,s0 ) is denoted by (ΩX, ωx0 ), where ωx0 is 1 the constant loop in X that maps every point of S to x0 , which is the base point of the loop space. (Ωn X, ω0 ) is called the n−loop space of (X, x0 ). Also, due to lemma 1.31, the loop space functor can be though of to be:

T

1

Ω ∶ hTop∗ → hTop∗ .

DR AF

Then, theorem 1.32 implies that reduced suspension functor is a left adjoint to loop space functor, as functors of the homotopy categories, when being restricted to Hausdorff spaces, as follow from the following corollary. Corollary 1.36. Let (X, x0 ), (Y, y0 ) be pointed spaces, such that X is Hausdorff. Then, there is a natural isomorphis: ⋅

(14)

[(ΣX, ∗), (Y, y0 )] ≅ [(X, x0 ), (ΩY, ωy0 )]

Corollary 1.37. Let (X, x0 ) be a pointed space, n ≥ 0. Then, corollary 1.36 and example 1.34 impels that (15) πn+1 (X, x0 ) = [(S n+1 , s0 ), (X, x0 )] = [(ΣS n , ∗), (X, x0 )] = [(S n , s0 ), (ΩX, ω0 )] = . . . = π1 (Ωn X, ω0 ) = π0 (Ωn+1 X, ω0 ) Since π0 (X, x0 ), as a set, is the set of path connected components of X, hence X is connected iff π0 (X, x0 ) is trivial for every x0 ∈ X. Then, it is natural to give the following definition in spirit of corollary 1.37. Definition 1.38 (n−connected). Let X be a non-empty topological space, n ≥ 0. Then, X is called n−connected iff all πk (X, x0 ) are trivial for all 0 ≤ k ≤ n, x0 ∈ X. Similarly, to the above construction, one may consider the pointed unit interval instead of the pointed unit circle to obtain cone and path functors. Definition 1.39 (Cone). The cone functor is defined to be the functor C ∶ Top∗ → Top∗ given by C = (I, 0) ∧ −. Let (X, x0 ) be a pointed space. Then, C(X, x0 ) is denoted by (CX, ∗). Also, due to lemma 1.29, the cone functor can be though of to be: C ∶ hTop∗ → hTop∗ . Definition 1.40 (Path Space). The path space functor is defined to be the function space functor P ∶ Top∗ → Top∗ given by

P = −(I,0) as in lemma 1.27. Let (X, x0 ) be a pointed space. Then, (X, x0 )(I,0) is denoted by (P X, px0 ), where px0 is the constant path in X that maps every point of I to x0 , which is the base point of the path space. (P n X, p0 ) is called the n−path space of (X, x0 ). Also, due to lemma 1.31, the path space functor can be though of to be: P ∶ hTop∗ → hTop∗ .

14

Hence, the cone functor is a left adjoint to path space functor, as functors of the homotopy categories, when being restricted to Hausdorff spaces, as follow from the following corollary. Corollary 1.41. Let (X, x0 ), (Y, y0 ) be pointed spaces, such that X is Hausdorff. Then, there is a natural isomorphis: ⋅

(16)

[(CX, ∗), (Y, y0 )] ≅ [(X, x0 ), (P Y, py0 )]

1.3. Relative Homotopy Groups. Let Top2∗ be the category of pointed pair spaces, i.e. the category with objects being triple (X, U, x0 ), where (U, x0 ) ⊆ (X, x0 ) is a pointed topological sub-space; and morphisms f ∶ (X, U, x0 ) → (Y, V, y0 ) for f is a continuous map f ∶ X → Y that preserves the sub-space and the base point. Then, one can generalise the notions of homotopy and homotopy groups to the settings of Top2∗ . These notations are related to the ordinary one, as shown in lemma 1.49. Definition 1.42 (Homotopy of pointed pair spaces). Let f0 , f1 ∶ (X, U, x0 ) → (Y, V, y0 ) be pointed maps of pair spaces. Then, a homotopy from f0 to f1 is a continuous map H ∶ (X, U, x0 ) × I → (Y, V, y0 ) such that H(−, t) ∶ (X, U, x0 ) → (Y, V, y0 ) is a pointed map of pair spaces, H(−, 0) = f0 and H(−, 1) = f1 . If such H exist, then f0 is said to be homotopic to f1 , and we write f0 ∼H f1 .

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One sees that the homotopy of pointed pair spaces is an equivalence relation on Top2∗ ((X, U, x0 ), (Y, V, y0 )), which is compatible with composition. Hence, one can consider the category hTop2∗ with objects being pointed pair spaces, and morphism being the homotopy classes of pointed maps of pair spaces. We carry out the notation, and write [(X, U, x0 ), (Y, V, y0 )] ∶= hTop2∗ ((X, U, x0 ), (Y, V, y0 )). In the definition of the ordinary homotopy groups, we considered homotopy classes of maps from the pointed n−sphere into pointed topological spaces. However, intuitively speaking, if we open the n−sphere from the base point and flatten it, we get the pointed pair space (Dn , S n−1 , s0 ), with s0 ∈ S n−1 the boundary of Dn . Then, similar to the ordinary case the representable functors [(Dn , S n−1 , s0 ), −] in hTop2∗ gives rise to relative homotopy groups as follows. Definition 1.43 (Relative Homotopy Groups). Let n ≥ 1. Then, we define the n−relative homotopy ’group’ functor to be πn (−, ∗) ∶ hTop2∗ → Set n

,S n−1 ,s0 )

D

given by the representable functor πn (−, ∗) = h(D

= [(Dn , S n−1 , s0 ), −].

The 0−relative homotopy group will be made sense of later on. Lemma 1.44. Let n ≥ 1, then the solid diagram Set ; ∗ US

hTop2∗

πn

 / Set

admits a canonical dotted lift, where US ∶ Set∗ → Set is the forgetful functor. Proof. Since πn = [(Dn , S n−1 , s0 ), −] is a representable functor on hTop2∗ to Set, we only need to show that πn (X, U, x0 ) is a pointed set for any pointed pair space (X, U, x0 ) and that πn ([f ]) is a pointed map for every continuous pointed map of pair spaces f ∶ (X, U, x0 ) → (Y, V, y0 ). There exists a canonical constant map of pair spaces (Dn , S n−1 , s0 ) → (X, U, x0 ) that sends all the points of Dn to x0 , and it is usually denoted by ex0 , hence [ex0 ] ∈ πn (X, U, x0 ). The map πn ([f ]) ∶ πn (X, U, x0 ) → πn (Y, V, y0 ) preserves base points that πn ([f ])([ex0 ]) = h(S

D

,S n−1 ,s0 )

([f ])([ex0 ]) = [f ][ex0 ] = [f ex0 ] = [ey0 ].

Then, one sees readily that hTop2∗

(X, U, x0 ) [f ] is the desired functor.

Ð→ Set∗ ↦ ([(Dn , S n−1 , s0 ), (X, U, x0 )], ex0 ) n n−1 ↦ h(D ,S ,s0 ) ([f ]) 

15

One can abuse notation and uses πn to denote the above lift hTop2∗ → Set∗ . Notice that every pointed space (X, x0 ) gives rise to the pointed pair space (X, x0 , x0 ). Then it is easy to see that πn (X, x0 ) = πn (X, x0 , x0 ) for n ≥ 1. We can extend the definition of relative homotopy groups for n = 0 by setting π0 (X, U, x0 ) = π0 (X, x0 )/i∗ π0 (U, x0 ) in Set∗ for the inclusion i ∶ U ⊆ X, i.e. identifying only the path connected components of (X, x0 ) that intersects with U gives π0 (X, U, x0 ). This extension becomes natural in view of lemma 1.49. Lemma 1.45. Relative Homotopy groups commute with filtered colimits. Proof. Similar to the proof of lemma 1.20, that Dn is compact space.



Lemma 1.46. Let (X, U, x0 ) be a pointed pair space, such that X is Hausdorff and locally compact. Then, the representable functor h(X,U,x0 ) lifts to Top∗ , i.e. the below solid diagram (17)

Top∗

<

UT

T

Top2∗

 / Set )

h(X,U,x0

DR AF

admits the dotted lift, where UT is the forgetful functor. Proof. TBC.



Definition 1.47 (Relative Path Spaces). Let (X, U, x0 ) be a pointed pair space. Then, the relative path space functor is defined to be P ∶ Top2∗ → Top∗ given by

P = −(I,{0,1},0)

as in the above lemma 1.46.

The relative path space could have defined similarly as well as −(I,{0,1},0) ∶ hTop2∗ → hTop∗ .

Let (X, U, x0 ) be a pointed pair space, then P (X, U, x0 ), as a set, is the set of paths in X starting at x0 ending in U . Remark 1.48. Let (X, U, x0 ) be a pointed pair space, n ≥ 0. Since πn (X, x0 ) = πn (X, x0 , x0 ). Then, the the identity map induces a map of pair spaces j ∶ (X, x0 , xx ) → (X, U, x0 ) that x0 ∈ U . Hence, there is a pointed map (homomorphism for n ≥ 2) j∗ ∶ πn (X, x0 ) → πn (X, U, x0 ). i.e. j∗ regards homotopy classes of n−loops in X passing through x0 (live in πn (X, x0 )) as homotopy classes of n−paths in X starting at x0 and ending in U (live inπn (X, U, x0 )). Also, there is a there is a pointed map (homomorphism for n ≥ 1) ∂ ∶ πn+1 (X, U, x0 ) → πn (U, x0 ) given by ∂[g] = [∂g] = [g∣S n ]. I.e. ∂ is induced by sending classes (n + 1)−paths in X that starts at x0 and ends in U to the classes of its end boundary. Lemma 1.49. Let (X, U, x0 ) be a pointed pair space. Then, there exist a long sequence j∗

(18)

j∗

i∗

∂

i∗

∂

j∗

⋯ → πn+1 (X, U, x0 ) → πn (U, x0 ) → πn (X, x0 ) → πn (X, U, x0 ) → πn−1 (U, x0 ) → πn−1 (X, x0 ) → ⋯ ∂

i∗

j∗

∂

i∗

∂

⋯ → π1 (U, x0 ) → π1 (X, x0 ) → π1 (X, U, x0 ) → π0 (U, x0 ) → π0 (X, x0 ) → π0 (X, U, x0 ) → ∗ where i ∶ U ⊆ X. Proof. See [Swi75, P38, Prop.3.9]



16

Remark 1.50. The above lemma suggests that one can think of the relative homotopy group as the homotopy group of the the resulting space when contracting U into a point, i.e. the homotopy group of the space X/U . That when πn (U, x0 ) is trivial for n ≥ 0 then all ?? are isomorphisms. The above lemma also provides an alternative definition of weakly equivalent homotopic sub-spaces. Corollary 1.51. Let (X, U, x0 ) be a pointed pair space. Then, X and U are weakly equivalent iff relative homotopy groups πn (X, U, x0 ) vanishes for all n ≥ 0. Proof. Straightforward result of the exactness of (18). That, assuming that X and U are weakly equivalent, then for n ≥ 1, i∗ ’s are isomorphism hence ker j∗ = im i∗ = πn (X, x0 ) i.e. j∗ = ∗ and im j∗ = ∗. Then, ker ∂ = im j∗ = ∗ i.e. ∂’s are injective. Then, πn (X, U, x0 ) ≅ im ∂ = ker i∗ = ∗. Simpler argument applies for n = 0. On the other hand, if πn (X, U, x0 )’s are trivial. Then, ker i∗ = im ∂ = ∗ , and im i∗ = ker j∗ = πn (X, x0 ).

T

Hence, i∗ ’s are isomorphisms.



DR AF

Definition 1.52 (Retraction). Let i ∶ K → L be a subspace. A a retraction from L onto K is a continuous map r ∶ L → K such that ri = idK . We say that K is a retract of L if there exists a retraction r ∶ L → K. Definition 1.53 (Deformation Retraction). Let i ∶ K → L be a subspace. A deformation retraction from L onto K is a homotopy from idL to a retraction ri. We say that K is a deformation retract of L. Definition 1.54 (Strong Deformation Retraction). Let i ∶ K → L be a subspace. A strong deformation retraction from L onto K is a homotopy from idL to a retraction ri relative to K. We say that K is a strong deformation retract of L. Remark 1.55. Let (X, U, x0 ) be a pointed pair space, such that U is a retract of X. Then, there exist a retraction r such that ri = idU , hence i∗ are monomorphism for n ≥ 0, and (19)

Πn (X, x0 ) ≅ im i∗ ⊕ ker r∗ ≅ Πn (U, x0 ) ⊕ Πn (X, U, x0 )

for n ≥ 2. Moreover, if r is a strong deformation retract, then ı∗ is an isomorphism, i.e. Πn (X, U, x0 ) is trivial for n ≥ 0. 1.4. Poincar´ e Groupoid (Fundamental Homotopy Groupoids). Definition 1.56. Let X be a topological space, not necessary pointed. Then, the fundamental homotopy groupoid, or Poincar´e groupoid, is defined to be the groupoid with the set of objects being the underlying set of X, and morphisms being end-point preserving homotopy classes of paths in X. It is denoted by capital Π1 (X). Notice that for any pointed spaces (X, x0 ) π1 (X, x0 ) = AutΠ1 (X) (x0 ) = Π1 (X)(x0 , x0 ). 1.5. Fibrations and Cofibrations. This section recalls fibrations and cofibrations for being useful tools to understand homotopy category. Let K be a subspace of L, having a homotopy between maps from K to Y , a natural question arises whether we can extend the homotopy to a homotopy between maps from L to Y , that restricts to the original homotopy. This question give rise to the below definition: Definition 1.57 (Homotopy Extension Property). Let i ∶ K → L be a continuous map, we say that i has the homotopy extension property with respect to X, if every solid commutative diagram in Top, admits a dotted lift: K i

 L

f

h g

/ XI > p0

 /X

17 ∗

where X I is the space of paths in X, and p0 = ∣ ∂11 ∣ gives the initial point of paths, i.e. the evaluation at 0. Notice that f is a homotopy, and if i is an inclusion, then g∣K = f (0, −). Hence, if the homotopy extension property hold for an inclusion i, then the homotopy f between maps from K to X can be extended to a homotopy between maps from L to X, which explain the name. Definition 1.58 (Hurewicz Cofibrations). Let i ∶ K → L be a continuous map, we say that i is Hurewicz cofibration if it satisfy the homotopy extension property with respect to all topological spaces X. Example 1.59. Inclusions of compactly generated Hausdorff spaces are Hurewicz cofibrations. Proof. Let i ∶ K ↪ L be an inclusion of compactly generated Hausdorff spaces, and for any commutative solid diagram: K i

 L

/ XI >

f

p0

h

 /X

g

See [Hat02].

T



DR AF

However, not every conclusion of topological spaces is a Hurewicz cofibration, as you can see in the following counter example. Counter Example 1.60.

In a dual analogue of the above treatment, having a homotopy between maps from L to Y , and a continuous map X → Y , a natural question arises whether we can lift the homotopy to a homotopy between maps from L to X, whose composition with p coincide with the original homotopy. Definition 1.61 (Homotopy Lifting Property). Let p ∶ X → Y be a continuous map, we say that p has the homotopy lifting property with respect to L, if every solid commutative diagram in Top, admits a dotted lift: L

i0

 L×I

g

h

f

/X <  /Y

p

where i0 = idL × ∣ ∂11 ∣. The homotopy f between maps from L to Y is lifted to a homotopy to X. Definition 1.62 (Hurewicz Fibrations). Let p ∶ X → Y be a continuous map, we say that p is Hurewicz fibration if it satisfy the homotopy lifting property with respect to all topological spaces L. We say that X is a Hurewicz fibrant if the unique map X → ∗ is a Hurewicz fibration. Example 1.63. Covering spaces are Hurewicz fibrations. Example 1.64. Serre fibrations are Hurewicz fibrations. Lemma 1.65. Every topological space is a Hurewicz fibrant. Proof. Let X be a topological space, and for any commutative solid diagram L i0

 L×I

g

/X < p

h

f

 /∗

then, obviously h ∶= gP r1 ∶ L × I → X is a continuous maps, and it lifts the homotopy f to X. Counter Example 1.66. Counter example of a fibration.



18

2. CW Complexes Roughly speaking CW complexes are topological spaces that can be constructed through gluing cells (open unit disks) Definition 2.1 (CW Complexes). A topological space X is called a cell complex or CW complex if it can be constructed using the following procedure, through constructing its skeletons: (1) Let X 0 be a discrete topological space, called the 0−skeleton. (2) The n−skeleton X n is constructed inductively from X n−1 , as follows: Let Cn be the indexing set of n−cells to be glued, ϕnα ∶ S n−1 → X n−1 continuous maps, ∀α ∈ Cn . Then, X n is defined to be the quotient space X n−1 ⊔( ⊔ Dn )/ ∼ where ∼ is the smallest equivalence relation α∈Cn

on X n−1 ⊔( ⊔ Dn ) such that x ∼ ϕnα (x) for x ∈ S n−1 = ∂Dn . α∈Cn

Then X is defined to be ⋃ X n , equipped with the final topology, i.e. it is topologised such that all the inclusions n≥0

X n ↪ ⋃ X n are continuous. n≥0

α∈Cn

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Notice that the αth Dn is glued to X n−1 by identifying its boundary to the image of ϕnα . Hence, X n = X n−1 ⊔( ⊔ enα ), where enα is an n−cells (open n−disk), form α ∈ Cn . 3. Compactly Generated Hausdorff Spaces The category of topological spaces fails to satisfy some of the desired properties that one needs to ..., the main issue we face in Top is that it is not a closed Cartesian category, and some of the topological spaces fails to satisfy some of the below properties. Therefore, one looks into a subcategory of a ’nicer’ topological spaces (nicer in the sense that it satisfy some interesting properties of containing CW-complexes, being Cartesian closed, complete and cocomplete). For most applications, the sucbcategory of compactly generated Hausdorff spaces, recalled below, is good enough to be considered rather than Top.

D

Recall that given two topological spaces X and Y , one can consider the topological space Y X with the underlying set of continuous maps from X to Y , endued with the compact-open topology. One would like to see if this topological spaces behave as an exponential object in the category Top. It does so for many topological spaces including compactly generated Hausdorff spaces, but not for all topological spaces. That is one of the main reasons to consider the sucbcategory of compactly generated Hausdorff spaces. Definition 3.1 (Compactly Open). Let X be a topological spaces. We say that subset U ⊆ X is compactly open, if for every continuous map f ∶ K → X with K a compact Hausdorff space, f −1 (U ) is open. Every open set of a topological space is compactly open, but the inverse does not hold in general Counter Example 3.2. An uncountable product of copies of R is not compactly generated. That the subsetU defined below is a compcatle open set, but not open. Definition 3.3 (Compactly Generated Hausdorff Spaces). Let X be a topological space, we say that X is compactly generated if every compactly open subset of X is open. If X is Hausdorff in addition, then we say that X is a compactly generated Hausdorff space. We denote the full subcategory Top with objects being compactly generated Hausdorff space by CGHaus. Example 3.4. CW-Complexes are compactly generated Hausdorff spaces. Notice that the category of compactly generated Hausdorff spaces is a closed Cartesian category with the inner hom given by the compact-open topology on the set of continuous maps of topological spaces, i.e. we have a natural bijection of sets: ≅ Top(X × Y, Z) Ð→ Top(X, Z Y )

CHAPTER 2

Model Categories

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Model categories was developed by Quillen in [Qui67] as a framework to study homotopy theories. That in the study of a homotopy theory, in a homotopy category, one wants to localise a given category with respect to an appropriate class of morphisms, like homotopy equivalences or weak homotopy equivalences in topology. Hence, this class of morphisms is usually refereed to by weak equivalences, to distinguish this localisation from quotienting out with respect to an honest equivalence relation. However, in general, when one wants to localise a category with respect to a class of morphisms, few issues arise. That is mainly due to the fact that formal localisation, as in def 8.1, does not always produce a category living in the same universe as of the original category on started with. Particularly, starting with a locally small category, i.e. a category C with C(X, Y ) are sets for all objects X, Y ∈ C, and a class S of morphisms of C, then C[S −1 ](X, Y ) might be a proper class, in general. Also, morphisms of C[S −1 ] are equivalence classes of strings of zigzags as in ??, hence the morphisms C[S −1 ] is not easy to understand what it actually represents, even for nice categories C. Therefore, it natural to look for a more understood ’representation’ of homotopy categories, and this is provided through model categories. Hence, we start by recalling the main notions and properties needed to understand model categories. Then, in following section 2, we recall homotopy categories. Afterwards, we distinguish special model categories that would be of a special importance in the study of motivic homotopy theory, particularly left proper, cellular and simplicial model categories, and left Bousfield localisation of model categories. This chapter is based mainly on Hovey [Hov99], Goerss-Jardine [GJ09], Hirschhorn [Hir03], Quillen [Qui67], Morel-Voevodsky [Voe98] and explanations and discussions with my supervisor Dr Guletski˘ı. 1. Preliminaries

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A model category are pairs of an underlying category and model structure on the classes of morphisms of the underlying category. Before recalling the definition of model categories, and recalling their properties, we need to recall morphisms properties, operators and related notions needed to understand model categories, namely the left and right lifting properties, two-out-of-three property, retract, ans classes of injections and projections of a class of morphisms. The reader who is familiar with these concepts can skip to the definition of model structure ??. Definition 1.1. Let C be a category. Then, the category of morphisms of C is defined to be the category with objects f ∶ X → Y in C, and morphisms commutative squares: X f

 Y

/ X′

d0

g

D

 / Y′

d1

in C, with the evident horizontal composition of squares. We denote the category of morphisms of C by MorC, so f, g ∈ MorC, and D ∶ f → g in MorC. Definition 1.2 (retract). Let C be a category, f, and g in C. Then, the morphism f is said to be a retract of g, if it is a retract of g as objects in MorC, i.e. if there exists commutative squares D ∶ f → g, and R ∶ g → f in C such that RD = idf in MorC. X f

 Y

d0

/ X′

r0

D

g

R

d1

 / Y′

r1

/X  /Y 19

f

X =

f

 Y

idX idf

idY

/X  /Y

f

20

Definition 1.3 (Lifting property). Let C be a category, i ∶ U → V and p ∶ X → Y in C. Then, i is said to have the left lifting property (LLP) with respect to p, p to have the right lifting property (RLP) with respect to i and the pair (i, p) to be an extension-lifting pair if and only if for any solid commutative square in C e0

U i

h

 V

e1

/X >  /Y

p

there exists a dotted lift h ∶ V → X, not necessary unique, that makes the whole diagram commutative. Let P be a class of morphism of C, we say that a morphism f has the LLP with respect to P if has the LLP with respect to every morphism of P , dually it is said to have the RLP with respect to a class I of morphism of C if has the RLP with respect to every morphism of I. Moreover, we define an I-right lifting problems of p to be such above solid commutative square with i ∈ I, and P -left lifting problems of i to be such above solid commutative square with p ∈ P . Remark 1.4. If either i is a monomorphism, or p is an epimorphism. Then, if the lift h exists, it is unique. That is the case in simplicial sets, the injective model structure of simplicial (pre-)sheaves on a Grothendieck site.

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Remark 1.5. If no such commutative solid square exist in C, then the lifting condition is satisfied automatically for not being violated, and hence p has the RLP with respect to i, and i has the LLP with respect to p. Example 1.6. Let C be a category. Then, isomorphisms in C have both the RLP and LLP with respect to any morphism in C. Proof. Let i ∶ U → V be an isomorphism in C, and p ∶ X → Y in C. It is enough to show that i has either the LLP or the RLP with respect to p and the other holds by duality. Notice that either there is no commutative solid square: e0

U

i

 V

D

e1

/X  /Y

p

and then, by remark 1.5, i has the LLP with respect to p; or, such diagram exist, then there is a lift h ∶ V → X, given by h = e0 i−1 , and i has the LLP with respect to p.  Classes of morphisms defined using left and right lifting properties are fundamental in the study of model category. Definition 1.7. Let C be a category, I a class of morphisms of C, we define: -proj) I-projectives morphisms of C to be those which have the LLP with respect to I. The class of I-projective morphisms is denoted I- proj. -inj) I-injectives morphisms of C to be those which have the RLP with respect to I. The class of I-injective morphisms is denoted I- inj. -cof) I-cofibrations morphisms of C to be those which have the LLP with respect to I- inj. The class of I-cofibration morphisms is denoted I- cof. I-cof ibrants objects X of C for which the unique morphism ∅ → X is an I- cof. -fib) I-fibrations morphisms of C to be those which have the RLP with respect to I- proj. The class of I-fibration morphisms is denoted I- fib. Remark 1.8. Let I ⊆ I ′ be two classes of morphisms of C, then one sees readily that: (1) I- cof = (I- inj)- proj, and I- fib = (I- proj)- inj. (2) I ⊆ I- cof, and I ⊆ I- fib. (3) These operations are stable in the sense that ((I- inj)- proj)- inj = I- inj,

, and

((I- proj)- inj)- proj = I- proj .

That - inj and - proj operators to (2) gives ((I- inj)- proj)- inj ⊂ I- inj,

, and

((I- proj)- inj)- proj ⊂ I- proj .

Then, substituting I = I- proj and I = I- proj, respectively, in (2) give the opposite inclusions.

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(4) I ′ - inj ⊆ I- inj, and I ′ - proj ⊆ I- proj. Hence, I- cof ⊆ I ′ - cof, and I- fib ⊆ I ′ - fib. Remark 1.9. Let I be a class of morphisms of C, then by the example 1.6, above, every isomorphism of C is in both I- inj, and I- proj. Hence, in I- cof, and I- fib. As it usually happens right problems can be combined into a problem by coproducts and left problems can be combined into a problem by products. This general principle holds here in the sense of the below lemma. Lemma 1.10. Let C be category. ● If C is cocomplete and I = {i ∶ Ui → Vi } is a sets of morphisms of C, then a morphism p ∶ X → Y of C belongs to I- inj if and only if it has the RLP with respect to the morphism ∐ i ∶ ∐ Ui Ð→ ∐ Vi . i∈I

i∈I

i∈I

● If C is complete and P = {p ∶ Xp → Yp } is a sets of morphisms of C, then a morphism i ∶ U → V of C belongs to P - proj if and only if it has the LLP with respect to the morphism ∏ p ∶ ∏ Xp Ð→ ∏ Yp . p∈P

i∈I

p∈P

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Proof. The proof of the above statement is dual. Therefore, we prove the first one, and the second is obtained by duality. Assume that p ∶ X → Y has the RLP with respect to ∐ i, and consider a commutative diagram D i∈I

/X

e0

U

i

p

D

 V

 / Y.

e1

D

[Bou77, P.213] shows the first direction. On the other hand, let p ∶ X → Y belongs to I- inj, i.e. it has the RLP with respect to I, and consider the commutative diagram D ∐ Ui

e0

/X

i∈I

∐i

p

D

 ∐ Vi

i∈I

i∈I

e1

 / Y.

Then, for every i ∈ I, the above diagram, can be extended to the commutative diagram Ui

/ ∐ Ui

e0

/X

i∈I i

∐i

 Vi

i∈I

 / ∐ Vi i∈I

p

e1

 / Y.

which admits a lift hi ∶ Vi → X. Then, by the universal property of coproduct we see that ∐ hi is a left for i∈I

D.



Lifting problems may admit multiple solution. However, in some situation it is useful to have a unique solution of a set of lifting problems. The below lemma gives a criteria to distinguish when the lift is unique. Definition 1.11. Let C be a category. ● If C is cocomplete. Let i ∶ U → V be a morphism in C, then fold at i is defined to be the unique morphism ∇i ∶ V ∐ V → V induced morphism by the push-out universal property for the commutative U

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diagram (20)

/V

i

U

iv

i

 V

⌟  / V ∐V

ih

idV

U ∇i

!  - V.

idV

● If C is complete. Let p ∶ X → Y be a morphism in C, then diagonal at p is defined to be the unique morphism ∆f ∶ X → X ×Y X induced morphism by the pull-back universal property for the commutative diagram (21)

X

idX ∆f

# X ×Y X ⌜

# /X

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πh

idX

p

πv

$  X

 / Y.

p

Lemma 1.12. Let C be a category.

● If C is cocomplete. A morphism p ∶ X → Y has the unique right lifting property (URLP) with respect to a morphism i ∶ U → V if and only of it has the RLP with respect to both i and ∇i . ● If C is complete. A morphism i ∶ U → V has the unique left lifting property (ULLP) with respect to a morphism p ∶ X → Y if and only of it has the LLP with respect to both p and ∆p .

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Proof. The proof of the above statement is dual. Therefore, we prove the first one, and the second is obtained by duality. On the one hand, assume that p has the URLP with respect to i, then in particular p has the RLP with respect to i. Consider a commutative diagram V ∐V

/X

e0

U ∇i

p

D

 V

 / Y.

e1

Then, each of the following two diagrams admit a unique lift V → X U

iv i

/ V ∐V

e0

/X

, and

U

ih i

/ V ∐V

U

 V

idV

 /V

/X

U

∇i

i

e0

p

e1

∇i

i

 /Y

 V

idV

 /V

p

e1

 / Y.

These lifts coincides because of the commutativity of (20) and p having the URLP with respect to i. Call such lift of the above two diagrams h ∶ V → X, then the universal property of push-out shows that h is a lift of D. On the other hand, assume that p has the RLP with respect to both i and ∇i , and consider a commutative diagram U i

 V

e0

D e1

/X p

 / Y.

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Since p has the RLP with respect to both i, then D admits at least a lift. Assume that hv , hh ∶ V → X are lifts of D. Then, by the universal property of push-outs we have a commutative diagram hv

∐ hh U

V ∐V U ∇i

/X = p

h

 V

e1

 / Y.

Then, by the assumption the above diagram admits the dotted lift h ∶ V → X. Then, composing with the canonical injections, we have commutative diagrams V

V iv

hv ∐ hh ! ! U /X V ∐V =

idV

U

∇i

!  V

e1

 / Y.

hv ∐ hh ! ! U /X V ∐V = U

∇i

p

h

hv

h

!  V

T

ih

idV

, and

hh

e1

p

 / Y.

DR AF

Then, by the commutativity of the above diagrams and the definition of hv ∐ hh , we have U

hh = (hv ∐ hh )ih = h = (hv ∐ hh )iv = hv . U

U

Hence, p has the URLP with respect to i.



The definition of model category we follow is Hovey’s, presented in [Hov99], which is ’relatively’ stronger than the original definition due to Quillen [Qui67], and we require the factorisation, CM5, to be functorial, therefore, we recall hereby functorial factorisation. Definition 1.13 (Factorisation System). Let C be a category. Then, a factorisation system on C is defined to be a pair of functions α, β ∶ Ob(MorC) → Ob(MorC) such that for every morphism f ∶ X → Y in C,

f = β(f ) ○C α(f ), i.e. we have the commutative diagram in C f

X α(f )

(

Z

6/ Y β(f )

where Z = dom β(f ) = codom α(f ). Definition 1.14 (Functorial Factorisation). Let C be a category. Then, a functorial factorisation on C is a pair of functors α, β ∶ MorC → MorC such that the their functions on objects form a factorisation system. Alternatively, the functorial factorisation (α, β) is given as a functor (α, β) ∶ C2 → C3 , where 2 and 3 are the categories :●

/●z

, and

:● ↺

 respectively.

●Z

/●z ?

,

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Remark 1.15. Avoid the confusion of thinking of α, β as functors C → C, they are not, but rather MorC → MorC, and they have to respect the composition of morphisms in MorC, i.e. the composition of

commutative diagrams of C, not the composition of morphisms of C. Also notice that for a D ∶ f → g in MorC, α(D) and β(D) are not necessary composable, and even if they do, it is not require to satisfy D = β(D) ○MorC α(D). Remark 1.16. The three concepts of factorisation, lifting properties and retract are interactively connected, and this might be best shown in the retract argument and its consequences, recalled below. Theorem 1.17 (The Retract Argument). Let C be a category, f ∶ X → Z in C is factorised as /Z. ?

f

X

p

i

Y Then, if f has the RLP with respect to i then its a retract of p. Dually, if f has the LLP with respect to p then its a retract of i.

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Proof. It is enough to prove either of the statements, then the other follows by duality. Let f has the LLP with respect to p. By the factorisation f = pi, we have the solid commutative square i

X

f

h

 Z

/Y . > p

 Z

Since f has the LLP with respect to p, we have a dotted lift h ∶ Z → Y , i.e. ph = idZ . Then, we have the commutative diagram X  Z

D

f

X

X . f

i

h

 /Y

 /Z

p

with the horizontal composite being the identities. Hence, f is a retract of i.



We see later that the retract argument is very useful is identifying the relation between classes of (weak) cofibrations and (weak) fibrations, lemma 1.27, among other useful implications. Lemma 1.18. Let C be a category. Then: (1) Morphisms that have the RLP with respect to a morphism i ∶ U → V are closed under retract. Therefore, I- proj, and hence I- cof, is closed under retract, for any class I of morphisms of C. (2) Morphisms that have the LLP with respect to a morphism p ∶ X → Y are closed under retract. Therefore I- inj, and hence I- fib, is closed under retract, for any class I of morphisms of C. Proof. It is enough to prove either of the statements, then the other follows by duality. Let g in C has the RLP with respect to i, and f be a retract of g. Then, we have a commutative diagram: X f

 Y

d0

/ X′

r0

D

g

R

d1

 / Y′

r1

/X  /Y

f

with the horizontal composite being the identities. Consider an arbitrary commutative square: U i

 V

e0

E e1

/X f

 / Y.

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Then, we can extend it to the solid commutative diagram: U i

 V

e0

/X

d0

/7 X ′

r0

E

f

D

g

R

e1

 /Y

d1

 / Y′

r1

/X  /Y

f

with the horizontal composite being e0 and e2 . Since g has the RLP with respect to i, there exists a lift h′ ∶ V → X ′ such that h′ i = d0 e0 and gh′ = d1 e1 . Hence, there exists h = r0 h′ ∶ V → X such that hi = r0 h′ i = r0 d0 e0 = e0 and f h = f r0 h′ = r1 gh′ = r1 d1 e1 = e1 , i.e there exist a lift of the original diagram E.  The above lemma, will be particularly useful showing the fibration and cofibration stability under retract in axiom CM3. In addition to stability under retract, the classes of I- inj and I- proj satisfy some other, standard, stability conditions, we recall in the below lemma. A similar straightforward verification proves the below two lemmas. Lemma 1.19. Let C be a category, I a class of morphisms of C. Then, the classes of I- proj and I- inj form subcategories of C, i.e. closed under composition, and contain all isomorphisms. Thus, I- cof, and I- fib are so.

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Proof. It is enough to prove that either the classes of I- proj or I- inj forms a subcategory of C, then the other follows by duality. Example 1.6 shows that the class of I- inj contains all isomorphisms. Let p ∶ X → Y , q ∶ Y → Z belong to I- inj. Then for any commutative square U

i

D

 V

/X

e0

 >Y

E

p

q

h

 /Z

e1

with i ∈ I, there exist a dotted ’partial’ lift h ∶ V → Y because q ∈∈ I- inj, i.e. pe0 = hi

, and

qh = e1 .

Hence, we have a commutative square U i

 V

/X

e0

E′

 /Y

h

p

which in turn admits a lift h′ ∶ V → X, i.e. e0 = h′ i

ph′ = h.

, and

Therefore, we have

e0 = h′ i , and qph′ = qh = e1 , which implies that h′ is a lift for E. Hence, qp ∈ I- inj.



Lemma 1.20. Let C be a category, I a class of morphisms of C. Then, I- inj is closed under pull-backs, and hence so is I- fib. Dually, I- proj is closed under push-outs, and hence so is I- cof. Proof. It is enough to prove either of the statements, then the other follows by duality. Let i ∶ U → V belong to I- proj, and let f ∶ U → K be a morphism of C. Since C is cocomplete, the below push-forward square exists U

f

iK

i

 V

/K

iL

⌟  /L

26

where L = V ∐ K. Then, we need to show that iK ∈ I-proj. Consider a commutative square U

iK

/X

e0

K

E

 L

 /Y

e1

p

with p ∈ I. It can be extended to the below solid commutative diagram /K

f

U

e0 h

iK

 V

iL

⌟  /L

 /Y

p

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i

7/ X .

e1

Since i ∈ I- proj, it has the LLP with respect to p, hence a lift h ∶ K → X for the extended solid commutative diagram exists. Hence, we have a commutative square U i

 V

/K .

f

e0

 /X

h

D

Then, by the universal property of push-outs, there exists a unique a morphism h′ ∶ L → X, that makes the evident diagram commutative, in particular, h′ iK = e0 . Repeating the same argument for the commutative square U

/K

f

i

 /Y

1. 23 .

 V

ph

pe0

′′

Le m m a

yield the existence of a unique a morphism h ∶ L → Y , that makes the below evident diagram commutative U

f

.

iK

i

 V

/K

iL

⌟  /L

pe0 g

 .Y

ph

Since h′′ is a unique such morphism, then on the one hand g = ph′ because ph′ makes that diagram commutative. On the other hand, e1 (iL i) = e1 (iK f ), hence g = e1 . Then, ph′ = e1 and h is a lift for E, i.e. iK ∈ I-proj.  Another central notion in homotopy theory is two-out-of-three property, the main identifier for weak equivalence. Definition 1.21 (two-out-of-three property). Let C be a category, W a class of morphisms of C. Then, W is said to satisfy the two-out-of-three property, if for any commutative diagram in C /6 Z

gf

X f

(

Y

g

with two of f, g and gf are in W , then the third is so. Definition 1.22 (Category with weak equivalences). Let C be a category, W be a class of morphisms of C. Then, (C, W ) is said to be a category with weak equivalences W , if W includes all isomorphisms of C, and it satisfies the two-out-of-three property.

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In the study of model categories, particular adjunction, namely Quillen adjunction will play the role of morphism in the ’category’ of model categories. Hence, it is useful to understand how adjunctions are related to lifting, retractions and functorial factorisation. Let (F, G, ϕ) ∶ C → D be an adjunction, i ∶ U → V be a morphism in C and p ∶ X → Y be a morphism in D. Then, (F (i), p) is an extension-lifting pair in D if and only if (i, G(p)) is an extension-lifting pair in C. Proof. The adjunction (F, G, ϕ) implies by lemma ?? F (U ) F (i)

 F (V )

one to one correspondence between left and right solid squares, /X =

and

p

hF

/ G(X) =

U hG

i

 /Y

 V

G(p)

 / G(Y )

.

DF DG Moreover, by the adjunction, one sees easily that if the dotted lift hF ∶ F (V ) → X of DF exists, then ϕV,X (hF ) is a left for DG , and if the dotted lift hG ∶ V → G(X) of DG exists, then ϕ−1  V,X (hG ) is a left for DG .

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Corollary 1.24. Let (F, G, ϕ) ∶ C → D be an adjunction, I be a class of morphisms in C and P be a class of morphisms in D. Then, F takes I- cof to P - cof if and only if G takes P - fib to I- fib. Proof. F takes I- cof to P - cof if for every i ∈ I- cof and p ∈ P - fib (F (i), p) is an extension pair in D, and G takes P - fib to I- fib if for every p ∈ P - fib and i ∈ I- cof (i, G(p)) is an extension pair in C. Then, the satement of the corollary holds by the previous lemma 1.23 

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Definition 1.25 (Model Structures). Let C be a category, C, F, W be classes of morphisms in C, and (α, β), (γ, δ) be functorial factorisations on C. Then, (C, F, W, (α, β), (γ, δ)) is called a model structure on C if the following axioms are satisfied: CM2 (two-out-of-three) Morphisms of W satisfy the two-out-of-three property, i.e. for morphisms f and g in C. If any two of f, g and gf are in W , then the third is so. CM3 (Stability under retract) The classes of morphisms C, F, and W are closed under retract, i.e. for g a morphism in C that is in C, F, or W , then retracts of g are in C, F, or W , respectively. CM4 (Lifting) Morphisms in C have the LLP with respect to elements of F ⋂ W , and morphisms in C ⋂ W have the LLP with respect to elements of F . CM5 (Factorisation) For every morphism f ∶ X → Y in C: a) β(f ) ∈ F ⋂ W , and α(f ) ∈ C. b) δ(f ) ∈ F , and γ(f ) ∈ C ⋂ W . Z > f γ(f )

X

>

δ(f )

∼ f ∼

α(f )

/ Y. >> β(f )

Wf Definition 1.26 (Model Categories). Let C be a category, (C, F, W, (α, β), (γ, δ)) be a model structure on C. Then, (C, C, F, W, (α, β), (γ, δ)) is called a model category if it satisfy CM1 C is complete and cocomplete. We usually shorten notation and refer to the model category with the notation of its underlying category, if no confusion is caused. When the above conditions are satisfied, we call morphisms in C, F, W, C ⋂ W and F ⋂ W , cofibrations, fibrations, weak equivalences, weak (also acyclic or trivial) cofibrations, and morphisms in weak (also, acyclic or trivial) fibrations, respectively. We refer to such morphisms by ↣, two−out−of −threeproperty headrightarrow, ∼

∼

∼

→, ↣, and two − out − of − threeproperty headrightarrow, respectively. Also, an object Y ∈ C is called cofibrant if the unique morphism ∅ → Y is a cofibration in C1, and an object X ∈ C is called fibrant if the unique morphism 1∅ is the initial object of C. It is the colimit of the empty functor to C and it exists because C is cocomplete.

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X → ∗ is a fibration in C 2. The full subcategories Cc , Cf , Ccf of cofibrant, fibrant, and cofibrant and fibrant objects, respectively, play an important role in the study of homotopy category. It worth mentioning that when Quillen introduced the closed category CM1 required C to only have finite limits and colimits, rather than all small limits and colimits. Also, factorisations in CM5 were not required to be functorial, however, in the all interesting examples they can be made functorial. Also, Model categories used to be referred to as closed model category, but several authors dropped the word closed in recent works. Still, it is useful to notice that the term closed refers to the integrability between the axioms of the model structure, which is illustrated in the below lemma. Actually, this interactivity starts with the retract argument, theorem 1.17. Lemma 1.27. Let C be a model category. Then, (1) Cofibrations, weak cofibrations are precisely morphisms of C with LLP with respect to the class of weak fibrations, fibrations, respectively. (2) Fibrations, weak fibrations are precisely morphisms of C with RLP with respect to the class of weak cofibrations, cofibrations, respectively.

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Proof. It is enough to prove either of the statements, then the other follows by duality. By axiom CM4, we see that cofibrations, weak cofibrations have the LLP with respect to the class of weak fibrations, fibrations, respectively. To prove the converse, let i ∶ K → M has the LLP with respect to the class of weak fibrations, fibrations, we need to show that its a cofibration, weak cofibration. Among the axioms of model category, only stability under retract can imply that a morphism is a cofibration, if it is a retract of one. Also, it can imply that its weak cofibration, if it is a retract of one. Therefore, we need to express i as a retract of a cofibration, weak cofibration, respectively, which can be only produced by the factorisation axiom, as seen below. Assume that i has the LLP with respect to the class of weak fibrations. Using the factorisation axiom, i factorises as /M K > α(i)

β(i)

D

L with α(i) a cofibration, and β(i) a weak fibration. By the assumption, i has the LLP with respect to β(i), then by the retract argument, theorem 1.17, i is a retract of α(i), hence i is a cofibration by the stability under retract. On the other hand, assume that i has the LLP with respect to the class of fibrations. Using the factorisation axiom, i factorises as /M K > γ(i)

δ(i)

L with γ(i) a weak cofibration, and δ(i) a fibration. By the assumption, i has the LLP with respect to β(i), then by the retract argument, theorem 1.17, i is a retract of α(i), hence i is a weak cofibration by the stability under retract.  Hence, isomorphisms are cofibrations, fibrations, as well as being weak equivalences, that they have the RLP and LLP with respect to all morphisms. The above lemma 1.27 also shows that C- cof, C ⋂ W - cof, F - fib and F ⋂ W - fib are the classes of cofibrations, weak cofibrations, fibrations and weak fibrations, receptively. Then, as a result, lemmas 1.18, 1.19 and 1.20 extends to theses classes, as seen in the below corollaries. Corollary 1.28. Let C be a model category. Then, classes of cofibrations, weak cofibrations, fibrations and weak fibrations are closed under retract. Corollary 1.29. Let C be a model category. Then, the classes of cofibrations, weak cofibrations, fibrations and weak fibrations form subcategories of C, i.e. closed under composition, and contain all the isomorphisms of C. Corollary 1.30. Let C be a model category. Then, the classes of cofibrations and weak cofibrations are closed under push-outs. Dually, the classes of fibrations and weak fibrations are closed under pull-backs. 2∗ is the terminal object of C. It is the limit of the empty functor to C and it exists because C is complete.

29

Later on we see that these are important results, particularly corollary 1.30 can be use to substitute partially the lifting axiom, as to be seen in Joyal’s trick, lemma 4.1. Lemma 1.27 shows that in a model category two classes of cofibrations and weak equivalences or fibrations and weak equivalences determines the third by the lifting properties. In fact, any two classes of of the model structure determine the third, as seen in the below proposition, which in turn justify the name of closed model category. Moreover, the retract axiom in the definition of closed model categories can be replaced by the requirement that any two of the three classes determines the third, as in [GM03, Ch V, §1.4, p293]. Proposition 1.31. Let C be a model category. Then, any two classes of cofibrations, fibrations, and weak equivalences determines the third. Proof. Lemma 1.27 shows that classes of cofibrations and weak equivalences or fibrations and weak equivalences determines the third by the lifting properties. Notice that any weak equivalence can be factorise as cofibration and weak fibration or weak cofibration and a fibration. Since weak equivalences satisfy the two-out-of-three property then in both cases any weak equivalence can be factorise as weak cofibration and a weak fibration. Lemma 1.27 shows that weak cofibrations and weak fibrations are determined by the classes of fibrations and cofibrations, respectively, hence the class of weak equivalences is determined by the classes of cofibrations and fibrations. 

RA FT

A given model category induces several natural model categories in different way. Hereby, we recall some of these natural constructions. Example 1.32. Let C be a model category. Then, inclusions Cc → C, Cf → C, and Ccf → C induce model structure on Cc , Cf and Ccf , respectively, setting cofibrations fibrations, and weak equivalences to be precisely these morphisms whose images are cofibrations fibrations, and weak equivalences, respectively; with functorial factorisations being the restriction of the functorial factorisations of C.

D

Proof. Axioms CM1-CM4 are obvious. To show that functorial factorisations of C can be restricted to Cc , all what is need to be shown is that domβ(f ) = codα(f ), and domδ(f ) = codγ(f ) are cofibrations in C for every morphism f ∶ X → Y between cofibrant objects, i.e. in Cc . Let f ∶ X → Y be a morphism in Cc . Then, ∅ → X, α(f ), and γ(f ) are cofibrations in C. Then, compositions ∅ → codα(f ) and ∅ → codγ(f ) are cofibrations in C. Hence, domβ(f ) = codα(f ), and domδ(f ) = codγ(f ) are cofibrations in C. The statement for Cf is proven by duality, ad for Ccf using thetwo statement for Cc and Cf , that Ccf = Ccf = Cf c .  Example 1.33. Let C be a model category, U, Y ∈ C. Then, evident forgetful functors U ↓ C → C, C ↓ Y → C and U ↓ C ↓ Y → C are model categories induce natural model structures on U ↓ C, C ↓ Y and U ↓ C ↓ Y , respectively, setting cofibrations fibrations, and weak equivalences to be precisely these morphisms whose images are cofibrations fibrations, and weak equivalences, respectively; with functorial factorisations being the ’restriction’ of the functorial factorisations of C. These model categories are particularly useful to understand relative and fibre-wise homotopies in subsection 2.1. 1.1. Cofibrant and Fibrant Approximations. In the study of homotopy category of a model category, the subcategory of cofibrant fibrant objects plays the essential role in realising the homotopy category as quotient category with respect to equivalence relation. Since weak equivalences are mapped into isomorphisms in the homotopy category, it is particularly useful to approximate objects of the model category by cofibrant or fibrant objects up to weak equivalences, as in the sense of the below definition. Definition 1.34. Let C be a model category. Then, ̃ , p), where U ̃ is a cofibrant object and (1) A cofibrant approximation of an object U ∈ C is a pair (U ̃ → U is a weak equivalence. p∶U (2) A cofibration cofibrant approximation of a morphism i ∶ U → V in C is a triple consisting of cofibrant ̃ , pU ) of U , cofibrant approximation (Ṽ , pV ) of V and a cofibration ̃i ∶ U ̃ → Ṽ that approximation (U makes the evident diagram commutative. ̃ , p) of U such that p is a (3) A cofibrant resolution of an object U ∈ C is cofibrant approximation (U fibration as well. (4) A cofibration cofibrant resolution of a morphism i ∶ U → V in C is a cofibration cofibrant approximation ̃ → Ṽ , pU , pV ) of i such that pU and pV are fibration. (̃i ∶ U ̂ i), where X ̂ is a fibrant object and i ∶ X → X ̂ (5) A fibrant approximation of an object X ∈ C is a pair (X, is a weak equivalence.

30

(6) A fibration fibrant approximation of a morphism p ∶ X → Y in C is a triple consisting of fibrant ̂ iX ) of X, cofibrant approximation (Ŷ , iY ) of Y and a fibration p̂ ∶ X ̂ → Ŷ that approximation (X, makes the evident diagram commutative. ̂ i) of X such that i is a fibration (7) A fibrant resolution of an object X ∈ C is fibrant approximation (X, as well. (8) A fibration fibrant resolution of a morphism p ∶ X → Y in C is a fibration fibrant approximation ̂ → Ŷ , iX , iY ) of p such that iX and iY are fibration. (̂ p∶X

DR AF

T

Definition 1.35. Let C be a model category. Then, (1) A cofibrant approximation functor on C is an augmented functor (F, η) on C such that (F (U ), ηU ∶ F (U ) → U ) is a cofibrant approximation of U for every U ∈ C. (2) A cofibration cofibrant approximation functor on C is a cofibrant approximation functor (F, η) on C such that F (i) is a cofibration for every morphism i in C. (3) A cofibrant resolution functor on C is a cofibrant approximation functor (F, η) on C such that ηU ∶ F (U ) → U is a week fibration for every U ∈ C. (4) A cofibration cofibrant resolution functor on C is a cofibrant resolution functor (F, η) on C such that F (i) is a cofibration for every morphism i in C. (5) A fibrant approximation functor on C is an coaugmented functor (G, ) on C such that (G(X), X ∶ X → G(X)) is a fibrant approximation of X for every X ∈ C. (6) A fibration fibrant approximation functor on C is a fibrant approximation functor (G, ) on C such that G(p) is a cofibration for every morphism p in C. (7) A fibrant resolution functor on C is a fibrant approximation functor (G, ) on C such that U ∶ X → G(X) is a week cofibration for every X ∈ C. (8) A fibration fibrant resolution functor on C is a fibrant resolution functor (G, ) on C such that G(p) is a cofibration for every morphism p in C. Notice that all the above defined approximations and approximation functors preserve weak equivalences by the two-out-of-three property of weak equivalences. Remark 1.36. The importance of the above approximations, which always exist in model categories, becomes clear when passing to the homotopy category, as they assign to each object of the homotopy category a well-behaved ’representative’ in the model category that becomes isomorphic to it in the homotopy category. By well-behaved we mean cofibrant or fibrant objects which are crucial to the understanding of the homotopy category of a model category. The functorial factorisation provides a canonical way of approximating any object of a model category by a fibrant or cofibrant up-to weak equivalence,as seen below. Cofibrant Replacement. Definition 1.37 (Cofibrant Replacement). Let C be a model category. Define, the functor Q ∶ C → Cc that sends each morphism i ∶ U → V to ⎛ codomMorC α ⎜ ⎜U h ⎝

i

∅

⎞ ⎛ ⎜U h = dom β 6/ V ⎟ Mor C ⎟ ⎜ ⎠ ⎝

i

∅

⎞ 6/ V ⎟ ⎟ ⎠

Since (α, β) is a functorial factorisation on C, then Q is a well-defined functor. It is called the cofibrant replacement functor of C. ∀U ∈ C, Q(U ) is a cofibrant object and β(∅ → U ) ∶ Q(U ) → X is a weak fibrations, i.e. (Q(U ), β(∅ → U )) is a cofibrant resolution of U . Moreover, it is easy to see that ic Q is a cofibrant resolution functor, which explains the name of the cofibrant replacement.
Q(U ) O ∅

/V < G

i ∼

Q(i)

/ Q(V ) 9

31

Q(U ) is weekly equivalent to any cofibrant resolution of U (see [Hir03, Cor.8.1. 8]). Notice that the cofibrant replacement of any morphism i ∶ U → V is between cofibrant object. However, it is not necessary a cofibration. We see later that cofibrations between cofibrant objects play an important role in the localisation of model categories. Therefore, we introduce the below variation of the cofibrant replacement. Definition 1.38 (Cofibration Cofibrant Replacement). Let C be a model category, i ∶ U → V a morphisms in C. Then, define the cofibration cofibrant replacement for i to be the morphism α(Q(i)), and denote it by ̃ Q(i). ̃ ) Q(V 8 α(Q(i))

β(Q(i))

̃ ) = Q(U ) Q(U

# / Q(V )

Q(i)

 U

 /V

i

RA FT

̃ ) is a cofibrant object, α(Q(i)) is a cofibration with Notice that for every morphisms i ∶ U → V in C, Q(U the domain Q(U ). Since cofibrations are closed under composition for being given by the LLP with respect ̃ ) is also a cofibrant object in C. Hence, Q(i) ̃ to weak fibrations, then Q(V is a cofibration between cofibrant ̃ objects, also Q(V ) → Q(V ) → V is a weak fibration, and that explains the name. Fibrant Replacement. Definition 1.39 (Fibrant Replacement). Let C be a model category. Define, the functor R ∶ C → Cf that sends each morphism p ∶ X → Y to ⎛ codomMorC γ ⎜ ⎜X ⎝

⎛ ⎞ / Y ⎟ = domMorC δ ⎜ X ⎜ ⎟ ⎝ ⎠

p

)∗u

p

)∗u

⎞ /Y ⎟ ⎟ ⎠

Since (γ, δ) is a functorial factorisation on C, then R is a well-defined functor. It is called the fibrant replacement functor of C.

D

∀X ∈ C, R(X) is a fibrant object and γ(X → ∗) ∶ X → R(X) is a weak cofibrations, i.e. (R(X), γ(X → ∗) ∶ X → R(X) is a fibrant resolution of X. Moreover, it is easy to see that if R is a fibrant resolution functor, which explains the name of the fibrant replacement. /Y

p

X ∼

| R(X)

∼ R(p)

| / R(Y )

 ∗ mr Similarly, R(X) is weekly equivalent to any fibrant resolution of X (see [Hir03, Cor.8.1.8]). Also, the fibrant replacement of any morphism p ∶ X → Y is between fibrant object. However, it is not necessary a fibration. Therefore, we introduce the below variation of the fibrant replacement. Definition 1.40 (Fibration Fibrant Replacement). Let C be a model category, p ∶ X → Y a morphisms in ̂ C, define the fibration fibrant replacement for f to be the morphism δ(R(p)), and denote it by R(p).

 R(X) γ(R(p))

/Y

f

X

R(p)

#

 ̂ ) / R(Y ) = R(Y 8 δ(R(p))

̂ R(X)

32

̂ Notice that for every morphisms p ∶ X → Y in C, R(p) is a fibrant object, δ(R(p)) is a fibration with the codomain R(Y ). Since fibrations are closed under composition for being given by the RLP with respect to weak ̂ ̂ cofibrations, then R(X) is also a fibrant object in C. Hence, R(p) is a fibration between fibrant objects, also ̂ X → R(X) → R(X) is a weak cofibration, and that explains the name. Cofibrant and fibrant replacements are very important in deducing functors on the level of homotopy categories, starting with ’nice’ functors between model categories (in most occasions they will be Quillen adjunction). Remark 1.41. Avoid some confusion that might arises when dealing with Cofibrant and fibrant replacements for first time. The definition of functorial factorisation doses not require α(∅ → U ) to coincide with ∅ → U for cofibrant objects U ∈ C. Hence, having a cofibrant object X, priorly Q(X) does not necessary coincide with X, it is priorly merely weekly equivalent to X. Similarly, having a fibrant object X, R(X) does not necessary coincide with X. This is seen in the below example. Also cofibrant and fibrations replacements are not necessary idempotent functors. Example 1.42. In the category of simplicial sets sSet endued with the classical Quillen’s model structure. ∆1 is a Kan fibrant. We use Kan Ex∞ functor3 to show that the fibrant replacement of ∆1 does not coincide with ∆1 . Recall that for a simplicial set X, Ex∞ (X) is the colimit of an injective system jX

/ Ex(X)

jEx(X)

/ Ex2 (X)

jEx2 (X)

j

/ Ex3 (X) Ex

3 (X)

RA FT

X

/ ...

of inclusions, where Ex is defied using the subdivision functor sd, by

Ex(X)n = sSet(∆n , Ex(X)) = sSet(sd ∆n , X).

Recall that sd ∆n = N nd ∆n , where N is the nerve functor, and the nd is the functor of poset of nondegenerate simplices. ∆1 is a Kan fibrant, but Ex(∆1 ) ≠ ∆1 as seen below. Since ∆[n] = N[n], we have

Ex(∆1 )1 = Set(sd ∆1 , ∆1 ) = Set(N nd ∆1 , N[1])

D

Since N is fully faithful functor, then

Ex(∆1 )1 = Set(N nd ∆1 , N[1]) ≅ PoS(nd∆1 , [1]).

nd ∆1 = ∂01 → id[1] ← ∂11 , then #PoS(nd∆1 , [1]) = 5 corresponding to the five monotic maps from nd ∆1 to [1] illustrated by the five diagrams id ∂01 / id[1] id o ∂11 ∂01 / id[1] o ∂11 ; [1] c b < [1] ∂01 / id[1] o ∂11 ∂01 ∂11 ∂11 ∂01 where lower position represent mapping to 0 and upper position represent mapping to 1. Hence # Ex(∆1 )1 = 5 > 3 = #{id[1] , ∂01 σ00 , ∂01 σ00 } = #(∆11 ), then Ex(∆1 ) ≠ ∆1 . ∆1 is a Kan fibrant, then the small object argument implies that ∆1 ⊆ Ex(∆1 ), then ∆1 ⊂ Ex(∆1 ) and Ex∞ (∆1 ) ≠ ∆1 . The small object argument4, recalled below in 4.2, shows that if the model category is cofibrantly generated then there exist infinity many functorial factorisations that are priorly different. This fact, in addition to the above remark and example, gives rise to the following questions. Question 1.43. Let C be a complete and cocomplete category, C, F, W three classes of morphisms of C that satisfy axioms CM2-CM4 of model category. Also, assume for simplicity that there exists at least a pair of functorial factorisations that satisfy the axiom CM5. Then, is there a ’minimal’ or canonical functorial factorisations that satisfy CM5. Where ’minimal’ or canonical means that the cofibrat replacement Q(U ) of U coincides with U , for a cofibrant object U , and the fibrat replacement R(X) of X coincides with X, for a fibrant object X. If such functorial factoriasation exist, then its cofibrant replacement Q is a right adjoint to the inclusion ic ∶ Cc → C, and its fibrant replacement R is a left adjoint to the inclusion if ∶ Cf → C. 3Kan Ex∞ functor is the fibrant replacement functor in Quillen’s model category of simplicial sets. 4It is a canonical method of construing functorial factorisations due to Quillen [Qui67].

33

2. Homotopy in Model Categories Recall that model categories was introduced as a ’representation’ of a homotopy category, that later is not always well-understood by its-own, especially in terms of its existence and the nature of its morphisms. To make this more precise recall that homotopy category of a category with weak equivalences is the localisation of the category with respect to the class of weak equivalences. This procedure might result in a category living in higher universe than the one started with. Also, morphisms of the resulting category, if exist5, are quotient of formal strings of zigzags, hence it looses the intuition one has about morphisms of the original category. Model categories are particular cases of categories with weak equivalence with an extra structure that enables us to define a ’honest’ notion of homotopy in the sense it produces an equivalence relation on a category equivalent to the one starting with. That makes it possible to show that homotopy category of a model category exists (in the same universe), and shows that it is equivalent to a more understood quotient category with respect to an equivalence relation obtained from the homotopy in the model category. hence, model categories provide useful and ’relatively’ easier tools to deal with and understand homophony theories. In fact, when Quillen introduced model category in [Qui67], it was called ”a category of model for homotopy theory”, where he provided the axiomatic treatment needed to study homotopy theory of abstract categories.

RA FT

In this section, we start by recalling homotopy category of a category with weak equivalence, recall the notions of homotopies in model category. Then show the main goal of this section which is the existence homotopy category of a model category exist in theorem 2.16, in the same universe, and link it to an appropriate quotient category with respect to an honest equivalence relation. Definition 2.1 (Homotopy Category). Let (C, W ) be a category with weak equivalences. Then, the homotopy category of (C, W ) is defined to be the localisation of C with respect to W , if exists, i.e. a pair (LW C, η), where LW C is a category, and η ∶ C → LW C is a functor that sends morphisms of W to isomorphisms in LW C such that any functor θ ∶ C → D that sends morphisms of W to isomorphisms in D factorises through η, hence there exists a unique functor θW ∶ LW C → D such that the following diagram of functors commutes η

C

/ LW C

↺

D

θ

θW

"  D.

If the homotopy category of (C, W ) exists, then it is denoted by HC, if no confusion arises about the choice of the class of weak equivalences. Remark 2.2. The name and notation of homotopy category might be confusing at the first glance, especially that the resulting category, if exists, it is not priorly a quotient with respect to any homotopy relation! However, we see later when C is a model category, then HC exists, and is equivalent to, but does not coincide with, a quotient with respect to a homotopy relation, theorem2.16, which justifies the name to some extent. Still the author agrees with Hovey [Hov99] and thinks that C[W −1 ] would have been a more appropriate notation. That if C[W −1 ] exists then HC exists and coincides with C[W −1 ] as seen in [ibid., Def.1.2.1 and Lem.1.2.2]. In order to fully understand the notions of homologies in a model category, it is useful to recall the motivation notions in topological spaces in 1 that one want to define a homotopy relations on a model category C in an analogue of what was done in the case of topological spaces. In order to do so, one needs to examine the categorical properties of the cylinder space V × I and the path space X I , used to define homotopy relations, in the category of topological spaces Top or any more convenient ’nice’ category of topological spaces like CGHaus. That leads to the abstract definitions of cylinder and path objects. P r1

ic

On the one hand notice that for any topological spaces V and X, V × I → V and X → X I are weak equivalences, where ic is the constant path map. On the other hand, left and right homotopy relation on topological spaces are equivalence relations, in particular they are reflexive. Hence, every continuous map is left homotopic to itself, at least by the constant homotopy, in particular for a topological space V , idV is left homotopic to itself. Hence we have the 5Some author restrict the definition of category to the case when hom classes are set (what is usually called by others locally small categories). If the resulting localised ’category’ lives in a higher universe then it is does not exist in the sense of this definition.

34

commutativity of the bottom triangles. V< ∐ Vb i0 ∐ i1

V

 / V ×I o

i0

i1

V

pr1

) V u

idV

idV

Whereas the commutativity of the top triangles is induced by the universal property of V ∐ V , which also implies that pr1 pr1 ○ (i0 ∐ i1 ) is a factorisation of the map idV ∐ idV . These two properties of the cylinder space V × I make sense in an abstract cocomplete category with weak equivalences, i.e. having a morphism P r1 ∶ V × I → V being a weak equivalence and idV ∐ idV factorising as pr1 ○ (i0 ∐ i1 ) for two morphisms i0 , i1 ∶ V → V × I. Dually, the commutativity of the top triangles is due to idX being right homotopic to itself. X idX

idX ic

 X co

 XI

 /X ;

p1

RA FT

p0

p0 ∏ p1

 X ∏X

Similarly, the commutativity of the bottom triangles is induced by the universal property of X ∏ X, which also implies that (pi0 ∏ p1 )○iP is a factorisation of the map idX ∏ idX . Also analogue two properties of the path space X I make sense in an abstract complete category with weak equivalences, i.e. having a morphism ic ∶ X → X I being a weak equivalence and idX ∏ idX factorising as (p0 ∏ p1 ) ○ ic for two morphisms p0 , p1 ∶ X I → X. If we restrict ourselves to the category of compactly generated Hausdorff spaces CGHaus, then i0 ∐ i1 is a Hurewicz cofibration, and pi0 ∏ p1 is a Hurewicz fibration. However, that does not hold for general topological spaces.

D

Motivated by the above discussion, the cylinder object, path object, left homotopy and right homotopy are generalised and defined in any model category . Definition 2.3 (Cylinder and Path Objects). Let C be a model category, V, X ∈ C. Then, ● The fold morphism for V is defined by the universal property of coproducts to be the unique dotted morphism ∇V ∶= idV ∐ idV ∶ V ∐ V → V for which the below diagram commutes: / V ∐V o

V

V

∇V

) V u

idV

idV

● A cylinder object for V is a factorisation C of the fold morphism: C∶

V ∐V

i0 ∐ i1

/ Cyl(V )

pC

/V

with pC being a weak equivalence, and i0 , i1 ∶ V → Cyl(V ) being morphisms in C such that the induced morphism i0 ∐ i1 is a cofibration. ● The diagonal morphism for X is defined by the universal property of products to be the unique dotted morphism ∆X ∶= idX ∏ idX ∶ X → X ∏ X for which the below diagram commutes: X idX

 Xo

idX ∆X

 X ∏X

 /X

● A path object for X is a factorisation P of the diagonal morphism: P ∶X

iP

/ Cyl(X) p0 ∏ p1 / X ∏ X

35

with iP being a weak equivalence, and p0 , p1 ∶ Cyl(X) → X being morphisms in C such that the induced morphism p0 ∏ p1 is a fibration. In the case of topological spaces, i0 and i1 , are thought of as the closed embeddings of the topological space into the bottom and top ’faces’ of its cylinder space, respectively. Similarly, p0 and p1 are thought of as the projections of paths at its initial and final points, respectively. These analogies are useful for most purposes. In fact, some sources uses the notation of V × I and X I as an abstract notion for Cyl(V ) and Cyl(X), respectively, not meaning a product or power objects. Remark 2.4. Although the cylinder and path objects are not required to exist for every object in the model category and not required to be functorial, the functorial factorisations provide canonical functorial cylinder and path objects, we see later it is sufficient to consider homotpies defined using these canonical cylinder and path objects. Therefore, it is useful to distinguish them, where we denote the canonical cylinder object for V ∈ C that is induced by the functorial factorisation (α, β) CV ∶

V ∐V

i0 ∐ i1

/ CylC (V )

pC

/V

and the canonical path object for X ∈ C that is induced by the functorial factorisation (γ, δ) / P athC (X)p0 ∏ p1 / X ∏ X .

iP

RA FT

PX ∶ X

Definition 2.5 (Homotopies). Let C be a model category, f0 , f1 ∶ V → X in C. Then, ● A left homotopy from f0 to f1 is a pair (C, Hl ) where C is a cylinder object for V C∶

V ∐V

i0 ∐ i1

/ Cyl(V )

pC

/V

and Hl is a morphism Hl ∶ Cyl(V ) → X such that f0 = Hl i0 and f1 = Hl i1 , i.e. it Hl makes the following diagram commutes: i0

V

/ Cyl(V ) o

i1

V

Hl

f0

)  u X

f1

D

If there exist a left homotopy (C, Hl ) from f0 to f1 , we say that f0 is left homotopic to f1 , and we l

l ≃

write f0 ≃ f1 , or f0 Hl f1 if the particular left homotopy is to be emphasised. ● A right homotopy from f0 to f1 is a pair (P, Hr ) where P is a path object for X P ∶X

/ Cyl(X) p0 ∏ p1 / X ∏ X

iP

and Hr is a morphism Hr ∶ V → Cyl(X) such that f0 = p0 Hr and f1 = p1 Hr , i.e. Hr makes the following diagram commutes: V f0

f1

 Xo

Hr

p1

 Cyl(X)

p0

 /X

If there exist a right homotopy (P, Hr ) from f0 to f1 , we say that f0 is right homotopic to f1 , and we r

r ≃ f0 Hr

write f0 ≃ f1 , or f1 if the particular right homotopy is to be emphasised. ● If f0 is both left and right homotopic to f1 , then we say that f0 is homotopic to f1 and we write f0 ≃ f1 . ● A morphism f ∶ V → X in C is called homotopy equivalence if there exists a morphism g ∶ X → V such that f g ≃ idX and gf ≃ idV . If f ∶ V → X is a homotopy equivalence, then we say that V is homotopy equivalent to X, or that V and X of the same homotopy type. The fact that, in topological spaces, the cylinder functor − × I is left adjoint to the path functor −I allows the interchange between right and left topological homotopy. Also, it explains the terminology of left and right homotopy, where the left homotopy is the one defined by the left adjoint cylinder functor. However, even the ’nicest’ cinder functor Cyl(−) and path functor Cyl(−) induced by the functorial factorisations of a model

36

category do not have to be adjoint, and in general a distinguishing between left and right homotopy is needed. The definition of the homotopies did not fix any preferable choice of the cylinder of path objects. However, any left or right homotopy can be shown to be a homotopy with the cylinder or path object, obtained through the appropriate factorial factorisation of the fold or diagonal morphisms, respectively, as illustrated in the below lemma. Lemma 2.6. Let C be a model category, f0 , f1 ∶ V → X morphisms in C. (1) If f0 is left homoerotic to f1 , then there is a left homotopy (CV , Hl ) from f0 to f1 with the canonical choice of the cylinder object CV , i.e. being induced from the functorial factorisation of the model category, as in remark 2.4. (2) If f0 is right homoerotic to f1 , then there is a right homotopy (PX , Hr ) from f0 to f1 with the canonical choice of the path object PX , i.e. being induced from the functorial factorisation of the model category, as in remark 2.4. Proof. See [Hir03, Prop.7.3.4].



One of the most important implication of this lemma is that it allows inheriting homotopies along morphisms as seen below.

DR AF

Proof. Straightforward, by tracking diagrams.

T

Corollary 2.7. Let C be a model category, f0 , f1 ∶ V → X, i ∶ U → V and p ∶ X → Y morphisms in C. (1) A left homotopy (C, Hl ) from f0 to f1 induces the left homotopy (CU , pHl Cyl(i)) from pf0 i to pf1 i. (2) A right homotopy (P, Hr ) from f0 to f1 induces the right homotopy (PY , Cyl(p)Hr i) from pf0 i to pf1 i. 

Notice that in general neither of the left or right homotopy in a model category define an equivalence relation on arbitrary hom-set C(V, X) for any objects V, X ∈ C, particularly the left and right homotopy relations defined on C(V, X) by the left and right homotopies are not symmetric as seen in the below example. However, that can be remedied through restriction attention to the subcategories of fibrant and cofibrant objects as seen in the below proposition. Proposition 2.8. Let C be a model category, V, X be objects in C. (1) If V is a cofibrant object, then the left homotopy is an equivalence relation on C(V, X). (2) If X is a fibrant object, then the right homotopy is an equivalence relation on C(V, X). (3) If V is a cofibrant object and X is a fibrant object, then the homotopy is an equivalence relation on C(V, X). Proof. See [Hir03, Prop.7.4.5].



Having an equivalence relation, it is natural to consider equivalence classes with respect such relation. Hence, the below definition arises naturally. Definition 2.9. Let C be a model category, V, X be objects in C. (1) If V is a cofibrant object, define π l (V, X) to be the set of left homotopy classes of morphisms from V l

to X, i.e. C(V, X)/ ≃. (2) If X is a fibrant object, define π r (V, X) to be the set of right homotopy classes of morphisms from V r to X, i.e. C(V, X)/ ≃. (3) If V is a cofibrant object and X is a fibrant object, define π(V, X) to be the set of homotopy classes of morphisms from V to X, i.e. C(V, X)/ ≃. Proposition 2.10. Let C be a model category, V a cofibrant object in C, and X is a fibrant object in C. Then, left homotopy relation, right homotopy relation, and homotopy relation on C(V, X) coincide. Proof. See [Hir03, Prop.7.4.9].



Hence, restricting attention to the full sub-category of cofibrant-fibrant objects Ccf , one notice that the above defined sets of homotopy classes defines a graph G(O, A) with objects O coinciding with objects of Ccf and arrows A(V, X) = π(V, X), ∀V, X ∈ Ccf . Then, it becomes natural to ask if it is actually a category, i.e.if the homotopy equivalence classes respect cofibrant-fibrant objects respect composition. Theorem 2.11. Let C be a model category. Then, the composition is well-defined on homotopy classes between cofibrant-fibrant objects.

37

Proof. See [Hir03, Th.7.5.5].



Hence, G(O, A) is in fact a category with composition being induced from Ccf . Definition 2.12. Let C be a model category. The, the classical homotopy category πCcf of C is defined to be the category with objects the cofibrant-fibrant objects of C and morphisms the homotopy classes of morphism in C. Lemma 2.13. Let C be a model category. Then, inclusion functors induces equivalences of categories HCcf → HCc → HC

HCcf → HCf → HC

, and

Proof. The inclusion ic ∶ Cc → C preserve weak equivalences by the very definition of model structure on Cc . Hence, it induces the functor Hic ∶ HCc → HC. It is an equivalence of categories with inverse HQ, where Q is the cofibrant replacement of the model category C.  Notice that HQ is an inverse of Hic . However, Q is not necessary an inverse or adjoint of ic . Theorem 2.14 (Whitehead Theorem). Let C be a model category, f be a morphism between cofibrantfibrant objects in C, i.e. f in Ccf . Then, f is a weak equivalence if and only if it is a homotopy equivalence. Proof. See [Hov99, Prop.1.2.8] and [Hir03, Th.7.5.10].



RA FT

Proposition 2.15. Let C be a model category, (HCcf , ηho ∶ Ccf → HCcf ) be the homotopy category of Ccf , and (πCcf , ηπ ∶ Ccf → πCcf ) the classical homotopy category of C. Then, there exists a unique isomorphism of categories πCcf → HCcf that makes the following diagram commutative ηho

Ccf

ηπ

↺

/ LW C

 | D

Proof. See [Hov99, Cor.1.2.9].



This discussion concludes with the below lemma that provides a more natural description of the morphisms of the homotopy category. Theorem 2.16. Let C be a model category. Then, HC exists. Moreover, HC is equivalent to πCcf .

D

Proof. See [Hov99, Th.1.2.10.(i)].



2.1. Relative and Fibre-wise Homotopy. In analogy of the different homotopy notion in topology, one can formulate the relative and fibre-wise homotopy in an abstract model category. This notion are useful in the sense that they provide an insight into the nature of the which are priorly not easy to comprehend in most interesting cases. Definition 2.17 (Constant homotopy). Let C be a model category, V, X ∈ C. Then, (1) A left homotopy (C, Hl ) from V to X is said to be constant at f ∶ V → X if it factorises as Hl = f pC : Cyl(V )

pC

/V

f

/ X.

(2) A right homotopy (P, Hr ) from V to X is said to be constant at f ∶ V → X if it factorises as Hr = iP f V

f

/X

iP

/ Cyl(X)

Definition 2.18 (Relative Homotopy). Let C be a model category, f0 , f1 ∶ V → X and i ∶ U → V be morphisms in C. (1) A left homotopy (C, Hl ) from f0 to f1 is called a left homotopy relative to i if the induced left homotopy (CU , Hl Cyl(i)) is a constant homotopy, i.e. f0 i = f1 i and the below diagram commutes Cyl(U )

pCU

f0 i=f1 i

Cyl(i)

 Cyl(V )

/U

Hl

 / X.

If i is be understood and no ambiguity may arise, we say that (C, Hl ) is a left homotopy relative to U.

38

(2) A right homotopy (P, Hr ) from f0 to f1 is called a right homotopy relative to i if the right homotopy (P, Hr i) is a constant homotopy, i.e. f0 i = f1 i and the below diagram commutes f0 i=f1 i

U

/X iP

i

 V

Hr

 / Cyl(X).

If i is be understood and no ambiguity may arise, we say that (P, Hr ) is a right homotopy relative to U. A homotopy from f0 to f1 is called a homotopy relative to i if it is both left and right homotopy relative to i. Definition 2.19 (Fibre-wise Homotopy). Let C be a model category, f0 , f1 ∶ V → X and p ∶ X → Y be morphisms in C. (1) A left homotopy (C, Hl ) from f0 to f1 is called a fibre-wise left homotopy with respect to p if the left homotopy (C, pHr ) is a constant homotopy, i.e. pf0 i = pf1 and the below diagram commutes Hl

Cyl(V ) pC

p

 / Y.

RA FT

 V

/X

pf0 =pf1

If p is be understood and no ambiguity may arise, we say that (C, Hl ) is a fibre-wise left homotopy with respect to Y . (2) A right homotopy (P, Hr ) from f0 to f1 is called a fibre-wise homotopy with respect to p if the induced right homotopy (PY , Cyl(p)Hr ) is a constant homotopy, i.e. pf0 i = pf1 and the below diagram commutes Hr / Cyl(X) V pf0 =pf1

D

 Y

Cyl(p)

iPY

 / Cyl(Y ).

If p is be understood and no ambiguity may arise, we say that (P, Hr ) is a fibre-wise right homotopy with respect to Y . (3) A homotopy from f0 to f1 is called a fibre-wise homotopy with respect to p if it is fibre-wise both left and right homotopy with respect to i. In the above definitions of we notice that f0 , f1 ∶ i → f in U ↓ C for f = f0 i = f1 i and f0 , f1 ∶ f → p in C ↓ Y for f = pf0 = pf1 . Then, it is natural to ask about the relation between homotopies from f0 to f1 relative to i in C and homotopies from f0 to f1 under U , i.e. in U ↓ C. Similarly, the question extend to the relation between fibre-wise homotopies from f0 to f1 with respect to p and fibre-wise homotopies from f0 to f1 over Y , i.e. in C ↓ Y , and that leads to the below lemma. Lemma 2.20. Let C be a model category, f0 , f1 ∶ V → X be a morphism in C, i ∶ U → V be a cofibration in C and p ∶ X → Y be a fibration in C. (1) A left or right homotopy from f0 to f1 is a left or right homotopy relative to i if and only if it is a left or right homotopy under U . (2) A left or right homotopy from f0 to f1 is a fibre-wise left or right homotopy with respect to p if and only if it is a left or right homotopy over Y . Proof. TBC.



One of the useful notion in understanding weak equivalences in terms of homotopy relation is strong deformation retract, that will be recalled below. Definition 2.21. Let C be a model category. Then, (1) A morphism i ∶ U → V in C is called an inclusion of deformation retract and U is called a deformation retract of V if there is a deformation retract r ∶ V → U (i.e. ri = idU ) such that ir ≃ idV . Moreover, an inclusion of deformation retract is called a strong inclusion of deformation retract if ir ≃ idV under U .

39

(2) A morphism p ∶ X → Y in C is called a projection of deformation section and Y is called a deformation section of X if there is a deformation section s ∶ Y → X (i.e. ps = idY ) such that sp ≃ idX . Moreover, a projection of deformation section is called a strong projection of deformation section if sp ≃ idX over Y. Lemma 2.22. Let C be a model category. Then, (1) A weak cofibration i ∶ U → V between fibrant objects is a strong inclusion of deformation retract. (2) A weak fibration p ∶ X → Y between cofibrant objects is a strong projection of deformation section. Proof. We prove the first statement, then the proof of the second statement is obtained by duality. To obtain a retract of i, consider the commutative diagram U i

 V

idU

/U ?

r

 /∗

RA FT

in U ↓ C. Since i is a weak cofibration and U is a fibrant object in C, then they are weak cofibration and fibrant object in U ↓ C, respectively. Therefore, a lift r ∶ V → U of the above diagram exists in U ↓ C, hence ri = idU and r is a retract of i. Since i is a weak cofibration and V is a fibrant object in U ↓ C, then [Hir03, Prop.7.5.9], implies that the induced morphism of right homotopy classes i∗ ∶ π r (V, V ) → π r (U, V )

is a bijection. Notice that i∗ (idV ) = i = iri = i∗ (ir). Hence, idV and ir belongs to teh same right homotopy r class, i.e. idV ≃ ir in U ↓ C. In U ↓ C, U is evidently a cofibrant object and since i is a cofibration, then so is V . Therefore, U and V are cofibrant-fibrant objects in U ↓ C, hence right homotopy relation coincide with homotopy relation on U ↓ C(V, V ), by lemma 2.10, hence idV ≃ir and i is a strong inclusion of deformation retract.  3. Quillen’s Pair and Derived Functors

D

In categories, in general, morphisms are suable generalisation of maps that preserve objects’ structures. In this seance, one might require morphisms of model categories to be functors that preserve the model structure. However, in practice this requirement is to restrictive and it exclude lots of naturally arising functors between model categories, like the geometric realisation functor. Since the aim of model categories is to represent homotopy categories, i.e. to provide a tool to study their homotopy theories, then what one actually need is to define a morphism of model categories F ∶ C → D to be a functor that naturally induce a functor of homotopy categories HC → HD. This obviously can be done by requesting F to preserve weak equivalences, hence one has the induced morphism of homotopy categories HF ∶ HC → HD. However, this is also too restrictive and exclude many natural and interesting examples. The accepted notion of morphisms of model categories is what is called Quillen’s pair or Quillen’s adjunctions. That, they induce adjunctions between homotopy categories and they cover all the interesting examples that naturally arise in homotopy theory. Definition 3.1 (Quillen’s pair). Let C and D be model categories. Then, ● A functor F ∶ C → D is called a left Quillen functor if and only if it is a left adjoint and preserves cofibrations and weak cofibrations. ● A functor G ∶ D → C is called a right Quillen functor if and only if it is a right adjoint and preserves fibrations and weak fibrations. Lemma 3.2. Let (F, G, φ) ∶ C → D be an adjunction between model categories. Then, the following conditions are equivalent (1) F is a left Quillen functor. (2) F preserves cofibrations and G preserves fibrations. (3) F preserves weak cofibrations and G preserves weak fibrations. (4) G is a right Quillen functor. Such adjunction that satisfy either of the above equivalent conditions is called Quillen adjunction or Quillen pair. Proof. Since - proj and - inj operators are stable, then lemma 1.27 states that cofibrations and weak cofibrations are precisely (weak fibrations)-cof and fibrations-cof , respectively, and fibrations and weak fibrations

40

are precisely (weak cofibrations)- fib and cofibrations- fib, respectively. Hence, the statement of the lemma is a direct result of corollary 1.24.  Example 3.3. The adjunction ∣−∣ ∶ sSet → Top ∶ S is a Quillen pair, as seen in chapter 3. Remark 3.4. One can then consider the category of model categories in a give universe to be a category in a higher universe that have model categories as its objects and Quillen pairs between then as its morphisms. Notice that Quillen adjunctions do not have to preserve the functorial factorisations. Hence, two model categories that only differs by the choice of functorial factorisations are isomorphic in this sense. In order to see show that Quillen pairs induce adjunction on the level of homotopy categories, we need first to recall Ken Brown’s lemma 3.5. Lemma 3.5 (Ken Brown’s Lemma). Let C be a model category, and D be a category with weak equivalences WD , and F ∶ C → D be a functor. Then, if F takes weak cofibrations between cofibrant objects to weak equivalence, then it takes all weak equivalences between cofibrant objects to weak equivalence. Dually, if F takes weak fibrations between fibrant objects to weak equivalence, then it takes all weak equivalences between fibrant objects to weak equivalence.

DR AF

T

Proof. We prove the first statement, then the proof of the second statement is obtained by duality. Let f ∶ U → V be a weak equivalence between cofibrant objects in C. Then factorises the morphism (i, idV ) ∶ U ∐ V → V into a cofibration α((f, idV )) ∶ U ∐ V → W and weak fibration β((f, idB )) ∶ W → V , through applying the functorial factorisation (α, β). Since cofibrations are closed under push-outs and U and V are cofibrant objects, then the canonical injections iU ∶ U → U ∐ V and iV ∶ V → U ∐ V are cofibrations, hence U ∐ V and W are cofibrant objects in C. Let i ∶= α((f, idV )) and p ∶= β((f, idB )). Then, we have the commutative diagram /U ∅  V

iU

iV

 / U ∐V

f

i

idV

#

W

p

 . V.

Since each of f , idV and p are weak equivalences, then by the two-out-of-three property both of iiU and iiV are weak equivalences. They are also cofibrations, that cofibrations are closed under composition. Therefore, F (iiU ) and F (iiV ) are weak equivalence in D. Then, applying the two-out-of-three for the morphisms F (iiV ), F (p) and F (piiV ) = idF (B) yield that F (p) is a weak equivalence in D; then for the morphisms F (iiU ), F (p) and F (piiU ) = F (f ) yield that F (f ) is a weak equivalence in D.  Corollary 3.6. Let (F, G, φ) ∶ C → D be a Quillen adjunction. Then, F preserves all weak equivalences between cofibrant objects, and G preserve all weak equivalences between fibrant objects. Proof. We prove the first statement, then the proof of the second statement is obtained by duality. Since F preserves weak cofibrations, then in particular it reserves weak cofibrations between cofibrant objects. Hence, it satisfies the condition of Ken Brown’s Lemma 3.5, and it preserves all weak equivalences between cofibrant objects.  Therefore, the restricted functors F ∣Cc ; Cc → D

, and

G∣Df ∶ Df → C

preserve all weak equivalences. Hence, they induce well-defined functors on homotopy category HF ∣Cc ∶ HCc → HD

, and

HG∣Df ∶ HDf → HC.

These functors in addition to fibrant and cofibrant replacements gave rise to an adjoint on the level of homotopy category HC ⇆ HD

41

as in the below definition Definition 3.7 (Total Derived Functors). Let (F, G, φ) ∶ C → D be a Quillen adjunction. Then, ● the total left derived functor L F ∶ HC → HD is defined to be the composition HC

HQ

/ HCc

HF

/ HD

where Q is the cofibrant replacement of C. Moreover, let τ ∶ F → F ′ be a natural transformation between left Quillen functors. Then, the total left derived natural transformation L τ ∶ L F → L F ′ is defined to be the horizontal composition Hτ ○ HQ. ● the total right derived functor R G ∶ HD → HC is defined to be the composition HD

HR

/ HDf

HG

/ HC

where R is the fibrant replacement of D. Moreover, let τ ∶ G → G′ be a natural transformation between left Quillen functors. Then, the total right derived natural transformation R τ ∶ R G → R G′ is defined to be the horizontal composition Hτ ○ HR.

RA FT

Hovey indicates in [Hov99] that the above total derived functors are the main reason beyond altering the definition of model categories and requiring the factorisation to be functorial and being fixed by the model structure. Fixing the functorial factorisation in the model structure makes the derived functors between homotopy categories dependent only on the Quillen adjunction (which implicitly contains the data of the model structures). Still, the total left and right derived functors can be defined for any cofibrant and fibrant approximation functors, but in that case specifying the total derived functor will require more data than encoded in Quillen adjunction, namely specifying the considered cofibrant or fibrant approximation functor, as seen in the below definition. This proves practical, especially that the cofibrant and fibrant replacement functors in the model categories of interest in motivic homotopy theory are far from being easy to compute. However, these model categories usually admit more nicely behaved cofibrant and fibrant approximation functors that are not obtain from a functorial factorisation on the model category, as seen especially in localised model categories, hence the below definition.

D

Definition 3.8 (Derived Functors with Respect to Approximation Functors). Let (F, G, φ) ∶ C → D be a Quillen adjunction. ● If Q′ ∶ C → Cc is a cofibrant approximation functor, the total left derived functor of F with respect to Q′ is defined to be the composition HC

HQ′

/ HCc

HF

/ HD

and denoted by LQ′ F ∶ HC → HD. Moreover, let τ ∶ F → F ′ be a natural transformation between left Quillen functors. Then, the total left derived natural transformation of τ with respect to Q′ is defined to be the horizontal composition Hτ ○ HQ′ and denoted by LQ′ τ ∶ L F → L F ′ . ● If R′ ∶ D → Df is a fibrant approximation functor, the total right derived functor of G with respect to R′ is defined to be the composition HD

HR′

/ HDf

HG

/ HC

and denoted by RR′ G ∶ HD → HC. Moreover, let τ ∶ G → G′ be a natural transformation between right Quillen functors. Then, the total right derived natural transformation of τ with respect to R′ is defined to be the horizontal composition Hτ ○ HR′ and denoted by RR′ τ ∶ R G → R G′ . In particular, we have LQ F = L F , and RR G = R G. What is the use of finration/cofibration? In the above sense simplicial homotopy is a left homotopy. The conditions of Quillen adjunctions can be formulated in several ways. Each of which might be more convenient in different occasions. Therefore, we recall some of these equivalent conditions, that will be used throughout this text. Lemma 3.9 (Dugger). Let (F, G, φ) ∶ C → D be an adjunction between model categories. ● If F preserves weak cofibrations, then it is a left Quillen functor if and only if it preserves cofibrations between cofibrant objects.

42

● If G preserves fibrations, then F is a left Quillen functor if and only if it preserves cofibrations between cofibrant objects. ● If F preserves cofibrations, then G is a right Quillen functor if and only if it preserves fibrations between fibrant objects. ● If G preserves weak fibrations, then it is a right Quillen functor if and only if it preserves fibrations between fibrant objects. Proof. The above statements are either dual or can be deduce from each other by lemma 1.24. Therefore, we prove the first, then the rest follows by duality and lemma 1.24. One direction is immediate, for the other assume that F preserves weak cofibrations and cofibrations between cofibrant objects. Then will show that F is a left Quillen by showing that G preserves all weak fibrations. Let p ∶ X → Y be a weak fibration in D. Since F preserve weak cofibrations, and fibrations are precisely (weak cofibrations)- fib, then corollary 1.24 shows that G preserves fibrations, and hence G(p) ∶ G(X) → G(Y ) is a fibration in C. Consider the cofibration cofibrant replacement for G(p), then we have the commutative diagram

̃ Q(G(p))

 Ỹ

pX

D

pY

/ G(X) G(p)

 / G(Y )

T

̃ X

DR AF

̃ with Q(G(p)) a cofibration between cofibrant objects, and pX and pX are weak fibrations. Since weak equivã lences satisfy the two-out-of-three property then it is enough to show that Q(G(p)) is a weak equivalence. The adjunction induces the diagram ̃ F (X)

̃ F (Q(G(p)))

 F (Ỹ )

p′X

/X =

p

h

p′Y

 /Y

̃ with F (Q(G(p))) a cofibration because F preserves cofibrations between cofibrant objects. The above diagram admits the dotted lift h ∶ F (Ỹ ) → X because p is a weak fibration. Then the adjunction induces a lift h′ ∶= ϕỸ , G(X)(h) for D. Then, considering the left upper triangle of D induced by the lift h′ , on has the diagram ̃ X

̃ Q(G(p))

 Ỹ

̃
r

h′

pX

 / G(X).

̃ i.e. a X ̃ is a retract of Ỹ for having Since pX is a weak fibration, there exists a dotted lift r ∶ Ỹ → X, l ̃ ̃ ̃ rQ(G(p)) = idX̃ . In fact it is deformation retract, that Q(G(p))r ∼ idỸ because pY Q(G(p))r = pY idỸ , Ỹ is a ̃ cofibrant object and pY is a weak fibration (see [Hir03, Prop.7.5.9]). Therefore, the images of both Q(G(p))r ̃ ̃ and rQ(G(p)) in HC are identifies, hence the images of both Q(G(p)) and r in HC are isomorphisms, and ̃ hence Q(G(p)) is a weak equivalence.  One may consider generalising the above lemma and ask for F preservation of all cofibrations as a result of its preservation of cofibration between cofibrant objects. However, that does not hold in general. 4. Existence of Model Structures In general it is difficult to verify directly that a certain classes of morphisms define a model structure on a given category, at least for the interesting model categorises. Therefore, we recall hereby some alternative methods used to verify the usually most difficult axioms, namely the lifting and factorisation axioms. We see below that Joyal’s trick enables us to substitute part of the lifting axiom with closeness with respect to composition and push-outs, which are usually easier to prove. Also, the small object argument provides a canonical way to verify the existence of the functorial factorisation.

43

4.1. Joyal’s trick. The lifting axiom is hard to prove in general. However, knowing that some of the other axioms are satisfied, in addition to a usually simpler condition - that is supposed to be satisfied for any model category - simplify the requirement of the lifting axiom. That is given precisely in Joyal’s trick below. Lemma 4.1 (Joyal’s trick). Let C be a cocomplete category (part of CM1), and C, F, W be classes of morphisms of C satisfy the two-out-of-three axiom (CM2), in addition to the following properties: (1) Morphisms of F has the RLP with respect to morphisms of C ⋂ W (part of the lifting axiom CM4). (2) All morphisms f in C factorises as e = qj, with q ∈ F ⋂ W and j ∈ C (weaker version of part of the factorisation axiom a). (3) The class C is closed under composition and pushouts. Then, the lifting axiom (CM4) is satisfied. Proof. Let E be a commutative square of morphisms of C e0

K i

E

 L

p

 /Y

RA FT

e1

/X

where i ∈ C and p ∈ F ⋂ W . Since C is cocomplete, there exist a pushout square /X

e0

K

iX

i

 L

 / L∐X

iL

K

Then, by the universal property of pushout squares, there exists a unique morphism e ∶ L ∐ X → Y that K

K

D

makes the below solid diagram commutative. Then, by 2, e factorise as e = qj, with q ∶ Z → Y ∈ F ⋂ W and j ∶ L ∐ X → Z ∈ C. K

e0

/X jiX iX

i

 L

iL

/7 X .

idX

 / L∐X

j

(

h′

8Z

e

q

p

$/  7Y

K e1

Since i ∈ C, then by 3, iX ∈ C. Hence, also by 3, jiX ∈ C. Notice that p = qjiX , where p, q ∈ F ⋂ W ⊂ W , then by CM2 jiX ∈ W . Hence, jiX ∈ C ⋂ W , and we already have p ∈ F ⋂ W ⊂ F . Then, 1 implies the existence of the doted lift h′ ∶ Z → X of the commutative square X jiX

 Z

idX

h q

/X . >  /Y

p

Therefore, E admits a lift h ∶= h′ jiL ∶ L → X. That, hi = h′ jiL i = h′ jiX e0 = idX e0 = e0 , and ph = ph′ jiL = qjiL = eiL = e1 . Then, the above argument and 1 implies that the lifting axiom (CM4) holds.



Remark 4.2. In some occasions, it is easier to verify the axioms of model category leaving the lifting axiom to the end. That, when the others are verified, all what is left to use Joyal’s trick is proving that the class C is closed under push-outs and composition.

44

4.2. Small Object Argument. Finding a functorial factoisation is not a easy task in general. But, if it is already know that the given category is cocomplete, and the domain of a class of morphisms satisfy a certain condition, called ’smallness’, then the the small object argument, explained below, give a canonical way of finding the desired functorial factorisations though morphisms induced from the given class. The small object argument is a canonical method, and actually it is the only way we use in these notes to prove the existence of functorial factorisations. 4.2.1. Sequences. In order to recall the small object argument, we need the notion of transfinite composition, to be recalled below. This arguments require familiarity with basics of ordinals and cardinals, we refer the reader to [Jec03] and [Hir03, §10]. Definition 4.3 (λ-sequence). Let C be a cocomplete category, and λ be an ordinal. A λ-sequence in C is defined to be a colimit-preserving functor Z ∶ λ → C. It is usually written as: Z0 → Z1 → ⋯ → Zξ → ⋯ Moreover, let I be a class of morphisms in C, such that Zξ → Zξ+1 is contained in I, for ξ + 1 < λ, then we say that Z is a λ-sequence of morphisms of I.

RA FT

Let ξ < λ be ordinals, iλξ ∶ ξ → λ be the evident embedding. Then recall that ξ ≅ colim iλξ , if and only if ξ = 0 or ξ is a limit ordinal. Whereas, if ξ = υ + 1 is a successor ordinal, then colim iλξ ≅ υ, hence ξ ≅ colim iλξ+1 . Therefore, having Z to be colimit-preserving, is equivalent to having Z0 = ∅ (the initial object in C) for being a colimit of the empty functor to C, and for limit ordinals ξ < λ the canonical morphism colim Z ○ iλξ → Zξ

is an isomorphism. No restriction is required on successor ordinals, that being colimit-preserving at a successor ξ = υ + 1 is equivalent to the tautological statement Zξ ≅ colim Z ○ iλξ+1 ≅ Zξ . Still, the requirement for Z ∶ λ → C to be colimit-preserving is very restrictive. For instance, Z0 ≅ ∅ for λ > 0. Therefore, there exists a unique constant λ-sequences in C for any given ordinal λ > 0. The same is evidently satisfied for λ = 0 that the category 0 is the empty category (the initial object in Cat). However, that can be remedied by choosing the suitable category to work with. For example, having a functor F ∶λ→C

D

that is colimit-preserving at all non-zero ordinals ξ < λ, but not at 0. One can consider the induced functor Z∶ λ ξ υ<ξ

→ ↦ ↦

F0 ↓ C (Fξ , f0ξ ) fυξ

which is a λ-sequence in F0 ↓ C. Also another way to remedy that is that for every X ∈ C, one can consider the unique λ-sequence CX (λ) in C with CX (λ)0 = ∅ and CX (λ)ξ = X for 0 < ξ < λ. The only difference between the two solution is that if the ’constant’ sequence is needed to be or morphisms of some class of morphisms I, then the second approach require ∅ → X to be in I, whereas teh first does not. In general if the above functor F fails to be colimit-preserving at some limit ordinals in λ, that also can be remedied as seen in the reindexing lemma 4.11. Remark 4.4. Since λ-sequences in C are in particular functors, then it is natural to define morphisms of λ-sequences to be natural transformations between them. Denote the category of λ-sequences in C by λ-Seq(C), then λ-Seq(C) is a full sub-category of Cλ . Since C is cocomplete, then Cλ is so with colimits given object-wise. Also, λ-Seq(C) is a cocomplete category with colimits given object-wise. To see that, let J be a small category, and θ ∶ J → λ-Seq(C) ⊂ Cλ be a small diagram in λ-Seq(C), then it is a small diagram in Cλ , hence colim θ exists in Cλ . Let ξ < λ be 0 or a limit ordinal. Since colimits in Cλ are given object-wise, then we have (colim θ)ξ = colim θξ J

∗ colimλ (iλξ ) θ

where θξ = is the evident induced functor J → C sends j to θ(j)ξ . On the other hand, since small colimts commute with each other, we have ∗

colim(colim θ)iλξ = colim colim ((iλξ ) θ) = colim colim (θ(−)iλξ ) = colim θξ ξ

∗ (iλξ )

J

ξ

J

J

ξ

J

where is the pre-composition with (iλξ ). Hence, Z is a λ-sequence, i.e. λ-Seq(C) is a cocomplete category with colimits given object-wise.

45

On the other hand, if C is complete, then λ-Seq(C) is not necessary so because small limits and small colimits do not always commute. Still, if C admits a terminal object ∗ ∈ C, then λ-Seq(C) admits the terminal object C∗ (λ). Example 4.5. Notice that conventional ’categorical’ sequences are nothing but ω-sequences. Throughout these notes, we denote the unique morphism of Zυ → Zξ in C by zυξ for any ordinals υ < ξ < λ. However, for ξ = υ + 1 we avoid redundancy and we write zυ . Similarly, we denote the unique morphism (up to choice of colimit, i.e. composition with isomorphism) Zξ → colim Z by zξλ for every ordinal ξ < λ. We see later, mainly in the proof of the small object argument that we need to construct some λ-sequences for a given ordinal λ; for that purpose we can use a modified version of the transfinite recursion, based on a modified version of the transfinite induction lemma below. This is motivated by [Jec03, Th.2.14 and Th.2.15]. Lemma 4.6 (λ-transfinite Induction). Let λ > 0 be an ordinal, P a property defined on λ. Suppose that

DR AF

Then, P (ξ) holds for all ordinals ξ < λ.

T

(1) P (0) holds. (2) For successor ordinal ξ = υ + 1 < λ, if P (υ) holds, then P (ξ) holds. (3) For limit ordinals ξ < λ, if P (υ) holds for all ordinals υ < ξ, then P (ξ) holds.

Proof. Define the subset

S = {ξ < λ∣P (ξ) does not hold}.

Assume for the sake of contradiction that S is non-empty. Since λ is well-ordered, then ξ = min S < λ exists. Then, applying the above conditions shows that P (ξ) holds, contradiction. Hence, S is empty and P holds for all ordinals less than λ.  λ-transfinite induction is used to show that certain property hold for all ordinals strictly less that λ. Also, it can be used as well to construct λ-sequences, for a given λ. However, it might be useful, as in the small object argument theorem 4.48, to construct sequences with an arbitrary length, or ’Ord-sequence’ of sequences; for that purpose we use the transfinite recursion, based on the transfinite induction lemma below. Lemma 4.7 (transfinite Induction). Let P a property defined on all ordinals. Suppose that (1) P (0) holds. (2) For successor ordinal λ = ξ + 1, if P (ξ) holds, then P (λ) holds. (3) For limit ordinals λ, if P (ξ) holds for all ordinals ξ < λ, then P (λ) holds. Then, P (λ) holds for all ordinals λ. Proof. Define the subclass S = {λ ∈ Ord∣P (λ) does not hold}. Assume for the sake of contradiction that S is non-empty. Since Ors is a well-ordered class, then λ = min S ∈ S ⊂ Ord exists. Then, applying the above conditions shows that P (λ) holds, contradiction. Hence, S is empty and P holds for all ordinals.  A slightly different version of the above lemma is given with the assumption of successor ordinal λ = ξ + 1 being replaced by P (υ) holds for all ordinals υ < λ. Definition 4.8 (Transfinite Composition). Let C be a cocomplete category, λ an ordinal, and Z a λ-sequence in C. Then, the transfinite composition of Z is defined to be the canonical injection of colimit zξλ ∶ Z0 → colim Z. Moreover, let I be a class of morphisms in C, such that zξ ∈ I, for ξ + 1 < λ, then the transfinite composition Z0 → colim Z is called a transfinite composition of morphisms of I. Furthermore, a class of morphisms I of C is said to be closed under transfinite composition if it contained the transfinite compositions of all λ-sequences in X ↓ C of morphisms of I, for every X ∈ C and for every ordinal λ.

46

Remark 4.9. Every ξ-sequence Z in C can be extended canonically to (ξ + 1)-sequence in C setting Zξ ∶= colim Z. Also, for every limit ordinal λ, and ξ < λ, every ξ-sequence Z in C can be extended canonically to λ-sequence in C setting Zυ = colim Z for ξ ≤ υ < λ. In fact every λ-sequence can be extended to ’Ord-sequence’ similarly. However notice that Ord − Seq(C) is not necessary cocomplete, that Ord is not a small category. In all cases we abuse notation and use the same notation for the original sequence and the extended one. It is easy to see that in any cocomplete category Lemma 4.10. Let C be a cocomplete category, I a class of morphisms of C. Then, I- proj is closed under transfinite compositions, and hence so is I- cof. Proof. TBC.



Since cofibrations in a model category are given as the cofibrations class of weak fibrations, then above lemma indicates that cofibrations in model categories are closed under transfinite compositions. Hence, a necessary condition form a class of morphisms in a category to be a class of cofibrations of a model structure, is to be closed under transfinite compositions.

RA FT

It is useful to be able to construct λ-sequences for a given ordinal, that they occur naturally in cocomplete category as in lemma 4.14, below. Also, not every functor from a given ordinal λ is necessary colimit preserving. Since such factors fail to be colimit preserving for 0 or for limit ordinals ξ < λ, then a shift of indices, or reindexing as in [Hir03] gives a canonical way for remedying this flaw and constructs a λ-sequence. Where a λ-sequence is obtained from a functor λ → C by shifting its values by 1, and assigning the colimits to limit ordinals less than λ. Lemma 4.11 (Reindexing). Let C be a cocomplete category, λ a limit ordinal, and F ∶ λ → C be a functor. Suppose that Z ∶ λ → F0 ↓ C, defined on objects by Z0 = (F0 , idF0 ), Zξ = colim(F0 ↓ F ) ○ iλξ for every limit ordinal ξ < λ and Zξ+1 = (Fξ , f0ξ ) for ξ + 1 < λ, and on morphisms by z0 = idF0 , zξ = colim F iλξ → Fξ for every limit ordinal ξ < λ and zξ+1 = fξ for ξ + 1 < λ.

D

Then, Z is a functor, called the reindexing of F . Moreover, it is a λ-sequence in F0 ↓ C, and colim Z = (colim F, f0λ ).

Proof. See [ibid., Lem.10.2.4]



Caution 4.12. The above lemma obtains λ-sequence in F0 ↓ C from a given functor λ → C, for a limit ordinal λ. However, for a class I of morphisms of C, if fβ ∶ Fξ → Fξ+1 is in I for all β + 1 < λ, that does not imply that the constructed λ-sequence in the above lemma is a λ-sequence of morphisms of i, unless I is closed under transfinite composition. That, for each limit ordinal ξ < λ, the morphism zξ ∶ colim F ○ iλξ → Fξ is not required to be in I, unless I is closed under transfinite composition. Lemma 4.13. Let C be a cocomplete category, I a class of morphisms of C, f ∶ X → Y be a morphism in C. If exist an ordinal λ and a functor F ∶ λ → C such that F0 = X, colim F = Y , f0λ ∶ F0 → colim F coincides with the morphism f , and morphisms colim F ○ iλξ → Fξ are in I for every ξ + 1 < λ; then the morphism f is a transfinite composition (in X ↓ C) of elements of I. Moreover, if λ is infinite, then f is a transfinite composition of of elements of I, indexed by a limit ordinal whose cardinal equals the Card λ. Proof. See [ibid., Prop.10.2.6]



Transfinite compositions are natural in cocomplete category. For instance, the induced morphisms by small colimits are transfinite composition of push-outs, as seen in the below lemma. Lemma 4.14. Let C be a cocomplete category, J be a set, S = {gj ∶ Xj → Yj ∣j ∈ J} be a set of morphisms in C. Then, the coproduct ∐ gj ∶ ∐ Xj → ∐ Yj j∈j

j∈j

j∈j

is a transfinite composition of push-outs of morphisms of S. Moreover, if J is infinite, then the transfinite composition is indexed by a limit ordinal whose cardinal equals Card J.

47

Proof. Notice that, for k ∈ J, the push-forward of the diagram / Yk

gk

Xk iXk

 ∐ Xj

j∈J

⎛ ⎞ ∐ Xj ∐ Yk with the evident injections, i.e. it replaces Xk in ∐ Xj with Yk through gk . That indicates ⎝j∈J∣j≠k ⎠ j∈J g can be obtained as a transfinite composition of push-outs of g ’s by replacing all Xk in ∐ Xj through gk ∐ j k

is

j∈J

j∈J

for all k ∈ J ’step by step’, as seen below. By Zermelo’s theorem, J can be made well-ordered. Let λ be the the ordinal isomorphic to a choice of well-order set of J. Then, under this isomorphism, we have S = {gξ ∶ Xξ → Yξ ∣ξ < λ}. Then, define F ∶ λ + 1 → C to be given on objects by

RA FT

⎛ ⎞ ⎛ ⎞ Fξ = ∐ Yυ ∐ ∐ Xυ for ξ < λ + 1 ⎝υ<ξ ⎠ ⎝ξ≤υ<λ ⎠

and on morphisms for every ξ, ξ ′ < λ + 1, fξξ is induced by the universal property of coproduct for the set of morphisms ′

{idYυ ∣υ < ξ} ⋃{gυ ∣ξ ≤ υ < ξ ′ } ⋃{idXυ ∣ξ ′ ≤ υ < λ + 1}.

The F is readily seen to be a functor, and for every ξ < λ we have a push-forward square gξ

Xξ

iXξ

iYξ

 Fξ

D

/ Yξ

fξ

⌟  / Fξ+1

Also, F0 = ∐ Xξ , colim F = ∐ Yξ , and f0λ ∶ F0 → colim F coincides with ∐ gξ ξ<λ

ξ<λ

X0 { ∐ Xξ = F0 ξ<λ

/ Y0

g0

⌟ f0

ξ<λ

 / F1

Xξ

...

/ Fξ

gξ

/ Yξ  } / Fξ+1

Xξ+1

⌟ fξ

gξ+1

/ Yξ+1  / Fξ+2

⌟ fξ+1

...

/ colim F = ∐ Yξ 3 ξ<λ

∐ gξ ξ<λ

For last axiom, if we show it is λ-sequence, we avoid reindexing...TBC Hence, the statement of this lemma holds by the previous lemma 4.13.



4.2.2. Relative I- cell Complex. Coproducts are natural in cocomplete categories, hence the above construed of morphisms of transfinite composition of push-outs are natural, which give rise to the generalisation given in the below definition. Definition 4.15 (Relative I- cell Complex). Let C be a cocoplete category, I a class of morphisms of C, f ∶ A → B in C. We say that f is a relative I- cell complex if it is a transfinite composition of pushouts of elements of I. I.e. f ∶ A → B is a relative I- cell complex if there exists an ordinal λ, a λ-sequence Z in A ↓ C, and for every ordinal ξ with ξ + 1 < λ there exists morphisms gξ ∶ Xξ → Yξ in I and Xξ → Zξ in C and

48

push-forward square Xξ  Zξ

gξ

zξ

/ Yξ ⌟  / Zξ+1

such that f is the transfinite composite of Z. X0 { A = Z0

z0

g0

/ Y0 ⌟  / Z1

gξ

Xξ+1

...

}

/ Zξ

/ Yξ

Xξ+1

⌟  | / Zξ+1

zξ

gξ+1

zξ+1

/ Yξ+1 ⌟  / Zξ+2

...

/ colim Z = B 3

f

RA FT

We denote the class of relative I- cell complexes by I- cell. And, we say that B is an I- cell complexes if the morphism ∅ → B is a relative I- cell complex. Also, we say that a relative I-complex f ∶ A → B is an inclusion of I- cell complexes if A is an I- cell complex. This notation is an abstraction of gluing cells in topology, hence the name. Example 4.16. In the settings of lemma 4.14 above, that lemma shows that ∐ gj is a relative S- cell complex.

j∈J

Remark 4.17. One sees that for any class of morphism I in C, every isomorphism f ∶ A → B in C is a transfinite composition of 1-sequence in A ↓ C, with value A. Hence, the above condition is satisfied automaticly (for not being violated), and f is a relative I- cell complex.

D

Let C be a cocomplete category, I a class of morphisms of C. Notice that relative I- cell complexes might be expressed in different ways as a transfinite composition of push-forward of morphisms of I or a transfinite composition of push-forward of products of morphisms of I, but in some applications especially in cellular model categories, it is important to fix a ’representation’ of the I- cell complexes as transfinite composition of push-forward of morphisms of I, hence the below notion of presentations of relative I- cell complex is recalled below. Definition 4.18. Let C be a cocomplete category, I a class of morphisms of C, f ∶ A → B. Then a presentation of f is a pair (Z, (Sξ , gξ , iξ )ξ<λ ) where Z is an λ-sequence in A ↓ C for some ordinal λ, with transfinite composition isomorphic to f , such that for every ξ < λ ● Sξ is a set (they indexes cells). ● gξ is a function gξ ∶ Sξ → I (they chooses cells). ● iξ is a function iξ ∶ Sξ → ob(C ↓ Zξ ) (they glues cells). ● dom gξ (sξ ) = dom iξ (sξ ), for every sξ ∈ Sξ such that there is a push-forward diagram for every ξ + 1 < λ ∐ gξ (sξ )

∐ X(sξ )

sξ ∈Sξ

/ ∐ Y (sξ )

sξ ∈Sξ

sξ ∈Sξ

∐ iξ (sξ )

sξ ∈Sξ

 Zξ

zξ

⌟  / Zξ+1 .

for X(sξ ) ∶= dom(iξ (sξ )) and Y (sξ ) ∶= codom(iξ (sξ )). Particularly, if A = ∅, then a presentation of f is called presentation of B. Moreover, a pair of relative I- cell complex and its presentation is called presented relative I- cell complex. Particularly, if A = ∅, a presented presented relative I- cell complex is called a presented I- cell complex. For a presented relative I- cell complex (f ∶ A → B, (Z, (Sξ , gξ , iξ )ξ<λ ))

49

● λ is called the presentation ordinal. ● A cell of the presentation of f is a pair (ξ, sξ ), where ξ < λ and sξ ∈ Sξ . ● The set ⊔ Sξ = {(ξ, sξ )∣ξ < λ, sξ ∈ Sξ } is called the set of cells of the presentation of f . ξ<λ

● ● ● ●

The cardinality of the set of cells of the presentation of f is called the size of the presentation of f . For ξ < λ, Zξ is called the ξ th -skeleton of the presentation of f . (Sξ , gξ )ξ<λ is called choice of cells of the presentation of f . (iξ )ξ<λ is called the gluing data of the presentation of f . particularly, iξ (sξ ) is called the gluing morphism of the cell (ξ, sξ ), for a cell (ξ, sξ ) ∈ ⊔ Sυ . υ<λ

The above distinguishing between presentation of relativeI- cell complexes and presented relativeI- cell complexes is due to the fact that presentations determine the relative I- cell complexes only up to an isomorphism, whereas it is uniquely determined for presented relativeI- cell complexes. However, if no confusion may arise we abuse notation and refer to the presented relative I- cell complex (f ∶ A → B, (Z, (Sξ , gξ , iξ )ξ<λ )) by f or Z, or simply by B if A = ∅.

RA FT

In Top one can talk of inclusions or topological spaces, particularly inclusions of CW complexes and CW sub-complexes. Although the notion of inclusions does not merely exist in arbitrary cocomplete category, the above notion of presentation of I- cell complexes provide an analogue way of topological spaces to talk about sub-complexes of relative I- cell complex, as follows. Definition 4.19 (Morphisms of Presented Relative I- cell complexes). Let C be a cocomplete category, I a class of morphisms of C. Then, define morphisms between presented relative I- cell complexes (f ∶ A → B, (Z, (Sξ , gξ , iξ )ξ<λ )) and (f ′ ∶ A → B ′ , (Z ′ , (Sξ′ , gξ′ , i′ξ )ξ<λ )) to be a morphism between the λ-sequences Z and Z ′.

D

A morphism between presented relative I- cell complexes f and f ′ induces a morphism between f and f ′ in A ↓ C, induced by the universal property of colimits. One can see easily that the above definition gives rise to the category of presented relative I- cell complexes in C and their morphisms.

One may would like to consider morphisms between presented relative I- cell complex with different domains, in such case we do not have the notion of morphisms between Z and Z ′ , as Z lives in the category of functors from λ to A ↓ C and ′ Z lives in the category of functors from λ to A′ ↓ C, having f ′ a relative I- cell complex from A′ to B ′ . Moreover, such morphisms are not needed in the development of localisation of model categories, hence we restrict ourselves to the above case. On the other hand, when considering presented relative I- cell complexes with different presentation ordinal, then the ’shorter’ one can be evidently extended to the presentation ordinal of the longer one.

We recall below the sub-complexes of presented relative I- cell complexes, which add a set-theoretical flavour to abstract cocomplete categories, especially when relative I- cell consists of effective monomorphisms as it is the case in cellular model categories. In such case they become an analogue of inclusions of sets, and admit similar operations of intersection and union that can be realised by their set-theoretical counterparts. Moreover, they have the advantage of being closed under pull-backs. Definition 4.20 (Sub-complexes of Presented Relative I- cell complexes). Let C be a cocomplete category, I a class of morphisms of C, (f ∶ A → B, (Z, (Sξ , gξ , iξ )ξ<λ )) be a presented relative I- cell complex. Then, a relative I- cell sub-complex of f is defined to be a presented relative I- cell complex (f ′ ∶ A → B ′ , (Z ′ , (Sξ′ , gξ′ , i′ξ )ξ<λ )) such that ● Sξ′ ⊆ Sξ and gξ′ = gξ ∣Sξ′ , for every ξ < λ. ● There exists a morphism of presented relative I- cell complexes ϕ ∶ f ′ → f (i.e. ϕ ∶ Z ′ → Z a natural transformation) such that iξ (s′ξ ) factories as iξ (s′ξ ) = ϕξ i′ξ (s′ξ ) for every ξ < λ, s′ξ ∈ Sξ′ , i.e. the below

50

diagram commutes for every ξ < λ Zξ′

′ zξ

/

^ ′ ′ ∐ iξ (sξ ) ′ s′ξ ∈Sξ

/

′ ′ ∐ dom(iξ (sξ )) ′ s′ξ ∈Sξ

ϕξ

 ∐ dom(iξ (sξ ))

/

sξ ∈Sξ

∐ iξ (sξ ) sξ ∈Sξ



/

RA FT

Zξ

zξ

where the vertical right morphism is the coprojection induced the the inclusion Sξ′ ⊆ Sξ , the restriction gξ′ = gξ ∣Sξ′ and the universal property of coproducts. We call such morphism ϕ ∶ f ′ → f a relative I- cell sub-complex morphism, and we call colim ϕ an inclusion of relative I- cell sub-complex. A morphism g ∶ W → B is said to factorise through the relative I- cell sub-complex at ξ < λ if it factorises through ϕξ and said to factorise through the inclusion of relative I- cell sub-complex if it factorises through colim ϕ. Moreover, when A = ∅ we call ϕ an I- cell sub-complex morphism, and we call colim ϕ an inclusion of I- cell sub-complex, and we write colim ϕ ∶ B ′ → B.

D

The first condition above is arresting that cells of the sub-complex are chosen to be cells of the complex, whereas the second condition asserts that the gluing data of the sub-complex should be compatible with the gluing data of original presented I- cell complex. This is seen clearly when C = Top and I = {∣∂ n ∣ ∶ ∣∂∆n ∣ ↪ ∣∆n ∣∣n ≥ 0}.

Then, I- cell complexes are princely CW complexes, and I- cell sub-complexes of presented I- cell complexes are sub-complexes of CW complex. Where the second condition above reads then that to choosing a cell for a sub-complex of a CW complex requires choosing its boundary in the original CW complex to be its boundary in the sub-complex as well (i.e. to choose as cells for the sub-complex its boundary should be chosen as well, and should be chosen as its boundary.). Notice that a relative I- cell sub-complex is not a sub-object in the category of presented relative I- cell complexes. Still, when I- cell consists of monomorphisms then each relative I- cell sub-complex gives rise to a sub-object of a presented relative I- cell complex, elements of the sub-object (element of that isomorphism class) do not necessary have the same set of cells, that is particularly obvious when the set I contains isomorphisms, hence the previous conditions are specific to the considered representative of the sub-object of the presented relative I- cell complex. The definition of relative I- cell sub-complex does not fix a choice of the relative I- cell sub-complex morphism ϕ ∶ f ′ → f , it merely asserts that such morphism exists. In fact, if I- cell complexes are momomorphism s for each relative I- cell sub-complex f ′ of f there is a canonical choice of ϕ ∶ f ′ → f for which ϕξ are relative I- cell complexes for every ξ < λ, as seen in the below lemma. Going back to the motivation example when C = Top, the category of topological spaces, and I = {∣∂ n ∣ ∶ ∣∂∆n ∣ ↪ ∣∆n ∣∣ n ≥ 0}. Choosing a sub-complexes of CW -complex is equivalent of choosing ’appropriate’ cells, by appropriate choice we mean that if a cell is chosen to be in the sub-complex then its boundary (along which it is glued) should be also chosen to be in the sub-complex. In Top the choice of cells of sub-complexes of a given CW -complex is sufficient to determine the subcomplex, i.e. to determine how they are to be glued in the sub-complex, that is due to the existence of a

51

canonical embedding (monomorphism) of the sub-complex in the original CW complex, which is a result of having relative I- cell complexes being monomorphisms in Top. However, in an arbitrary cocomplete category that does not not necessary hold, as in Rng for the canonical set of ring homomorphisms I = {Z → Z/n∣n ≥ 1} where relative I- cell complexes are not monomorphisms. We see below that restricting attention to cocomplete categories C and sets of morphisms I in C such that relative I- cell complexes are monomorphisms makes subcomplexes determined by the choice of the set of cells, of course an appropriate choice of cells in the above sense, where the sub-complex needed gluing data is recovered from the gluing data (iξ )ξ<λ of the original presented relative I- cell complexes, as seen in the below lemmas 4.23 and 4.24. In such settings, sub-complexes of relative I- cell complexes provide an analogue of inclusions of sets in abstract categories. Lemma 4.21. Let C be a cocomplete category, I a class of morphisms of C such that I- cell consists of monomorphisms. Then, for sub-complex (f ′ ∶ A → B ′ , (Z ′ , (Sξ′ , gξ′ , i′ξ )ξ<λ )) of a presented relative I- cell complex (f ∶ A → B, (Z, (Sξ , gξ , iξ )ξ<λ )), there exist a relative I- cell sub-complex morphism ϕ ∶ f ′ → f such that ϕξ are relative I- cell complexes for every ξ < λ. Proof. Straightforward by transfinite induction.



RA FT

Corollary 4.22. Let C be a cocomplete category, I a class of morphisms of C such that I- cell consists of monomorphisms. Then, inclusions of relative I- cell sub-complexes are inclusions of relative I- cell complexes. Proof. Direct result of lemma 4.21, above.



Lemma 4.23. Let C be a cocomplete category, I a class of morphisms of C such that I- cell consists of monomorphisms, and let (f ∶ A → B, (Z, (Sξ , gξ , iξ )ξ<λ )) be a presented relative I- cell complex. Then, every sub-complex of f is determined up-to isomorphism by its set of cells. Proof. Let (f 1 ∶ A → B 1 , (Z 1 , (Sξ1 , gξ1 , i1ξ )ξ<λ )) and (f 2 ∶ A → B 2 , (Z 2 , (Sξ2 , gξ2 , i2ξ )ξ<λ )) be sub-complexes of the presented relative I- cell complex f , and assume that Sξ1 = Sξ2 for every ξ < λ. Then, we will see that both sub-complexes are isomorphic. Since gξi is the restriction of Sξ to Sξi for every ξ < λ, for i = 1, 2. Then, gξ1 (sξ ) = gξ (sξ ) = gξ2 (sξ ), ∀sξ ∈ Sξ1 = Sξ2 , ∀ξ < λ

D

hence, gξ1 = gξ2 for every ξ < λ. Since I- cell consists of monomorphisms, then lemma 4.21 asserts that there exist relative I- cell sub-complex morphisms ϕ1 ∶ f 1 → f and ϕ2 ∶ f 2 → f with components being relative I- cell complexes. Then by the assumption, ϕ1 and ϕ2 are monomorphisms. Then, observing that ϕi0 = idA for i = 0, 1 and using the transfinite induction one can see that ∀ξ < λ ● There is an isomorphism ψ ∶ ϕ1 → ϕ2 given by the universal property of push-outs. ● For every ξ < λ, sξ ∈ Sξ1 = Sξ2 , then iξ (sξ ) factorises through ϕ1ξ and ϕ2ξ as iξ (sξ ) = ϕ1ξ i1ξ (sξ ) = ϕ2ξ i2ξ (sξ ) = ϕ2ξ ψξ i1ξ (sξ ). Since ϕ2ξ is a monomorphism, we have i2ξ (sξ ) = ψξ i1ξ (sξ ). Hence, the sub-complexes f 1 and f 2 are isomorphic.



Notice that a cell can be contained in a sub-complex of size κ + 1 is it can be attached to a sub-complex of size κ, i.e. if iξ factorises through an inclusion of relative I- cell sub-complex of size κ. Hence, in the settings where I- cell are monomorphisms this gives a rise to the formulation what is meant earlier by appropriate choice of cells, as in the below lemma. Lemma 4.24. Let C be a cocomplete category, I a class of morphisms of C such that I- cell consists of monomorphisms, and let (f ∶ A → B, (Z, (Sξ , gξ , iξ )ξ<λ )) be a presented relative I- cell complex. Then, a set of cells ∐ Sξ′ ⊂ ∐ Sξ determines a sub-complex of the presented relative I- cell complex f with set of cells ξ<λ

ξ<λ

′ ∐ Sξ if and only if: ξ<λ

For every ξ < λ, having that ∐ Sυ′ ⊂ ∐ Sυ determines a sub-complex (f ξ ∶ A → B ξ , (Z ξ , (Sυξ , gυξ , iξυ )υ<ξ )) υ<ξ

υ<ξ

of (f ∶ A → colim Z∣ξ , (Z∣ξ , (Sυ , gυ , iυ )υ<ξ )) with a relative I- cell sub-complex monomorphism ϕξ ∶ Z ξ → Z∣ξ , implies that Sξ′ ⊆ {sξ ∈ Sξ ∣ iξ (sξ ) factorises through ϕξ ∶ Zξ′ → Zξ }

52

where ● for a successor ordinal ξ = υ + 1, Zξ′ and ϕξ are induced by the push-out and the universal property of push-outs in the below diagram ′ zυ

ZυξX

/ Zξ′ > F

ξ ξ ∐ iυ (sυ ) ξ ξ sυ ∈Sυ

ξ ξ ∐ gυ (sυ ) ξ

ξ

●

/●



/ ●

sυ ∈Sυ

ϕξ

ϕξ υ

● ∐ gυ (sυ ) sυ ∈Sυ

∐ iυ (sυ ) sυ ∈Sυ

 

>

Zυ

zυ

  / Zξ

DR AF

T

● for a limit ordinal ξ, Zξ′ and ϕξ are induced by the colimit and the universal property of colimits in the evident diagram. Moreover, such sub-complex is determined uniquely up to isomorphism. Proof. See [Hir03, Prop.10.6.11].



Although, I-cofibrations and I- cell are not closed in general under pull-back. When I- cell consists of effective monomorphisms then relative I- cell sub-complexes are closed under pull-backs, moreover these pull-backs are understood in terms of Set pull-backs, i.e. intersection. More generally, if I- cell consists of monomorphisms then push-outs of relative I- cell sub-complexes are also understood in terms Set push-outs, i.e. unions. Lemma 4.25. Let C be a cocomplete category, I a class of morphisms of C such that I- cell consists of monomorphisms, let (f ∶ ∅ → B, (Z, (Sξ , gξ , iξ )ξ<λ )) be a presented I- cell complex, and f 1 and f 2 are I- cell sub-complexes of B with sets of cells ∐ Sξ1 ⊂ ∐ Sξ and ∐ Sξ2 ⊂ ∐ Sξ , respectively. Then, the push-out ξ<λ

ξ<λ

ξ<λ

ξ<λ

B 1 ∐ B 2 (push-out of ϕ1 along ϕ2 ) is an I- cell sub-complex of B with set of cells ∐ Sξ1 ⋃ Sξ2 , and is denoted B

ξ<λ

by B 1 ⋃ B 2 . Proof. A direct result of the above lemma 4.24.



Lemma 4.26. Let C be a cocomplete category, I a class of morphisms of C such that I- cell consists of effective monomorphisms, let (f ∶ ∅ → B, (Z, (Sξ , gξ , iξ )ξ<λ )) be a presented I- cell complex, and f 1 and f 2 are I- cell sub-complexes of B with sets of cells ∐ Sξ1 ⊂ ∐ Sξ and ∐ Sξ2 ⊂ ∐ Sξ , respectively. Then, the ξ<λ

ξ<λ

ξ<λ

ξ<λ

pull-back B 1 ×B B 2 (pull-back of ϕ1 along ϕ2 ) exists. Also, it is an I- cell sub-complex of B 1 ,B 2 , B 1 ⋃ B 2 and B with set of cells ∐ Sξ1 ⋂ Sξ2 , and is denoted by B 1 ⋂ B 2 . Moreover, the induced evident diagram ξ<λ

B1 ⋂ B2

/ B2

 B1

 / B1 ⋃ B2

is both cartesian and cocartesian. Proof. See [ibid., Prop.12.2.3].



I- cof and I- cell are defined by different concepts, the first by the lifting property whereas the second by push-outs and transfinite compositions. Nevertheless, the are highly liked, as seen below. Hereby, we recall basis stability properties of I- cell, and how this link for a general class of morphisms in a cocomplete category,

53

and later on, after introducing the small object argument, we recall the link for set of morphisms that satisfy the small object argument. Lemma 4.27. Let C be a cocomplete category, I a class of morphisms of C. Then, a. I- cell is closed under push-outs. b. I- cell is closed under transfinite composition. c. I- cell ⊆ I- cof. Proof. a and b. Direct result of the definition of I- cell, based on the fact that colimits commute with each other, See [Hir03, §10]. c. Since I ⊆ I- cof, I- cof is closed under transfinite composition and push-outs, then I- cell ⊆ I- cof. TBC.  Remark 4.28. We will see later, when I admits the small object argument, that I- cof are retracts of I- cell. In contrast to I- inj and I- proj, I- cell is not closed under retract. For example, in Top, let I consists of the canonical inclusions {∣∂ n ∣ ∶ ∣∂∆n ∣ ↪ ∣∆n ∣∣n ≥ 0}.

RA FT

We see later I admits the small object argument. A topological spaces X satisfy ∅ → X ∈ I- cell if and only if X is a CW complex, whereas ∅ → X ∈ I- cof is satisfied for any topological spaces, and it is a retract of ∅ → X ′ , for a CW complex X ′ , even if X is not so. Having a transfinite composition of a λ-sequence, as in the case of relative I- cell complex, it is interesting to see when morphisms to the colimit of the λ-sequence factorise through an element of the sequence. We will see that if λ is ’big enough’ or the λ-sequence is ’long enough’, then such factorisation occur. To make this is precise we start with the motivating example in Set, then recall the notions it gives rise too, namely κ-filtered ordinals and κ-small relative object.

D

Example 4.29. Let K be a set, λ be a limit ordinal, Z ∶ λ → Set be a λ-sequence in Set, and K ∶ A → colim X be a map of sets. It will be useful to know when such map factorises through Zξ for some ξ < λ. Recall that ⎛ ⎞ colim Z = ⊔ Zξ / ∼ ⎝ξ<λ ⎠ for the equivalence relation ∼ that identify xξ1 ∈ Zξ1 , and xξ2 ∈ Xξ2 , for ξ1 ≤ ξ2 < λ if there exists an ordinal ξ such that ξ1 ≤ ξ2 ≤ ξ < λ and zξξ1 (xξ1 ) = zξξ2 (xξ2 ); i.e. ∼ identify xξ1 , and xξ2 that are filtered to the same element. ∀k ∈ K, let ξk < λ be the smallest ordinal such that ∃xξk ∈ Zξk , zξλ (xξk ) = f (k). Then, consider the set Af = {ξk < λ∣k ∈ K}, and notice that ∣Af ∣ ≤ ∣K∣. Notice that, by assumption on elements of Af , Af ⊂ λ and sup Af ≤ λ. Then, we distinguish two cases: If sup Af < λ, then there exist ξ ∶= sup KA < λ such that ∀k ∈ K, zξξa (xξk ) ∈ Zξ and zξλ (zξξk (xξk )) = f (k). Thus, we can define the map fξ ∶ A → Zξ that sends k to zξξk (xξk ). Then, by the definition of the colimit of sets we see that f factorises through fξ at Zξ . On the other hand, if sup Af = λ such factorisation is not possible. That if such factorisation exists, say at Zξ for some ξ < λ. Then, ∀k ∈ K, ξk ≤ ξ, hence sup Af ≤ ξ < λ, which leads to contradiction. The argument in the above example gives rise to ’big enough’ ordinals and ’long enough’ sequences in Set, so such factorisations of f exists. When maps from K to a ’long enough’ transfinite composition of λ-sequences factorises, K is called small in analogy to factorisation in Z. That is made precise in the following definitions. Definition 4.30 (κ-filtered Ordinals). Let κ be a cardinal. We say that an ordinal λ is κ-filtered if it is a limit ordinal, and for every A ⊆ λ with ∣ A ∣≤ κ, then sup A < λ. Remark 4.31. Notice that for finite cardinal κ, sup A ∈ A, hence sup A < λ, then all limit ordinals are κ-filtered. However, for κ infinite, then κ-filtered ordinal are those limit ordinals greater than or equal κ+ . That, if λ < κ+ , then for any A ⊆ λ cofinal, ∣ A ∣≤ κ, and sup A = λ ≮ λ, i.e. λ is not κ-filtered. Wherase, if λ ≥ κ+ , then the cofinality of λ, co(λ) > κ. Therefore, having A ⊂ λ, ∣ A ∣≤ κ implies that A is not cofinal, hence sup A < λ, and λ is κ-filtered. Hence, the definition of κ-filtered ordinals can be rephrased to be limit ordinals with cofinality greater than κ.

54

Definition 4.32 (κ-small relative). Let C be a complete category, I a class of morphisms of C, and κ a cardinal. We say that an object K ∈ C is κ-small relative to I, if for all κ-filtered ordinals λ, the canonical map of sets, given by canonical post-composition ϑ ∶ colim C(K, Z− ) → C(K, colim Z)

(22)

is an isomorphism, for all λ-sequences Z ∶ λ → C of morphisms of I. We say that K is finite relative to I, if it is κ-small relative to I for some finite cardinal κ. We say that K is small relative to I, if it is κ-small relative to I for some cardinal κ. And, we say that K is small in C if it is small relative the class of all morphisms of C.

Notice that

RA FT

Avoid the confusion between small objects in C and compact objects in C. Compact objects in C are κ-small in C for all cardinals κ. However, in contrast to compact objects, co-Yoneda functors of κ-small objects in C do not have to commute with all filtered colimits, but rather transfinite compositions (colimits of λ-sequences, for big enough λ, namely κ-filtered).

⎛ ⎞ colim C(K, Z) = ⊔ C(K, Zξ ) / ∼ ⎝ξ<λ ⎠

for the equivalence relation that identify fξ1 ∶ K → Zξ1 , and fξ2 ∶ K → Zξ2 , for ξ1 ≤ ξ2 < λ if there exists an ordinal ξ, such that ξ1 ≤ ξ2 ≤ ξ < λ and zξξ1 ○ fξ1 = zξξ2 ○ fξ2 . I.e. when the two rectangles containing Zξ commutes, not necessarly the left one with vertices K, Zξ1 , and Zξ2 : K

fξ

D

fξ2

fξ1

Zξ 1

(

zξξ 1

Z ξ2

( zξξ 2

Zξ .

K

4. 40 .

Then, the surjectivity of the isomorphism (22) implies that morphisms f ∶ K → colim Z factors through some fξ ∶ K → Zξ , for some ξ < λ. I.e. there exists ξ < λ, and dotted morphism that makes the diagram commutes: f

Le m m a

fξ

Z0

⋅⋅⋅

'/

Zξ

⋅⋅⋅

* / colim Z. λ

Example 4.33. Recall example 4.29 to see that a set A is ∣ A ∣ --small relative to any class of morphisms of Set, i.e. small in sSet. Example 4.34. Every simplicial set is small. Proof. TBC.



Example 4.35. Let X be a simplicial set, with a graded set of generators S, and set of simplicial relations R. Let ∣ S ∣∶=∣ ⊔ Sn ∣, then X is ∣ S ∣ -small relative to any class of morphisms in sSet. Proof. TBC.



Corollary 4.36. ∀n ≥ 0, 0 ≤ r ≤ n, looking at the generators of Λr [n], ∂∆[n], and ∆[n], we see that they are n-small relative, (n + 1)-small relative, and 1-small relative, respectively, to any class of morphisms in sSet. Proof. TBC.



Example 4.37. The small topological spaces, in Top are precisely discrete topological spaces. Proof. See [Bou77, Ex.4.4].



Remark 4.38. One can see easily from the definition, that if κ < κ′ are two cardinals, then κ′ -filtered ordinals are κ-filtered. Whereas, the converse does not hold. That, for limit cardinals κ < κ′ , κ′ is κ-filtered, but κ′ -filtered. Hence, if A is κ-small relative to I, then it is κ′ -small relative to I.

55

Remark 4.39. The above definition make sense for any ordinal, not necessary cardinals. However, the property of being small relative is caused by the cardinality of the ordinal, and when it holds for some limit ordinal it holds for all limit ordinal with the same cardinality. On the other hand, having the definition this way makes every set K an ∣K∣-small relative in Set, which follows from the motivating example 4.29. Let C be a complete category, I a class of morphisms of C, and κ a cardinal. If A is a κ-small relative to I, then retracts of A are κ-small relative to I. Proof. Let B be a retract of A, i.e. there exist morphisms i ∶ B → A and r ∶ A → B such that ri = idB . Let λ be a κ-filtered ordinal, Z a λ-sequence in Z0 ↓ C of morphisms of I. Then, we have the commutative diagram of sets idcolim C(B,Z− )

colim C(B, Z− )

colim r

ϑB

/ colim C(A, Z− )

∗

colim i

ϑA

r∗

 / C(A, colim Z)

RA FT

 C(B, colim Z)

∗

, / colim C(B, Z− ) ϑB

i∗

 / C(B, colim Z) 2

idC(B,colim Z)

Since horizontal composites are identities, then particularly colim r∗ is an injective and i∗ is a surjective. Since A is κ-small relative to I, then ϑA is an isomorphism. The injectivity of colim r∗ and the isomorphism of ϑA implies the injectivity of ϑB . Also, the surjectivity of i∗ and the isomorphism of ϑA implies the surjectivity of ϑB . Hence, ϑB is an isomorphism, and B is κ-small relative to I. 

D

Recall that in topological spaces, given a cardinal κ, a spaces X is called κ-compact if every open covering of X contains a sub-covering of cardinality κ. The notion of κ-smallness, presented above, provides to some extent an analogue of κ-compactness, when morphisms of I are though of as open covering. However, sub-complexes of presented relative I- cell complexes present a more appropriate generalisation of open subsets, hence it can be used to defined what is called κ-compactness. To the author’s knoledge, this notion is due to Hirschhorn [Hir03, Def.10.8.1], and it is particularly used to define cellular model categories, in which [ibid.] proves the existence of the left Bousfield localisation. Definition 4.41 (κ-Compactness). Let C be a cocomplete category, I a class of morphisms in C, κ a cardinal. Then, an object A is called κ-compact relative to I if for every presented relative I- cell complex (f ∶ X → Y, (Z, (Sξ , gξ , iξ )ξ<λ )) morphisms A → Y factorises through a sub-complex of (f ∶ X → Y, (Z, (Sξ , gξ , iξ )ξ<λ )) of size at most κ. Moreover, A is called compact relative to I if it is κ-compact relative to I, for some cardinal κ. Remark 4.42. Avoid the confusion between the above notion of compactness and categorical compact objects, the co-Yoneda of the later commutes with filtered colimits. Also, avoid the confusion that might arise when comparing compactness with smallness. For κ-small objects,for a given λ-sequence Z, morphisms factorise through at Zξ for some ξ < λ, whereas for κ-compact objects, morphisms factorise through at the transfinite composition of the sub-complex of the presented relative I- cell complexes. It is straight forward from the definition if A is κ-compact relative to I and κ < κ′ , then A is κ′ -compact relative to I. Example 4.43. Let I be the set of canonical inclusions of simplicial sets {∂ n ∶ ∂∆n ↪ ∆n ∣n ≥ 0}. Then, every finite simplicial set (with finite set non-degenerate simplices) is ℵ0 -compact. The notion of compactness is stronger than smallness, when I- cell are monomorphisms, as seen in the below lemma. Lemma 4.44. Let C be a cocomplete category, I a set of morphisms in C such that relative I- cell complexes are monomorphisms, κ a cardinal. Then, an object A that is κ-compact relative to I is a κ-small relative to I- cell. Proof. Let λ be a κ-filtered ordinal, Z a λ-sequence in Z0 ↓ C of I- cell. Since I- cell are closed under transfinite composition, then zξλ is a relative I- cell complex for every ξ < λ, hence a monomorphisms by the

56

assumption. To show that A is a κ-small relative to I- cell, we need to show that the canonical morphism ϑ ∶ colim C(A, Z− ) → C(A, colim Z) is an isomorphism of sets. To show injectivity, let [f ], [g] ∈ colim C(A, Z− ) = ⊔ C(A, Z− ) Ò∼ such that ξ<λ

ϑ([f ]) = ϑ([g]). Then, there exists υ ≤ ξ < λ (without loose of generality) such that f ∶ A → Zυ and g ∶ A → Zξ such that zυλ f = zξλ g. Since zξλ is a monomorphism, then zυξ f = g, hence [f ] = [g], and ϑ is an injection. To show surjectivity, let f ∈ C(A, colim Z). Since A is κ-compact relative to I, then there exists a presentation for Z0 → colim Z such that f factorises at a sub-complex (Z0 → colim Z ′ , (Z ′ , (Sξ′ , gξ′ , i′ξ )ξ<κ )) of the presented (Z0 → colim Z, (Z, (Sξ , gξ , iξ )ξ<κ )) of size at most κ, as f = colim ϕf ′ , where ϕ′ → Z is the morphism of λ-sequences defining the sub-complex. Each cell in the this sub-complex corresponds to ordinal ξ such that Sξ′ ≠ ∅. Since the sub-complex if of size at most κ, then ∣{ξ < λ∣Sξ′ ≠ ∅}∣ ≤ κ. λ is a κ-filtered ordinal, hence ξ0 ∶= sup{ξ < λ∣Sξ′ ≠ ∅} < λ, hence, z ′ ξ0 ∶ Zξ′ 0 → colim Z ′ is an isomorphism. Therefore, f factors at Zξ0 as λ

−1

f = zξλ0 ○ (ϕξ0 (z ′ ξ0 ) f ′ ). λ

Z0′

z0′

ϕ0

 Z0 −1

/ Z′ 1 ϕ0

z0

 / Z1

RA FT

A

/

z1′

/ Z′ r ξ0

(z ′λ ξ )

/

zξ′

/ colim Z ′

0

/

 / Zξ o 0 zξ

t

colim ϕ

ϕξ0

z1

f

−1

0

′

f

 z / colim Z

/

0

−1

Hence, ∃[ϕξ0 (z ′ ξ0 ) f ′ ] ∈ colim C(A, Z− ) such that ϑ([ϕξ0 (z ′ ξ0 ) f ′ ]) = f . Therefore, A is κ-small relative to I- cell.  λ

λ

D

Lemma 4.45. Let C be a cocomplete category, I a class of morphisms in C, κ cardinals, A ∈ C a κ-compact relative to I. Then, retracts of A are κ-compact relative to I. Proof. Let B be a retract of A, i.e. there exists morphisms i ∶ B → A, r ∶ A → B in C such that ri = idB . Let (f ∶ X → Y, (Z, (Sξ , gξ , iξ )ξ<λ )) be presented relative I- cell complex, and let h ∶ B → Y be a morphism in C. Since A is a κ-compact relative to I, then hr ∶ A → Y factorise through a sub-complex of (f ∶ X → Y, (Z, (Sξ , gξ , iξ )ξ<λ )) of size at most κ, hence h = hri factorise through that sub-complex of (f ∶ X → Y, (Z, (Sξ , gξ , iξ )ξ<λ )) of size at most κ.  4.2.3. Understanding of the Small Object Argument. Motivation for the Small Object Argument. The main point of theorem 4.48, below, is to obtain a functorial factorisation (i.e. verify axiom CM5 of model categorises). This argument of the theorem was first used by Quillen [Qui67, §2.3,Lem.3] to show that topological spaces admit a the desired factorisation of fibrations and cofibrations, though at that point factorisation was not required to be functorial. For the first glance on theorem 4.48, its assumption seems rather artificial. Therefore, we start with what the author think it should be the motivation of the statement. Assume that the attempt to show that a given category C with a structure (C, F, W, (α, β), (γ, δ)) is indeed a model category, then every morphisms f ∶ X → Y factorises as X

f

α(f )

/Y X ? β(f )

Z

f

γ(f )

/Y > δ(f )

W

with β(f ) ∈ W F, α(f ) ∈ C and δ(f ) ∈ F, γ(f ) ∈ W C. Also, recall that lemma 1.27 implies that fibrations, weak fibrations, cofibrations, and weak cofibrations are precisely W C- inj, C- inj,W F - proj, F -Proj. Hence, having the above model structure on C implies that α(f ) ∈ C, β(f ) ∈ C-inj, and γ(f ) ∈ W C, δ(f ) ∈ W C- inj .

57

In either case there is a class of morphisms I (can be C for the first factorisation, and W C for the second one), for which f factorises as f = pi with i ∈ I and p ∈ I- inj. Of course, one on the other hand can consider to dual construction by fixing classes F and W F and taking W C and C to be their proj, respectively. However, since the categories we are interested in are cofibrantly generated we prefer the first approach. In both cases above, I contains all isomorphisms of C. Now, motivated by the above fact, let C be a category (not necessary a model category), I a class of morphisms of C, we will try to see what natural conditions6 we need to impose on C and I to be able to factorise morphisms f ∶ X → Y in C as f = pi with i ∈ I, and p ∈ I- inj. If f is in I- inj we get the factorisation f = f idX with idX ∈ I, f ∈ I- inj (It does not have to coincide with any such functorial factorisation, see example ?? below). Otherwise, there exist a commutative square E /X

e0

K g

E

 L

 /Y

e1

f

RA FT

with g ∈ I that does not admit a lift (i.e. an unsolved I-lifting problem for f ). We keep always in mind that the aim is to factorise f = pi, such that i ∈ I and p ∈ I- inj, in particular, all I-lifting problems for p should be solved, that each lifting problem of f induces evidently a lifting problem for p, should a factorisation f = pi exists. K

e0

/ X0

i?

 =Z?

g

h?

 L

e1

f

p?


/ Y.

D

Hence, if the desired factorisation exists, each such unsolved I-lifting problem E admits a ’partial lifting’ at p, for having p ∈ I- inj. For the sake of having coherent notation throughout the argument, put f0 ∶ X0 → Y for f ∶ X → Y , then we start with looking for a factorisation f0 = f1 x0 for some7 x0 ∶ X0 → X1 and f1 ∶ X1 → Y for which all I-lifting problems for f0 are solve by its very construction, i.e. instead of looking for a lift of E that does not have to exist, we look for a ’partial lifting’ of E that has to exist if we are to obtain the desired factorisation. In other words, we look for the factorisation f0 = f1 x0 such that there is a lift hE ∶ L → X1 for all I-lifting problems of f0 (Not to be confused with I-lifting problems of f1 ). (23)

K

e0



g hE ?

 L

/ X0

e1

x0 ?

X = 1?

f0

f1 ?


/ Y.

This condition imposed on f1 and x0 is natural, that if f1 is to be the desired p, it is necessary that they satisfy these condition. For C to be a model category, in particular it is necessary that it is cocomplete. Also, if C is locally small8 then the class of I-lifting problems for f0 is a ’small’ set9 if the class I is also a ’small’ set. Then, it is natural to require C to be cocomplete, and I to be a set. Under these assumptions, let I0 be the class of I-lifting problems

6Natural conditions in the sense they are supposed to hold for model categories. 7Ideally we would have asked for x ∈ I, however, we will see later that does not have to hold in general. 0 8Some authors require categories by definition to be locally small. 9’Small’ set is meant to be a set, not a proper class. Not to be confused with the smallness introduced above.

58

of f0 , then I0 is a set. Also, under the same assumptions, the below solid commutative diagram exists in C ∐ e0,E (24)

/ ∐ KE

iK

K

E∈I0

/ X0

E∈I0 x0 ?



∐ gE

g

7 X1 ?

E∈I0 h0 ?

 L

f1 ?



/ ∐ LE

iL

E∈I0

f0

  / Y. ∐ e1,E E∈I0

Notice that if the dotted morphisms in 24 exist, then all ’partial lifting’ problems 23 are solved. Then, using the commutativity of the solid diagram in 24, it would be natural to define X1 , x0 , h0 using push-forward of the diagram ∐ e0,E / X0

RA FT

∐ KE

E∈I0

E∈I0

∐ gE

E∈I0

 ∐ LE .

E∈I0

and f1 to be the unique morphism induced by the universal property of push-outs. Hence, we obtain the factorisation f0 = f1 x0 , such that all I-lifting problems of f0 are solved for f1 (admits ’partial lifting’ at f1 ). Also, notice that example 4.16 shows that ∐ gE ∈ I- cell. Also, since I- cell is closed under push-outs , by E∈I0

D

lemma 4.27(a), then x0 ∈ I- cell. Ideally, we would have wanted x0 ∈ I. However, since .... We iterate the above to obtain fk+1 and xk from fk in the same manner, obtaining a sequence, visualised in the below commutative diagram X0

f0

 } Y qo

x0

f1

/ X1

x1

/⋯

/ Xk

xk

/ Xk+1

xk+1

/⋯

fk fk+1

where for every integer k ≥ 0, xk ∈ I- cell, all I-lifting problems of fk are solved for fk+1 (admits ’partial lifting’ at fk+1 ), hence all I-lifting problems of fk are solved for fk′ , for k ′ > k. Taking colimit for the evidently chosen functor10, we obtain fω , and xω k for every integer k ≥ 0, such that all I-lifting problems of fk are solved for fω and xω ∈ Icell for every integer k ≥ 0 because I- cell is closed under transfinite composition, by lemma k 4.27(b). Having established the initial, successor11, and limit case, we can iterate the above construction for arbitrary ordinal λ to obtain fλ , xλξ from preceding fξ ’s and xξυ ’s for all υ < ξ < λ, such that xλξ ∈ I- cell and all I-lifting problems of fξ are solved for fλ for all ξ < λ. In order for the above procedure to be effective, regardless of being successful to give the desired factorisation, It should halt. Therefore, a natural question arise on at which ordinal should the above described procedure halt. One might be tempted to halt at ω 12, and that actually works and gives the desired factorisation in some occasions13, as in Quellin’s proof [Qui67, §2.3,Lem.3] of the factorisation in topological spaces, and simplicial sets as well. However, that does not give the required factorisation, even for topological spaces or simplicial, 10Also expressed in the proof of theorem 4.48. 11The same argument used for integers greater than 0 applies for any successor ordinal. 12The first infinite ordinal. 13Under extra assumption, namely finite smallness of domains of I relative to I cell.

59

unless I is ’chosen carefully’13. In order to realise if there is in general an ordinal λ for which fλ , xλ0 give the desired factorisation, assume that such limit ordinal λ exists, and see what natural conditions that imposes on C and I. In the argument here we might be a unsystematic, and highlight the motivation. However, a systematic proof is provided in 4.48. Consider an I-lifting problem for fλ e0 / Xλ K g

14

fλ

E

 L

 / Y.

e1

fλ , xλ0

It has to admit a lift in order for to be the desired factorisation. Since λ is a limit ordinal, then Xλ is given as a colimit for the evident functor. Most, if not all, categories of interest here are concrete, hence elements in the object of the colimits Xλ are ’coming’ from elements of objects the colimit cocone. In Set, we saw that each map to a ’long enough’ transfinite composition factorise at a ’preceding’ transfinite composition. Assume that e0 in E factorise at Xξ with ξ < λ, and see if that implies the existence of a lift of E. This assumption induces the below solid I-lifting problem for fξ , then by the very construction of fξ+1 , xξ , it admits the dotted ’partial lifting’ e′0

/ Xξ

RA FT

K

xξ



Xξ+1 F

g

fξ xλ ξ+1

hξ

 L

xλξ e0,ξ .

 Xλ

fλ

e1

  /Y

xλξ+1 hξ

D

for e0 = Then, we see that h = is a lift for E. Recall, X = X0 → Xξ is transfinite compositions of I- cell, for ξ ≤ λ. Hence, a natural choice for λ at which the above procedure halts would be to be ’big enough’ such that all morphism from domains of I factorise at Xξ , ξ < λ. Then, if domains of I are κ-small relative to I- cell, then it is sufficient to choose λ to be κ-filtered ordinal so that fλ ∈ I- inj. On the other hand, for any choice of λ, xλ0 ∈ I- cell. We see below that its sufficient to have domains of I to be small relative to I- cell. To summarise, having a category C and I a class of morphisms of C. Suppose, that C is cocomplete, I is a sets, domains of I are small relative to I- cell. Then, there exists a ’big enough’ limit ordinal λ such that every morphism f ∶ X → Y can be factorise as f = fλ xλ0 with xλ0 ∈ I- cell and fλ ∈ I- inj. This, is being proved systematically in 4.48. Definition 4.46. Let C be a cocomplete category, I a set of morphisms of C. Then, I is said to admit the small object argument, if the domains of elements of I are small relative to I- cell. Moreover, let κU be the smallest cardinal for which U is κU -small relative to I- cell for every U ∈ dom(I). Then, λI is defined to be the smallest ordinal that is κU -filtered for every U ∈ dom(I), where that κU is the smallest cardinal for which U is κU -small relative to I- cell. Proposition 4.47 (The Weak Transfinite Small Object Argument). Let C be a cocomplete category, I a set of morphisms of C that admits the small object argument. Then, there is a factorisation system (α, β) on C such that ∀f ∈ C ∶ f = β(f ) ○C α(f ) where α(f ) ∈ I- cell, and β(f ) ∈ I- inj. Theorem 4.48 (The Transfinite Small Object Argument). Let C be a cocomplete category, I a set of morphisms of C that admits the small object argument. Then, there is a functorial factorisation (α, β) on C such that ∀f ∈ C ∶ f = β(f ) ○C α(f ) where α(f ) ∈ I- cell, and β(f ) ∈ I- inj. 14It is a necessary, but not sufficient condition.

60

This theorem can be proven by transfinite induction. One can prove it directly hereby, however, in order to avoid a misconception the author sees that it is better to split the proof and introduce an auxiliary lemma first, then provide the proof of theorem. Remark 4.50 explains the advantage of this approach. One can thing of any factorisation f = pi in C as a morphism i ∶ f → p in C or simply as an object in f ↓ (C ↓ Y ). The later consideration simplify the argument, thus it would be used in the proof.

D

Then, P holds for all ordinals.

RA FT

Lemma 4.49. Let C be a cocomplete locally small category, I a set of morphisms of C, f ∶ X → Y be a morphisms in C, P be a property defined on all ordinals by ⎧ There exists a (λ + 1)sequence Z(λ) = ((X(λ), ρ(λ)), x(λ)) ∶ λ + 1 → f ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ↓ (C ↓ Y ) of morphisms of I- cell , every Ilifting problem E for ρ(λ)ξ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ e0 ⎪ / X(λ)ξ ⎪ K ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ g ⎪ ρ(λ)ξ E ⎪ ⎪ ⎪ ⎪   ⎪ ⎪ ⎪ /Y ⎪ L ⎪ e1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ with ξ < λ and g ∈ I, admits a ’partial lifting’ at ρ(λ)ξ+1 ⎪ ⎪ ⎪ ⎪ ⎪ e0 ⎪ / X(λ)ξ ⎪ K ⎨ ⎪ ⎪ P (λ) ⇔ ⎪ ⎪ x(λ)ξ ⎪ ⎪ ⎪ ⎪  ⎪ ⎪ ⎪ ⎪ X(λ)ξ+1 ⎪ ⎪ ⎪ C ⎪ ⎪ g ⎪ ρ(λ)ξ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ h ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ρ(λ)ξ+1 ⎪ ⎪ ⎪ ⎪ ⎪    ⎪ ⎪ ⎪ / Y. ⎪ L ⎪ e ⎪ 1 ⎪ ⎪ ⎪ λ+1 ⎪ ⎩ and ∀ξ < λ, P (ξ) holds, and Z(ξ) = Z(λ)iξ+1 .

Proof. We proof the lemma using transfinite induction, lemma 4.7 on the property P . (1) For λ = 0, the category 1 = ∗ consist of one object and its isomorphism, hence functors 1 → f ↓ (C ↓ Y ) correspond to objects of f ↓ (C ↓ Y ). Since we are seeking a 1-sequence Z(0), then Z(0) should commute with colimits, i.e. it must satisfy Z(0)0 ≅ colim Z(0)i10 . However, regardless of the choice of the functor Z(0), Z(0)i10 is just the empty functor to f ↓ (C ↓ Y ), hence colim Z(0)i10 is the initial of f ↓ (C ↓ Y ), namely ((X, f ), idX ). Hence, we take Z(0) to be the constant functor sending 0 to ((X, f ), idX ). It is colimit-preserving by its very definition, hence it is a 1-sequence in f ↓ (C ↓ Y ). ∄ξ ∈ Ord with ξ < 0, hence Z(0) is automatically a 1-sequence of morphism of of I- cell (or any other desired set of morphisms!). Also, it implies that the reset of the conditions are satisfied automatically, hence P (0) holds. (2) For a successor ordinal λ = ξ + 1, suppose that P (ξ) holds, then we need to show that P (λ) holds. Having P (ξ) holds, then there exist Z(ξ) a (ξ + 1)-sequence of morphisms of I- cell for which P (ξ) holds. Let Iξ be the class of I-lifting problems for ρ(ξ)ξ K g

 L

e0

E e1

/ X(ξ)ξ  /Y

ρ(ξ)ξ

with g ∈ I. Since C is locally small and I is a set, then coproducts indexed by Iξ exists in C. Let, the subscript Iξ denote evident coproducts indexed by Iξ , for instance LIξ ∶= ∐ LE . Then, we have the E∈Iξ

61

induced lifting problem e0,Iξ

KIξ gIξ

/ X(ξ)ξ

EI ξ

 LIξ

ρ(ξ)ξ

 / Y.

e1,Iξ

Consider the push-forward of the solid diagram below e0,Iξ

K Iξ

/ X(ξ)ξ

g Iξ

xξ

 LIξ

 / Xλ .

RA FT

hλ

Then, there exists a unique morphism ρλ ∶ Xλ → Y induced the universal property of push-outs that makes the below diagram commutative. KIξ

e0,Iξ

/ X(ξ)ξ xξ

 X C λ

g Iξ

ρ(ξ)ξ

D

hλ

 KIξ

e1,Iξ

ρλ

  / Y.

Hence, EIξ admits ’partial lifting’ at ρλ . Then, every E an I-lifting problem for ρ(ξ)ξ admits a partial lifting at ρλ , given by the composition hE = hλ iL . Put xλ = xξ x(ξ)ξ , and Zλ = ((Xλ , ρλ ), xλ ) then Zλ ∈ f ↓ (C ↓ Y ) and xξ ∶ Z(ξ)ξ → Zλ in f ↓ (C ↓ Y ). Moreover, ∀E ∈ Iξ , gE ∈ I, hence, by lemma 4.14, gIξ ∈ I- cell. Also, I- cell is closed under push-outs, by lemma 4.27(a), hence xξ ∈ I- cell. Define Z(λ) ∶ λ + 1

→ f ↓ (C ↓ Y ) Z(ξ)υ υ < λ υ ↦ { Zλ υ=λ υ x(ξ) υ<λ υ′ υ′ < υ ↦ { ξ xξ x(ξ)υ′ υ = λ Since Z(ξ) is a functor, and Zλ is given as push-forward, then Z(ξ + 1) is a functor. Let υ < λ + 1 be a limit ordinal, since λ = ξ + 1 is a successor ideal, then υ < ξ + 1. Then, Z(λ)υ = Z(ξ)υ ≅ colim(Z(ξ)iξυ ) = colim(Z(λ)iξυ ). Hence, Z(λ) is a (λ + 1)-sequence in f ↓ (C ↓ Y ). Moreover, ∀υ with υ + 1 < λ + 1, either υ + 1 < ξ + 1 or υ = ξ. If υ + 1 < ξ + 1, then x(λ)υ = x(ξ)υ ∈ I- cell that Z(ξ) is a (ξ + 1)-sequence of I- cell. Otherwise, when υ = ξ, x(λ)υ = x(λ)ξ = xξ ∈ I- cell, as seen above. Therefore, Z(λ) defined above is

62

a (λ + 1)-sequence of I- cell. Since P (ξ) holds, then P (υ) holds ∀υ < ξ, i.e. P (υ) holds ∀υ < ξ + 1 = λ. Also, based on the definition of Z(λ), it is straightforward to verify that Z(υ) = Z(λ)iλ+1 υ+1 , ∀υ < λ. Let E be an I-lifting problem for ρ(λ)υ K g

 L

e0

/ X(λ)υ

E

 /Y

e1

ρ(λ)υ

RA FT

with υ < λ. Then, if υ < ξ, then Z(λ)υ = Z(ξ)υ , hence E is an I-lifting problem for ρ(ξ)υ . Then, P (ξ) implies that E admits a ’partial lifting at ρ(ξ)υ+1 = ρ(λ)υ+1 . However, when υ = ξ, then the above construction of Z(λ) shows that E admits a ’partial lifting at ρ(λ)λ . Hence, P (λ) holds. (3) For limit ordinal λ, suppose that P (ξ) holds for all ξ < λ, then we need to show that P (λ) holds. Having P (ξ) holds ∀ξ < λ, let Z(ξ) be a (ξ + 1)-sequence of morphisms of I- cell for which P (ξ) holds, ∀ξ < λ. ∀υ < xi < λ, since P (ξ) holds, then Z(υ) = Z(ξ)iξ+1 υ+1 , hence then there is a nature monomorphism Z(ξ) of (λ + 1)-extended sequences Z(υ) ↪ Z(ξ), denoted by iZ(υ) . Define θ(λ) ∶ λ → (λ + 1)-Seq(f ↓ (C ↓ Y )) ξ ↦ Z(ξ) Z(ξ) υ < ξ ↦ iZ(υ) . where Z(ξ) here is the (λ + 1)-extended sequence. The existence of Z(ξ), above, for ξ < λ implies that θ(λ) is well-defined on objects and morphisms. Also, θ(λ) is a functor that for  < υ < ξ, on one hand Z(ξ) θ(λ)( < ξ) = iZ() , and On the other hand Z(ξ)

Z(υ)

Z(ξ)

D

θ(λ)(υ < ξ) ○ θ(λ)( < υ) = iZ(υ) ○ iZ() = iZ() .

Since (λ + 1)-Seq(f ↓ (C ↓ Y )) is cocomplete, by lemma ??, then colim θ(λ) exist. Put Z(λ) = colim θ(λ), then it is (λ + 1)-sequence in C. Since colimits in (λ + 1)-Seq(f ↓ (C ↓ Y )) are given object-wise, then Z(λ) is given by Z(λ) ∶ λ + 1 ξ

→ ↦

υ<ξ

↦

f ↓ (C ↓ Y ) G(λ)ξ ξ<λ { colim G(λ) ξ = λ G(λ)υ<ξ { G(λ)υ → colim G(λ)

. ξ<λ ξ=λ

for the evident functor G(λ) ∶ λ ξ υ<ξ

→ ↦ ↦

f ↓ (C ↓ Y ) Z(ξ)ξ x(ξ)ξυ .

Moreover, for every ordinal ξ with ξ + 1 < λ + 1, i.e. ξ < λ. Since, λ is a limit ordinal, then ξ + 1 < λ, hence x(λ)ξ = G(λ)ξ<ξ+1 = x(ξ + 1)ξ ∈ I- cell, because P (ξ + 1) holds for having ξ + 1 < λ. Hence, Z(λ) is a (λ + 1)-sequence of morphisms of I- cell. ∀ξ < λ, P (ξ) holds by the induction assumption. Also, ∀ξ < λ, and for υ < ξ + 1, we have ξ + 1 < λ, hence υ < λ Z(λ)υ = G(λ)υ = Z(υ)υ If υ = ξ, then Z(λ)υ = Z(ξ)υ . Also, when υ < ξ, we have Z(υ)υ = Z(ξ)υ because P (ξ) holds for ξ < λ. Therefore, we have Z(ξ) = Z(λ)iλ+1 ξ+1 , ∀ξ < λ.

63

Let E be an I-lifting problem for ρ(λ)υ K g

 L

e0

E e1

/ X(λ)υ  /Y

ρ(λ)υ

with υ < λ. Then, υ + 1 < λ, hence P (λ + 1) holds. υ < υ + 1, then Z(λ)υ = Z(υ + 1)λ , as seen above. Then, E is an I-lifting problem for ρ(υ +1)υ , hence E admits a ’partial lifting’ at ρ(υ +1)υ+1 . However, ρ(υ + 1)υ+1 = ρ(λ)υ+1 , hence E admits a ’partial lifting’ at ρ(λ)υ+1 . Hence, P (λ) holds. Therefore, using transfinite induction, lemma 4.7, the property P holds for all ordinals. 

D

RA FT

Remark 4.50. One might be tempted to state the property P at λ to be the existence of a (λ + 1)-sequence of morphisms of I- cell without specifying the core condition they should satisfy, then use the specificity of a particular inductive proof, when the result of the transfinite induction is to be used15. However, this is approach is not valid, the transfinite induction lemma implies that the property holds for all ordinals without specifying how it does that. It might be intuitive to think that if an inductive proof is constructed in some canonical way, then the result may assume that construction. However, that is not always correct. In fact, emphasising this point is the sole reason beyond spiting the proof of theorem 4.48 into two parts. The first proves that a property holds for all ordinals, whereas the other applies the fact that the property itself (not its proof) holds for a convenient choice of an ordinal, in order to obtain a factorisation system. To illustrate this point, say Q is property given for an ordinal λ by Q(λ) ⇔ There exists (λ + 1)sequence Z(λ) = ((X(λ), ρ(λ)), x(λ)) λ + 1 → f ↓ (C ↓ Y ) of morphisms of I cell, such that ∀ξ < λ, Q(ξ) holds and Z(ξ) = Z(λ)iλ+1 ξ+1 , ∀ξ < λ. without going into the trouble of specifying what conditions they satisfy. Then, the statement that Q holds for all ordinal can be proven ’from scratch’ using the different construction to these used in the proof of the previous lemma, which leads to differs factorisation that still suffiecient enough to proof the weak small object argument. However, the statement that Q holds for all ordinals (which the result transfinite induction gives) is not of much help by itself. That, not every sequences satisfying the above condition leads to the desired factorisation. In particular, one can take Z(λ) to be the constant (λ + 1)-sequence sending ξ < λ to ((X, f ), idX ), i.e. factorising f as f idX ! If the specificity of a particular construction is needed to be used when applying the results of the transfinite induction, then that should be captured by the statement of the property. Suppose some constructions is used to prove that a property holds by transfinite induction for all ordinals. If the property is ’extended’ to capture the feature of the inductive proof of the first, then priorly the validity of the first property does not imply the validity of the ’extended’ one. That depends on each particular situation. However, one good way to verify that the needed property is stated correctly, is to extract the part of the proof that uses transfinite induction into separate lemma. Then, if using the statement of the auxiliary lemma is sufficient to prove what is needed, then the needed property is stated correctly for what it matters then. Otherwise, if the statement of the auxiliary lemma is not sufficient and one needs also to use the way it has been proven, then the statement of the property is not correct and should be thought over! Same principle applies to verifying each step of the transfinite induction. For instance, for instance when showing that P (ξ) implies P (ξ + 1), one is only eligible to use the validity of the statement of P (ξ), not using any specificity of a particular proof of P (ξ). Remark 4.51. The essential part of statement defining the property P at λ is highlighted, with the core condition being highlighted with a darker colour. The non-highlighted part ’and ∀ξ < λ, P (ξ) holds, and Z(ξ) = Z(λ)iλ+1 ξ+1 ’ is technical, we only require it to ease the proof. However, it wont be of particular use in the proof of theorem 4.48. In contrast, the dark-highlighted parts capture the essence of the proof of the small object argument, that the factorisation will be obtained based on the smallness of objects of dom(I), which enables us to lift morphisms K → colim X(λ) with K ∈ dom(I) to morphisms K → colim X(ξ) = X(λ)ξ for ξ < λ, and then apply the ’partial lifting’ for the induced I-lifting problem for ρ(λ)ξ , hence deduce the lift of the original I-lifting problem for colim ρ(λ), as illustrated in the below proof. Although the actual 15Several references seem to follow this pattern.

64

construction of ’partial lifting’ is of interest to understand how we actually perform factorisation, i.e. important for computations, it is not curial for the below proof of the statement of the small object argument. However, it seems to be more difficult to prove the previous lemma directly without using these technical part. Therefore, the technical part of the property P was introduced only to ease the proof. Actually, the lemma could have been given such that, for its assumptions, there exists a fuction (not necessary given rise to a functor) Z(−) ∶ Ord → Ob(Ord-Seq(f ↓ (C ↓ Y ))) that satisfies the darkly highlighted condition. However, it seems hard to proof without including the above technical restrictions. Proof of proposition 4.47. Following the above explained motivation for the small object argument, we need to fix a ’big enough’ ordinal. We look for a limit ordinal λ for which morphisms from objects in dom(I) to λ-transfinite composition of I- cell factorises at some ordinal less than λ. λ can be choosing to be any ordinal greater than λI , that λI is κK -filtered for every K ∈ dom(I), such that κK is the smallest cardinal for which K is κK -small relative to I- cell. We choose λ = λI .

RA FT

Let f ∶ X → Y be a morphism in C, then there exists a (λ + 1)-sequence Z(λ) of morphisms of I- cell for which P (λ), in lemma 4.49, holds. Recall that objects of f ↓ (C ↓ Y ) are factorisation of f , and Z(λ)λ ∈ f ↓ (C ↓ Y ), then in particular f factorises as 7/ Y.

f

X

ρ(λ)λ

'

x(λ)λ 0

X(λ)λ

D

Since Z(λ) is a (λ + 1)-sequence of morphisms of I- cell, then x(λ)λ0 ∈ I- cell. In order to show that ρ(λ)λ ∈ I- inj, let E be an I-lifting problem for ρ(λ)λ e0

K g

/ X(λ)λ ρ(λ)λ

E

 L

 / Y.

e1

with g ∈ I, we need to show that E admits a lift. Notice that λ is chosen to be is κ-filtered, hence it is a limit ordinal, then Z(λ)λ is a transfinite composition of the λ-sequence Z(λ)iλ+1 of morphisms of I- cell. Also, K is κ-small relative to I- cell, then e0 factorises as λ

K

e0,ξ

/ Xξ

x(λ)λ ξ

/Z.

for ξ < λ. This induced the following I-lifting problem for ρ(λ)ξ K

e0,ξ

/ X(λ)ξ

x(λ)λ ξ

 X(λ)λ

g

 L

ρ(λ)λ e1

  / Y.

ρ(λ)ξ

65

Since ξ < λ, then lemma 4.49 implies that the above induced I-lifting problem admits a ’partial lifting’ at ρ(λ)ξ+1 e0,ξ

K

/ X(λ)ξ

x(λ)ξ

 X(λ)ξ+1 G

g

x(λ)λ ξ+1 h

ρ(λ)ξ

 X(λ)λ ρ(λ)λ

 L

e1


/ Y.

x(λ)λξ+1 h

RA FT

is a lift of E. Hence, ρ(λ)λ ∈ I- inj. Then, We should have used the index f to illustrate that the above sequence exists for each morphism f . TBC



Proof of theorem 4.48. It is similar to the proof of proposition 4.2.3, for a property P similar to the property P in lemma 4.49, but given functorially!  Throughout these notes, for a category C and a set of morphisms I of C taht admits the small object argument, (αI , βI ) will denote functorial factorisation obtain through the small object argument for I and the ordinal λI . If no confusion may arise, then for every morphism f ∶ X → Y in C, codom αI = dom βI will be denoted by YI .

D

Definition 4.52. Let C be a cocomplete category, I a set of morphisms in C that admits the small object argument. Then, for every f ∶ X → Y in C, the transfinite small object argument, theorem 4.48, for I and the cardinal gives the factorisation f

X αI (f )

!

/Y >

βI (f )

YλI

for the ordinal λI , Def 4.46, such that αI (f ) is an I- cell, and βI (f ) is an I- inj. If C is also a model category, with I cofibrations, avoid confusing (αI , βI ) with the functorial factorisation of C, as cofibrations and weak fibrations. Priorly, functorial factorisations of a given model category does not have to coincide with any of those obtained by the transfinite small object argument, even when the category is cofibrantly generated by I, J, see 5. Moreover, the transfinite small object argument gives priorly different functorial factorisations for different ordinals λ > λI . 4.2.4. Implications of the Small Object Argument. Hereby, we review some implications of the small object argument on facts related to the λ-sequences, small object argument and smallness, this will be particularly useful for Bousfield localisation of cellular model categories. Definition 4.53. Let C be a cocomplete category, I a class of morphisms of C, λ an ordinal. Then, a ̃ in Z0 ↓ C if Z is a retract of Z ̃ as objects λ-sequence Z in Z0 ↓ C is said to be a retract of a λ-sequence Z ⋅ ⋅ λ ̃ ̃ in functors category (Z0 ↓ C) , i.e. if there exists two natural transformations i ∶ Z → Z, r ∶ Z → Z such that ri = idZ . Lemma 4.54. Let C be a cocomplete category, I a set of morphisms of C that admits the small object argument, λ is an ordinal. Then, any λ-sequence Z in Z0 ↓ C of I- cof is a retract of a λ-sequence in Z0 ↓ C of morphisms of I- cell. Proof. We use transfinite induction to show that the below statement holds for every ordinal ξ ≤ λ: there exists a (ξ + 1)-sequence Z(ξ) ∶ ξ + 1 → Z0 ↓ C of morphisms of I- cell, such that Z∣ξ+1 is a retract of Z(ξ), and that this statement holds for every ordinal υ < ξ with Z(υ) = Z(ξ)∣υ + 1.

66

(1) For ξ = 0 ≤ λ, the category 1 = ∗ consist of one object and its isomorphism, hence functors 1 → Z0 ↓ C correspond to objects of Z0 ↓ C. Since we are seeking a 1-sequence Z(0), then Z(0) should commute with colimits, i.e. it must satisfy Z(0)0 ≅ colim Z(0)i10 . However, regardless of the choice of the functor Z(0), Z(0)i10 is just the empty functor to Z0 ↓ C, hence colim Z(0)i10 is the initial of Z0 ↓ C, namely (Z0 , idZ0 ). Hence, we take Z(0) to be the constant functor sending 0 to (Z0 , idZ0 ). It is colimit-preserving by its very definition, hence it is a 1-sequence in Z0 ↓ C. ∄ξ ∈ Ord with ξ < 0, hence Z(0) is automatically a 1-sequence of morphism of of I- cell (or any other desired set of morphisms!). Put i0 = r0 = idZ0 . Then, we see that Z(0) is retract of Z∣1 . Also, the rest of the conditions are satisfied automatically. (2) For a successor ordinal ξ = υ + 1 < λ, suppose that there exists a (ξ + 1)-sequence Z(ξ) ∶ ξ + 1 → Z0 ↓ C of morphisms of I- cell, such that Z∣ξ+1 is a retract of Z(ξ), and that this statement holds for every ordinal υ < ξ with Z(υ) = Z(ξ)∣υ + 1, then we need to show that there exists a (ξ + 2)-sequence Z(ξ + 1) ∶ ξ + 2 → Z0 ↓ C of morphisms of I- cell, such that Z∣ξ+2 is a retract of Z(ξ + 1), and that this statement holds for every ordinal υ < ξ + 1 with Z(υ) = Z(ξ + 1)∣υ + 1. Since Z∣ξ+1 is a retract of ⋅ ⋅ Z(ξ), then there exists morphisms of (ξ + 1)-sequence i ∶ Z∣ξ+1 → Z(ξ) and r ∶ Z(ξ) → Z∣ξ+1 such that ri = idZ∣ξ+1 , as illustrated in the below commutative diagram z0

i0

 Z(ξ)0

z1

i1

z(ξ)0

r0

 Z0

/ Z1

z0

/ Z2

/

/ Zξ

zξ

/ Zξ+1

/

zξ

/ Zξ+1

/

RA FT

Z0

 / Z(ξ)1

z2

iξ

i2

z(ξ)1

 / Z(ξ)2z(ξ)2 /

rξ

r1

r2

 / Z1

 / Z2

z1

 / Z(ξ)ξ  / Zξ

/

z2

D

with vertical composite being identities. Since I admits the small object argument, then the morphism zξ rξ factorises as zξ rξ = pz ′ where z ′ = αI (zξ rξ ) ∈ I- cell and p = βI (zξ rξ ) ∈ I- inj. Also, notice that zξ = zξ rξ iξ = pz ′ iξ , which induce the below solid commutative square Zξ zξ



Zξ+1

z ′ iξ

h

/ Z′ < p



Zξ+1

where Z ′ ∶= dom p = codom z ′ . Since, zξ ∈ I- cof by assumption, then P ∈ I- inj has the RLP with respect to zξ , hence the above diagram admits the dotted lift h ∶ Zξ+1 → Z ′ . Particularly, h satisfies hzξ = z ′ iξ and ph = idZξ+1 . Therefore, Z(ξ) can be extended into an (ξ + 2)-sequence in Z0 ↓ C of I- cell Z(ξ + 1) putting Z(ξ + 1)∣ξ+1 = Z(ξ), Z(ξ + 1)ξ+1 = Z ′ , and z(ξ + 1)ξ+1 = z ′ . Also, i and r ⋅ ⋅ can be extended into morphisms of (ξ + 2)-sequence i′ ∶ Z∣ξ+2 → Z(ξ + 1) and r′ ∶ Z(ξ + 1) → Z∣ξ+2 ′ putting i′ ∣ξ+1 = i, i′ξ+1 = h, r′ ∣ξ+1 = r, and rξ+1 = p. Then, it is seen readily that Z∣ξ+2 is a retract of a (ξ + 2)-sequence in Z0 ↓ C of I- cell, namely Z(ξ + 1). The rest of the conditions are automatically satisfied based on the inductive assumption and the very definition of Z(ξ + 1). (3) For limit ordinal ξ ≤ λ, suppose that for every ordinal υ < ξ there exists a (υ+1)-sequence Z(υ) ∶ υ+1 → Z0 ↓ C of morphisms of I- cell, such that Z∣υ+1 is a retract of Z(υ), and that this statement holds for every ordinal  < υ with Z() = Z(υ)∣ + 1, then we need to show that there exists a (ξ + 1)-sequence Z(ξ) ∶ ξ + 1 → Z0 ↓ C of morphisms of I- cell, such that Z∣ξ+1 is a retract of Z(ξ), and that this statement holds for every ordinal  < ξ with Z() = Z(ξ)∣+1 . That can be done in the same manner as in the poof of small object argument, proof 4.2.3, as a colimit of the evident functor of (ξ+1)-sequences. Then, by the transfinite induction, the above statements holds particurally for the ordinal λ, which proves the lemma.  Corollary 4.55. Let C be a cocomplete category, I a set of morphisms of C that admits the small object argument. Then, I-cofibrations are retracts of relative I- cell complexes, with the same domain, and I-cofibrant objects are retracts of relative I- cell complexes.

67

Proof. Let i ∶ U → V be an I-cofibration, then f is the composition of 2-sequence Z in U ↓ C of I- cof, with Z0 = (U, idU ), Z1 = (V, i), and z0 = i. Hence, by the above lemma 4.54 Z is a retract of a 2-sequence Z ′ in U ↓ C of morphisms of I- cell, hence i is a retract of a relative I- cell complex, with the same domain. Moreover, for U = ∅, one sees that I-cofibrant objects are retracts of relative I- cell complexes.  Theorem 4.56. Let C be a cocomplete category, I a set of morphisms of C that admits the small object argument, κ is an ordinal. If A ∈ C is κ-small relative to I- cell, then A is κ-small relative to the class of I- cof. Proof. Let λ be a κ-filtered ordinal and Z ∶ λ → Z0 ↓ C a λ-sequence of I- cof. Since I admits the small object argument, then by the lemma 4.54, above, Z is a retract of Z ′ a λ-sequence of morphisms of I- cell. ⋅ ⋅ Hence, there exists morphisms of λ-sequences i ∶ Z → Z ′ , r ∶ Z ′ → Z such that ri = idZ . Hence, we have the below commutative diagram f

A

Z0

f′

z0

i0

r0

 Z0

z1

i1 z0′

/ Z2

/

/ Zξ

z2

iξ

i2

 / Z′ 1

z1′

 / Z′ 2

zξ

z2′

  / Z′ ξ

/

/ Zξ+1 iξ+1

zξ′

z0

r1

r2

rξ

 / Z1

 / Z2

 / Zξ

z1

z2

/

) / colim Z

zξ+1

/

colim i



/ Z′

RA FT

 Z0′

/ Z1

 / colim Z ′

′ zξ+1

/

ξ+1

rξ+1

zξ



/ Zξ+1

colim r zξ+1

/

 / colim Z

with vertical composite being identities. Since Z ′ → colim Z ′ is a transfinite compositions of I- cell indexed by the the κ-filtered ordinal λ, then the canonical morphism colim C(A, Z−′ ) → C(A, colim Z ′ )

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is an isomorphism of sets. Also, the commutativity of the above diagram with vertical composite being identities implies that the canonical morphism is colim C(A, Z− ) → C(A, colim Z) is a retract of colim C(A, Z−′ ) → C(A, colim Z ′ ), with the below canonical retraction diagram. colim C(A, Z− ) colim i∗

 colim C(A, Z−′ ) colim r∗

 colim C(A, Z− )

/ C(A, colim Z) (colim i)∗

 / C(A, colim Z ′ )

(colim r)∗

 / C(A, colim Z)

Since colim i∗ and is an injection, then colim C(A, Z− ) → C(A, colim Z) is so, and since (colim r)∗ is a surjection, then colim C(A, Z− ) → C(A, colim Z) is so, i.e. an isomorphism of sets. Therefore, A is κ-small relative to I- cof.  Lemma 4.57. Let C be a cocomplete category, I a set of morphisms of C that admits the small object argument such that I- cof consists of monomorphisms, J a set of I-cofibrations whose domains are κ-compact relative to I, for some infinite cardinal κ. Then, every cell of relative J- cell complex f is contained in a sub-complex of f of size at maximum κ. Proof. Let f ∶ A → B be a relative J- cell complex. Then, there exists an ordinal λ and presentation (Z, (Sξ , gξ , iξ )ξ<λ ) for f of transfinite composition of push-outs of morphisms of J, i.e. with #Sξ = 1 and gξ ∶ Sξ → J for ξ < λ. To avoide redundant notation, we write ξ, gξ , and iξ for (ξ, sξ ), gξ (sξ ) and iξ (sξ ) for the unique sξ ∈ Sξ for ξ < λ. Let ξ be a cell of the presented relative J- cell complex (f ∶ A → B, (Z, (Sυ , gυ , iυ )υ<λ )) for some ξ < λ. Since κ is infinite then the existence of a sub-complex (f ′ ∶ A → B ′ , (Z ′ , (Sυ′ , gυ′ , i′υ )υ<λ )) of f of size at most κ that contains the cell ξ, i.e. sξ ∈ Sξ′ is equivalent to the existence of a sub-complex of f of size κ on which ξ can be glued, i.e. for which iξ is factorises through the ξ th -component of the choice of morphism of that sub-complex. We will use the λ-transfinite induction on ξ to show the existence of a sub-complex of f of size at most κ that contains ξ.

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Recall that since J ⊂ I- cof and I- cof is closed under transfinite composition and push-outs, the J- cell ⊂ I- cof. Hence, I- cell consists of monomorphisms. Then, by lemma 4.23 sub-complexes of f are determined up-to isomorphism by their set of cells. (1) For ξ = 0 ≤ λ, by lemma 4.24, f has the empty sub-complex with ϕ the evident choice of morphism of sub-complex. ϕ0 = idA , hence i0 factorises through ϕ0 and ξ can be glued to the empty sub-complex creating a sub-complex of f of size 1 < κ. (2) For a successor ordinal ξ = υ + 1 ≤ λ, suppose that the cell υ is contained in a sub-complex of f of size at most κ. Recall that Zξ is constructed as a push-out / Yυ

gυ

Xυ iυ

 Zυ

zυ

⌟  / Zξ .

Since J- cell ⊂ I- cof and I admit the small object argument, then lemma 4.54 implies that Z∣ξ is a retract of ξ-sequence W ′ in Z0 ↓ C of morphisms of I- cell, i.e. there exists natural transformations j ∶ Z∣ξ → W ′ and r ∶ W ′ → Z∣ξ such that rj = idZ∣ξ . / Yυ

gυ

RA FT

Xυ iυ

A = Z0 j0



A = W0′

/ Z2

z1

j1

w0′

r0

 A = Z0

/ Z1

z0

 / W′ 1

/ Zυ

j2

 / W′ 2

w1′

r1

z0

/

z2

 / Z1

z1

zυ

⌟  / Zξ /

jυ

w2′

 / W′ υ

/

r2

 / Z2

}

rυ

z2

 / Zυ

/

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Since domains of morphisms of J are κ-compact relative to I, then the morphism jυ iυ factorises through an I-sub-complex of A = W0′ → colim W ′ of size at most κ. Let such sub-complex of A = W0′ → colim W ′ be represented by (f ′′ , (W ′′ , (Sυ′′ , gυ′′ , i′′υ )υ<λ )), with ϕ′ ∶ f ′′ → f ′ a choice of relative I- cell sub-complex morphism. Then, we have the commutative diagram. A = W0′′ ϕ′0



A = W0′

w0′′

w1′′

ϕ′1 w0′

r0

 A = Z0

/ W ′′ 1  / W′ 1

z0

w2′′

/ W ′′ υ /

ϕ′2 w1′

r1

 / Z1

/ W ′′ 2  / W′ 2

ϕ′υ w2′

r2

/

z1

 / Z2

 / W′ υ

rυ z2

/

 / Zυ

We look for a factorisation of rυ ϕ′υ through an inclusion of relative J- cell sub-complex colim Z ′ → colim Z∣ξ . Non-empty cells (, t′ )) of W ′ is obtained for  < υ by construction...TBC . (3) TBC.  5. Cofibranty Generated Model Categories The classes of weak equivalences, cofibrations, and fibrations of model category are usually proper, i.e. not small sets, and the nature of their elements are not always easily understood. For any model category, if I is the class of cofibration, then I- inj is the class of weak fibrations, and if J is the class of weak cofibrations then J- inj is the class of fibrations. However, in some model categories these classes are encoded and can be recovered through left and right lifting properties from easier to understand small sets. When it is the case, that makes the model structure more understood and simplify the verification of the model category axioms. In addition, such model categories possess desired properties, to be recalled below. A very occurring examples of such model categories are cofibranty generated model categories. Therefore, we recall them below and recall

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some of their related results, especially the criterion theorem due to Kan, theorem 5.6, as well as introducing the notion of cofibrantly generated model categories. As the name suggests, the model structure is generated from classes of what will be cofibration in the resulting model structure. Also, in cellular model categories, particular cofibrantly generated model categories, inclusions of sub-complexes becomes a natural analogy of injective maps of sets. That provides an insight and provide an insight about an important class of morphism of C, hence it allows a careful generalisation of set-theoretical argument to categories of abstract nature. Definition 5.1 (Cofibranty Generated Model Categories). A cofibrantly generated model category is a triple (C, I, J), where C is a model category, I and J are sets of morphisms of C that both admit the small object argument such that (1) The class of fibrations is J- inj, or equivalently the class of weak cofibrations is J- inj. (2) The class of weak fibrations is I- inj, or equivalently the class of cofibrations is I- inj. If satisfied, then I is called generating cofibrations and J is called generating trivial cofibrations. In cofibrantly generated model categories, morphisms of the class I are cofibrations for having the left lifting property with respect to all weal fibrations and morphisms of the class J are weak cofibrations for having the left lifting property with respect to all fibrations. That, justifies the terminology used in the definition.

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Remark 5.2. One can consider the dual of the above notion and define what is called fibranty generated model categories, where cofibration will be K- and weak cofibrations L- for sets of generating weak fibrations K and generating fibrations L. However, since most model categories of our interest in motivic homotopy theory are cofibrantly generated, then we restrict the study to them. In particular the standard model structure on simplicial sets, the local injective model structure on simplicial pre-sheaves on a site, left Bousfield localisation of a cofibrantly generated model category are important examples of cofibrantly generated model categories. Notice that in the above example CW-complexes are precisely I- cell. This notation is curried out to other cofibrantly generated model categories as follows.

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Definition 5.3. Let (C, I, J) be a cofibrantly generated model category. Then, ● Relative I- cell complexes are called relative cell complexes, and I- cell complexes are called cell complexes. ● For a cell U , a relative cell complex i ∶ U → V is called an inclusion of sub-complex. ● A cell complex that is given as a finite composition of push-forward of morphisms of I is called finite cell complex. Definition 5.4. Let (C, I, J) be a cofibrantly generated model category, κ a cardinal. Then, κ-compact objects relative to I are called κ-compact objects. The definition of cofibrantly generated model categories expresses explicitly the classes of fibrations and weak fibrations in terms of the sets of generating cofibrations I and generating weak cofibrations J. However, in such categories, the classes of cofibrations and weak cofibrations have the nice property of being understood in terms of the sets I and J as seen in the below lemma. Lemma 5.5. Let (C, I, J) be a cofibrantly generated model category. Then, (1) The class of cofibrations of C is I- cof, and every cofibration is a retract of relative I- cell complexes. (2) The class of weak cofibrations of C is J- cof, and every weak cofibration is a retract of relative J- cell complexes. (3) Domains of I are small relative to all cofibrations. (4) Domains of J are small relative to all weak cofibrations. Proof. (1) The class of cofibrations in any model category consist of precisely morphisms that have the LLP with respect to weak fibration. Hence, in a cofibrantly generated model category, the class of cofibrations coincides with the class I- cof = (I- inj)- proj. Moreover, since I admits the small object argument, there is a functorial factorisation (αI , βI ) on C such that αI (f ) ∈ I- cell and βI (f ) ∈ I- inj for every morphism f in C. Then, every cofibration i ∶ U → V can be factorise as U

i

αI (i)

/V > βI (i)

VI

70

Since βI (i) ∈ I- inj, i.e. a weak fibration and i is a fibration, then βI (i) has the LLP with respect to i. Then, by the retract argument, i is a retract to αI (i) ∈ I- cell. (2) Similar to 1. (3) Since domains of I as small relative to I- cell, then by theorem 4.56, domains of I are small relative to I- cof, which was seen in 1 to be the class of all cofibrations. (4) Similar to 3.  Recall that the aim of model category is to study homotopy categories of category with weak equivalences. Therefore, it is useful if starting from category with weak equivalences and two sets of morphisms one can determine if that gives rise to a model category with the same weak equivalences. Kan provided the below tool that verifies that such data, if satisfied certain conditions, defines a cofibrantly generated model category.

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Theorem 5.6 (Kan Criteria or the ’Recognition Theorem’). Let C be a complete and cocomplete category, W a class of morphisms of C, and I and J are sets of morphisms of C. Then, (C, I, J) is a cofibrantly generated model category with generating cofibrations I, generating weak cofirations J, and weak equivalences W if and only if the following conditions hold: (1) The class W satisfies the two-out-of-three property, and is closed under retract. (2) I and J admit the small object argument. (3) J- cell ⊆ W ⋂ I- cof. (4) I- inj ⊆ W ⋂ J- inj. (5) Either W ⋂ I- cof ⊆ J- cof or W ⋂ J- inj ⊆ I- inj. Proof. Suppose that (C, I, J) is a cofibrantly generated model category with generating cofibrations I, generating weak cofirations J, and weak equivalences W . Then, both sets I and J admit the small object argument, W satisfies the two-out-of-three property, and is closed under retract, by the definition of cofibrantly generated model structure. Since C is cocomplete, then lemma 4.27.(c) shows that J- cell ⊆ J- cof. Lemma 5.5 shows that I- cof and J- cof are the classes of cofibrations and weak cofibrations, respectively, in C. Hence, J- cell ⊆ J- cof = W ⋂ I- cof. Also, since I- inj and J- inj are the classes of weak fibrations and cofibrations, respectively, in C. Then, I- inj ⊆ W ⋂ J- inj. In fact, both W ⋂ I- cof ⊆ J- cof, or W ⋂ J- inj ⊆ I- inj hold.

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On the other hand, assuming that the above conditions are satisfied. We will show that the classes of W as weak equivalences, I- cof as cofibrations and J- inj as fibrations, with functorial factorisation induced by the transfinite small object argument on the sets I and J define a model structure on C and that the resulting model category is a cofibrantly generated model category with generating cofibrations I, generating weak cofirations J. Notice that CM1 holds by the theorem hypothesis. Also, since W satisfy the two-out-of-three property, then CM2 holds CM3 W is closed under retract, by the assumption. Also, I- cof and J- inj are closed under retract, by lemma 1.18, for being defined the left and right lifting property. CM5 Applying the small object argument, theorem 4.48, on classes I and J yield functorial factorisation (α, β) and (γ, δ) such that ∀f ∈ MorC β(f ) ∈ I-inj, α(f ) ∈ I-cell, δ(f ) ∈ J-inj, and γ(f ) ∈ J-cell. Since I- cell ⊆ I- cof by lemma 4.27.(c) and I- inj ⊆ W ⋂ J- inj and J- cell ⊆ W ⋂ I- cof by the assumption, then β(f ) ∈ W ⋂ J-inj, α(f ) ∈ I-cof, δ(f ) ∈ J-inj, and γ(f ) ∈ W ⋂ I-cof. as required. CM4 Let E be a commutative square of morphisms of C A i

 B

e0

E e1

/X  /Y

p

where i ∈ I- cof, p ∈ I- inj, and either of i or p belongs to W . Then, we need to show that E admits a lift h ∶ B → X. Distinguish two cases either W ⋂ I- cof ⊆ J- cof or W ⋂ J- inj ⊆ I- inj. Suppose that W ⋂ I- cof ⊆ J- cof.

71

(a) Let i ∈ W , hence i ∈ W ⋂ I- cof. Then the lift h ∶ B → X exists because J- cof = (J- inj)- proj. (b) Let p ∈ W , hence p ∈ W ⋂ J- inj. Then, as seen in the proof of CM5 p factorises as p = β(p) ○ α(p) with β(p) ∈ I- inj ⊆ W ⋂ J- inj, α(p) ∈ I- cell ⊆ I- cof. Since W satisfy the two-out-of-three property then α(p) ∈ W ⋂ I- cof ⊆ J- cof. Then, by CM4a it has the LLP with respect to p. Hence, by the retract argument theorem 1.17 p is a retract of β(p) ∈ I- inj. Since I- inj is stable under retract, be lemma 1.18, then p ∈ I- inj. Therefore, the lift h ∶ B → X exists because I- cof = (I- inj)- proj. Otherwise, W ⋂ J- inj ⊆ I- inj. (a) Let p ∈ W , hence p ∈ W ⋂ J- inj. Then the lift h ∶ B → X exists because I- cof = (I- inj)- proj. (b) Let i ∈ W , hence i ∈ W ⋂ I- cof. Then, as seen in the proof of CM5 i factorises as i = δ(i) ○ γ(i)

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with δ(i) ∈ J- inj , γ(I) ∈ J- cell ⊆ W ⋂ I- cof. Since W satisfy the two-out-of-three property then δ(i) ∈ W ⋂ J- inj ⊆ I- inj. Then, by CM4a it has the RLP with respect to i. Hence, by the retract argument theorem 1.17 i is a retract of γ(i) ∈ J- cof. Since J- cof is stable under retract, be lemma 1.18, then i ∈ J- cof. Therefore, the lift h ∶ B → X exists because J- cof = (J- inj)- proj. Hence (C, I- cof, J- inj, (α, β), (γ, δ)) is a model category. The sets I and J admits the small object argument by assumption and the class of fibrations is defined to be J- inj. Also, I- inj ⊆ W ⋂ J- inj. Then, in order to see that (C, I, J) is cofibrantly generated we only need to show that W ⋂ J- inj ⊆ I- inj. If is already satisfied by the assumption then we are done. Otherwise, W ⋂ J- inj is the class of weak fibrations hence the class of cofibrations I- cof has the LLP with respect to W ⋂ J- inj, i.e W ⋂ J- inj = (I- cof)- inj = I- inj . 

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Question 5.7. Let (C, I, J) be a cofibrantly generated model category. Is there a minimal choice for I and J that result in the same model category, where minimal is the set theoretical sense on the size of I and J and their associated ordinals λI and λJ . 5.1. Generating (weak) Cofibrations and Partial Extensions an Liftings. In the study of some cofibrantly generated model categories, like the standard model category of simplicial sets, the local injective model category of simplicial (pre-)sheaves of a Grothendieck site, that core idea of examining the conditions of the cofibrantly generated model structure is to show that the classes of fibration and weak cofibration can be given as injections of a set of weak cofibrations or cofibrations, respectively. Hereby in this section, we extract a systematic method that simplifies the requirements of the verification, and it is based on Jardine’s construction of the local injective model category of simplicial (pre-)sheaves of a Grothendieck site in [Jar87] and Bousfield and Smith cardinality argument in [Hir03]. The argument is based on reducing lifting problems to partial lifting or extension problems, recalled below. Then, reducing the later to the condition that is referred to by bounded cofibration condition, in the contest of the local injective model category of simplicial (pre-)sheaves of a Grothendieck site. Therefore, we start by identifying what we mean by partial liftings or extensions. A weaker version of the lifting property is particularly useful in study of cofibrantly generated model categories (which cover most model categories one interested in in motivic homotopy theory), namely I-partial lifting property or P -partial extension property for a given classes of morphisms I and P . The main application of partial lifting or extension properties is illustrated in lemma 5.12 below, that if exists, it can be used to form a partially ordered set of partial liftings or extensions, which admits Zorn’s lemma, and in turn implies the existent of the desired lifting. Definition 5.8 (I-Partial Extension). Let C be a category, I a class of morphisms of C, D be a commutative diagram e0 /X U i

 V

D e1

 /Y

p

72

where i is a morphism of I, that is not an isomorphism. Then, p is said to have the right I-partial extension property with respect to i, or i is said to have the left I-partial extension property with respect to p, if there exists a commutative diagram pe ∶

j

i

h

 W  V

/X >

e0

U

p

k

 /Y

e1

such that j and k are morphisms of I, and j is not an isomorphism. If exists, such diagram is called I-partial extension of D, and we say that D admits the I-partial extension pe. Denote the class of I-partial extensions of D by P ED . Moreover, if p has the RLP with respect to all i ∈ I that are not isomorphism, then p is said to have the I-partial extension property. Definition 5.9 (Order of I-Partial Extension). Let C be a category, I a class of morphisms of C. Then, define the relation ≤ on the class of I-partial extensions of D /X

RA FT

e0

U

i

D

 V

 /Y

e1

p

such that for every I-partial extensions of D pe ∶

j

D

i

j′

p

k

i

 /Y

e1

 W′  V

/X =

e0

U

h

 W  V

/ X pe′ ∶ >

e0

U

h′

p

k′

 / Y.

e1

pe ≤ pe′ if there exist a morphism morphisms w ∶ W → W ′ in I that induces a morphism w ∶ j → j ′ in U ↓ C, w ∶ k → k ′ in C ↓ V and w ∶ h → h′ in C ↓ V , i.e. that makes the natural diagrams j

U

/W w

j′

 W′

W

h

/X =

w

W

/V . >

k

w

 W′

 W′

h′

k′

commutate. Notice that these conditions of ordering of I-partial extensions arise naturally when pe′ is obtained through composing pe with an I-partial extension of the bottom quadrangle of pe, as arises naturally through the construction of cofibrantly generated model structure, especially in left Bousfield localisation, and particularly in the local injective model structure of simplicial sheaves of a Grothendieck site, as in lemmas 2.66 and 2.67. Dual treatment gives rise to partial lifting and order of partial liftings. Definition 5.10 (P -Partial Lifting). Let C be a category, P a class of morphisms of C, D be a commutative diagram U i

 V

e0

D e1

/X  /Y

p

73

where p is a morphism of P , that is not an isomorphism. Then, p is said to have the right P -partial lifting property with respect to i, or i is said to have the left P -partial lifting property with respect to p, if there exists a commutative diagram e0 /X pl ∶ U r

 >Z

i h

 V

e1

p

q

 /Y

such that q and r are morphisms of P and q is not an isomorphism. If exists, such diagram is called P -partial lifting of D, and we say that D admits the P -partial lifting pl. Denote the class of P -partial liftings of D by ILD . Moreover, if i has the left partial lifting property with respect to all p ∈ P that are not isomorphism, then i is said to have the P -partial lifting property. Definition 5.11 (Order of P -Partial Lifting). Let C be a category, P a class of morphisms of C. Then, define the relation ≤ on the class of P -partial liftings of D e0

/X

RA FT

U

i

D

 V

e1

 /Y

p

such that for every P -partial liftings of D pl ∶ U

/X

e0

pl′ ∶ U

e0

r

 >W

i

D

 V

r′

p

i

q

h

 /Y

e1

/X

h′

 V

e1

 ′ >Z

p′

q

 /Y

pl ≤ pl′ if there exist a morphism morphisms z ∶ Z → Z ′ in P that induces a morphism z ∶ r → r′ in X ↓ C, z ∶ q → q ′ in C ↓ Y and z ∶ h → h′ in V ↓ C, i.e. that makes the natural diagrams r

X

/Z

h

V

z

r′

/Z

Z

z

 Z′

 Z′

h′

/Y . >

q

z

 Z′

q′

commutate. The below lemma 5.12 illustrates how lifting problems can be simplified to partial lifting or extension problems, which is usually easer to solve. Lemma 5.12. Let (C, W ′ ) be a cocomplete category with weak equivalences, p ∶ X → Y a morphisms in C, I a set morphisms of C such that relative I- cell are effective monomorphisms in C, I ′ be the intersection of the class of inclusions of I- cell sub-complexes with W ′ , such that I ′ is closed under transfinite composition. Then, p has the RLP with respect to I ′ if and only if it has the I ′ -partial extension property. Proof. One direction is evident. For the remaining direction, assume that p has the I ′ -partial extension property, and consider I ′ -lifting problem e0 /X U i

 V

′

D e1

 /Y

p

with i ∈ I . If i is an isomorphism, then D admits the lift e0 i−1 ∶ V → X. Otherwise, when i is not an isomorphism, then the assumption and the hypothesis implies that P ED is none-empty class. Every I ′ -partial

74

extension accounts to an inclusion of I- cell sub-complexes Z ⊂ V that since relative I- cell are effective monomorphisms then ?? shows that the choice of Z ⊂ V determines uniquely the morphism U → Z in the I ′ -partial extension of D. Every such inclusion of I- cell sub-complexes Z ⊂ V is determined uniquely by its set of cells, by ??, hence P ED is bounded by the set of subsets of cells of V , i.e. ∣P ED ∣ < 2∣V ∣ , hence P ED is a set. We will show that P ED admits Zorn’s lemma, and use Zorn’s lemma and the assumptions to prove the existence of the desired lift. It is readily seen that the order of I ′ -partial extension ≤ defines a partial ordering on the set P ED , in fact it is an inductive ordering because: Considering P ED as the category of its pre-ordered set, then every well-ordered chain C ⊂ P ED of increasing I ′ -partial extensions of D, indexed by the ordinal λ, is equivalent to a λ-sequence peC ∶ λ → peC,0 ↓ P ED , such that, for every ξ < λ, peC,ξ is the commutative diagram /X =

e0

U

peC,ξ ∶

jC,ξ

 ZC,ξ

i

hC,ξ p

kC,ξ

 /Y

RA FT

  V

e1

that is an I -partial extension of D for, with peC given on morphisms by the order of I ′ -partial lifting. peC induces λ- sequences ′

ZC ∶ λ → ZC,0 ↓ C, jC ∶ λ → jC,0 ↓ (U ↓ C), kC ∶ λ → kC,0 ↓ (C ↓ V ), andhC ∶ λ → hC,0 ↓ (C ↓ X). Since C is cocomplete category, then so is U ↓ C, C ↓ V , and C ↓ X. Therefore, there exist ZC,λ = colim ZC , jC,λ = colim jC , kC,λ = colim kC , and hC,λ = colim hC . Hence, there exist a diagram

peC,λ ∶

/X =

e0

U

D

jC,λ

 ZC,λ

i

hC,λ

p

kC,λ

  V

 /Y

e1

which is commutative by the universal property of colimits. Since zC,ξ ∈ I ′ for every ξ < λ, I ′ is closed λ under transfinite composition and jC,0 ∈ I ′ , then jC,λ ∈ I ′ . Also, Since kC,0 , zC,0 ∈ I ′ , then in particular λ ′ ′ kC,0 , zC,0 ∈ W , which satisfies the two-out-of-three property, hence kC,λ ∈ W . Since relative I- cell are effective monomorphisms, then colimits of inclusions of I- cell sub-complexes are inclusions of I- cell sub-complexes be lemma ??, i.e. kC,λ is an inclusion of I- cell sub-complexes and kC,λ ∈ I ′ . Consider the I- inj extension problem of kC,λ ZC,λ kC,λ

e′0

D′

 V

e′1

/X q

 /Y

i.e. with q ∈ I- inj. It induces a the below commutative diagrams for every ξ < λ through composing with the canonical injection of the colimit (CDξ )

ZC,ξ kC,ξ

 V

λ zC,ξ

/ ZC,λ

e′0

q

kC,λ

 V

/X

e′1

 /Y

75

Since each kC,ξ ∈ I- cof, then outside diagrams admit liftings hC,ξ ∶ V → X for every ξ < λ. Notice that hC,ξ is a lift for all the diagrams (CDυ ) for υ ≤ ξ. Then, consider the natural λ-sequence hC ∶ λ → hC0 ↓ (V ↓ C ↓ X) given above on objects, and defined on morphisms through ???, and let hC,λ = colim hC , then hC,λ is a lifting for all diagrams (CDξ ) for ξ < λ, then tracing the above diagrams we find λ λ e′0 zC,ξ = hkC,ξ = hkC,λ zC,ξ , ∀ξ < λ.

Then, by the universal property of colimits e′0 = hkC,λ , hence h is a lift of D′ , and kC,λ ∈ I- cof. Therefore, λ kC,λ ∈ P . Moreover jC,λ is not an isomorphism. That, if jC,λ is an isomorphism, then zC,ξ are epimorphisms for every ξ < λ because of the commutativity of the diagrams U jC,λ

jC,ξ

 ZC,ξ

# / ZC,λ

λ zC,ξ

(25)

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λ λ for every ξ < λ. Since zC,ξ ∈ I ′ which consists of monomorphisms, that would have implied that zC,ξ are isomorphisms for every ξ < λ, which contradict with jC,ξ being not isomorphisms for every ξ < λ. Therefore, peC,λ is an I ′ -partial extension of D. Moreover, peC,λ is an upper bound of C with peC,ξ ≤ peC,λ induced by the colimits cocoins and having I ′ closed under transfinite composition. Hence, P ED admits Zorn’s lemma, and there exists pe a maximal element in P ED . Assume for the sake of contradiction that k in pe is not an isomorphism, since k is morphism of I ′ and p has the I ′ -partial extension property, then there exists an I ′ -partial extension of the bottom quadrangle of pe, namely

/X >

h

Z

j′

 Z′

k

p

k′

  V

D

h′

 /Y

e1

′

Which induces the following I -partial extension of D pe′ ∶

e0

U

/X >

′

j j

i

 Z′   V

h′ p

k′ e1

 /Y

The commutativity of (25) implies that pe ≤ pe′ . Since j ′ is not an isomorphism then pe < pe′ which contradict with having pe a maximal element of P ED . Therefore, k is an isomorphism, hence D admits the lift hk −1 ∶ V → X.  Remark 5.13. Notice that the above lemma can be extended to the case when I ′ = W ′ ⋂ I- cof, replacing inclusions of I-sub-complexes with I- cof if W ′ ⋂ I- cof closed under colimit which is needed to show that kC,λ ∈ I ′ , otherwise the argument is almost identical. The main advantage of I-sub-complexes, when relative I- cell consists of effective monomorphisms, is that they are admit set-theoretical arguments that does not hold in an arbitrary abstract cocomplete category, particularly the colimit of inclusions of I-sub-complexes being an inclusions of I-sub-complex due to having colimits of inclusions of sets being inclusions of sets. However, when I- cof admit an analogy set-theoretical arguments, then the generalisation of the above lemma is straightforward as shown in the study of the local injective model category of simplicial pre-sheaves on a site in lemmas 2.67 and 2.66. One can consider the dual statement of the above lemma as well. However, it is not of particular use in the study of motivic homotopy theory.

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Lemma 5.12, above, reduces right lifting problems with respect to I ′ to right I ′ -partial extensions problems, for nicely behaved class of morphisms I ′ . The below theorem 5.15 also reduces the lifting problems further to the κ-bounded sub-complexes condition (the name is a modification of bounded cofibrations condition, due to Jardine [Jar87], where it was presented for simplicial (pre-)sheaves, whereas the formalisation is due to Hirschhorn [ibid.]). Therefore, start with recalling the κ-bounded sub-complexes condition. Therefore, we start by recalling the κ-bounded sub-complexes condition. Definition 5.14 (κ-Bounded sub-complexes W ′ -condition). Let (C, W ′ ) be a cocomplete category with weak equivalences, I a set morphisms of C such that relative I- cell are effective monomorphisms in C, I ′ be the intersection of the class of inclusions of I- cell sub-complexes with W ′ , such that I ′ is closed under transfinite composition, κ a cardinal. Then, we say that I ′ satisfies the κ-Bounded sub-complexes W ′ -condition if for every i ∶ U → V in I ′ that is not an isomorphism, there exists a inclusion of I- cell sub-complexes j ∶ L → V that is not an isomorphism such that (1) The size of L is at most κ. (2) The L is not a sub-complex of U . (3) The inclusions of I- cell sub-complexes U ⋂ L → L and U → U ⋃ L belong to W ′ .

DR AF

T

Theorem 5.15. Let (C, W ′ ) be a cocomplete category with weak equivalences, p ∶ X → Y a morphisms in C, I a set morphisms of C such that relative I- cell are effective monomorphisms in C, I ′ be the intersection of the class of inclusions of I- cell sub-complexes with W ′ , such that I ′ is closed under transfinite composition and satisfies the κ-Bounded sub-complexes W ′ -condition for a cardinal κ. Then, p has the RLP with respect to I ′ if and only if it has the RLP with respect to representatives of isomorphism classes of morphisms of I ′ of size at most κ. Proof. On direction is evident. For the other, assume that p has the RLP with respect to representatives of isomorphism classes of morphisms of I ′ of size at most κ, and consider the I ′ -lifting problem U

i

 V

e0

D

e1

/X  /Y

p

with i ∈ I ′ . If i is an isomorphism, then D admits the lift e0 i−1 ∶ V → X. Otherwise, i.e. when i is not an isomorphism, since I ′ satisfies the κ-Bounded sub-complexes W ′ -condition, then there exists inclusion of I- cell sub-complexes j ∶ L → V that is not an isomorphism with L is not a sub-complex of U , and of of size at most κ. Also,the inclusions of I- cell sub-complexes pL ∶ U ⋂ L → L and iU ∶ U → U ⋃ L belong to W ′ , hence belong to I ′ . Hence, we have the commutative diagram L⋂U pL

 L

pU

/U

h′ h

iU

iL

36/ X

e0

⌟  / L⋃U

i j

p

"

V

e1

 / Y.

Notice that pL ∈ I ′ , L is of size at most κ. Then, assumptions that p has the RLP with respect to morphisms of I ′ of size at most κ, for a representative of the isomorphism class of PL , implies the existence of the dotted lift h ∶ L → X that makes the diagram commutative. Then, since L ⋃ U ≅ L ∐ U , the universal property of L⋂U

colimits implies the existence of the morphism h′ ∶ L ⋃ U → X that makes the diagram commutative. Since L is not contained in U , then iU ∶ U → L ⋃ U is not an isomorphism. Moreover, j is an inclusion of I- cell sub-complexes as seen in ??. Also, since W ′ satisfies the two-out-of-three property and iU , i ∈ W ′ then j ∈ W ′ , hence j ∈ I ′ . Therefore, there exist I ′ -partial extension of D at L ⋃ U . Then, lemma 5.12 above implies that p has the RLP with respect to I ′ .  5.2. Cellular Model Categories. Cellular model categories are common cofibrantly generated model categories. It strengthens the definition of cofibrantly generated model categories, and produces declarable results, what will be particularly useful for Bousfield localisation. Particularly, it allows set-theoretical treatment

77

to for the class of inclusions of sub-complexes (see 5.16.3) and provides a desired boundedness on the size inclusions of sub-complexes (in 5.16.1 and 5.16.2). Definition 5.16 (Cellular Model Categories). Let (C, I, J) be a cofibrantly generated model category, with I the set of generating cofibration and J the set of generating weak cofibrations. Then, (C, I, J) is called cellular if it satisfies (1) Domains and codomains of morphisms of I are compact relative to I. (2) Domains of of morphisms of J are small relative to I- cell. (3) Cofibrations of C are effective monomorphisms (equalizers of parallel morphisms). In addition to the previously mentioned results on inclusions of sub-complexes when I- cell consists of monomorphisms, one of the important results in cellular model categories is that all cofibrant objects are small relative to the class of all cofibration of the model category, as seen below. This result is very important (at least technically) for Bousfield localisation. Theorem 5.17. Let (C, I, J) be a cellular model category. Then, cofibrant objects of C are small relative to the class of all fibrations.

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Proof. Since the set I admits the small object argument, then cofibrant objects are I-cofibrant objects which are retracts of I- cell complexes, by corollary 4.55, hence, by lemma 4.40, it is enough to show that I- cell complexes are small relative to the the class of all fibrations. Also, the class of all cofibrations in a cellular model category is the class I- cof, by lemma 5.5. Therefore, by theorem 4.56, it is enough to show that I- cell complexes are small relative to I- cell. Since I- cell ⊆ I- cof, and I- cof consists of effective monomorphisms, then in particular I- cell consists of monomorphisms. Also, domains and codomains of morphisms of I are compact relative to I, then they are small relative to I- cell by lemma 4.44. Let A be an I- cell complex, i.e. the unique morphisms i ∶ ∅ → A is a relative I- cell complex, i.e. a transfinite composition of a λ-sequence Z in C for an ordinal λ such that for ordinals ξ with ξ + 1 < λ there exists a morphism gξ ∶ Xξ → Yξ in I and xξ ∶ Xξ → Zξ in C and push-forward square gξ

Xξ

/ Yξ

yξ

xξ

D

 Zξ

⌟

zξ



/ Zξ+1

such that f is the transfinite composite of Z. X0

x0

| ∅ = Z0

/ Y0

g0

Xξ xξ

y0 ⌟ z0

 / Z1

gξ

...

/ Zξ

/ Yξ yξ

~

⌟

Xξ+1 xξ+1

/ Yξ+1 yξ+1

 |

/ Zξ+1

zξ

gξ+1

 / Zξ+2

⌟ zξ+1

...

/ colim Z = A 3

i

We fix the above presentation of i, and use the transfinite induction to prove that each of colim Ziλξ is a small relative to I- cell, for every ξ ≤ λ. Notice that ∅ is κ-small relative to any class of morphisms in C for any cardinal κ, so in particular it is small relative to I- cell any cardinal κ. Also, domains and codomains of I are small relative to I- cell, then there exists a cardinal κ > ∣λ∣ such that domains and codomains of I, and ∅ are κ-small relative to I- cell. Let λ′ be a κ-filtered ordinal, Z ′ be a sequence in Z0′ ↓ C of morphisms of I- cell. (1) For ξ = 0 ≤ λ, i is the isomorphism id∅ ∶ ∅ → ∅, and as argued before ∅ is κ-small relative to I- cell. (2) For a successor ordinal ξ = υ + 1 ≤ λ, suppose that Zυ = colim Ziλξ is κ-small relative to I- cell, and we need to show that Zξ = colim Ziλξ+1 is κ-small relative to I- cell, i.e. we need to show that the canonical morphism ϑZξ ∶ colim C(Zξ , Z−′ ) → C(Zξ , colim Z ′ ) is an isomorphism. Since I- cell consists of morphisms, then as seen in the proof of 4.44, the invectivity of ϑZξ is evident. For surjectivity, let f ∶ Zξ → colim Z ′ . Since Yυ and Zυ are κ-small relative to i- cell,

78

then the compositions f yυ and f zυ factorise at Zξ′ y′ and Zξz′ , respectively, for ξy′ , ξz′ < λ′ . Then, they factorise at Zξ′ ′ for ξ ′ = max{ξy′ , ξz′ } < λ′ as f yυ = z ′ ξ′ y ′ λ′

Xξ

gξ

λ′

/ Yξ yξ

xξ

 Zξ

f zυ = z ′ ξ′ z ′

, and

⌟ zξ



/ Zξ+1

y′ f z′

Z0′

f

′

⋅⋅⋅

'/*

Zξ′ ′

⋅⋅⋅

* / colim Z ′ . λ

From the commutativity of the push-forward square we have z ′ ξ′ z ′ xυ = f zυ xυ = f yυ gυ = z ′ ξ′ y ′ gυ . λ′

λ′

RA FT

Since the λ′ -sequence Z ′ is of morphisms of I- cell, I- cell is closed under transfinite composition by λ′ lemma 4.27.(b), and I- cell consist of monomorphisms, as seen above, then z ′ ξ′ is a monomorphism, hence we have z ′ xυ = y ′ gυ . Then, by the universal property of push-outs, there exists the unique dotted morphisms f ′ ∶ Zξ → Zξ′ that makes the above diagram commutative, i.e. y ′ = f ′ yυ

, and

z ′ = f ′ zυ′ .

Applying the universal property of push-outs again for the morphisms f yυ and f zυ yield that f = z ′ ξ′ f ′ , hence f factorises at Zξ′ ′ , and ϑZξ is surjective. Hence, ϑZξ is an isomorphism and Zξ is κ-small relative to I- cell. (3) For a limit ordinal ξ ≤ λ, suppose that colim Ziλυ is κ-small relative to I- cell for every υ < ξ, and we need to show that Zξ = colim Ziλξ is κ-small relative to I- cell, i.e. we need to show that the canonical morphism ϑZξ ∶ colim C(Zξ , Z−′ ) → C(Zξ , colim Z ′ )

D

λ′

is an isomorphism. Since ξ is a limit ordinal, then having colim Ziλυ is κ-small relative to I- cell for every υ < ξ is equivalent to having Zυ is κ-small relative to I- cell for every υ < ξ. Since I- cell consists of morphisms, the invectivity of ϑZξ is evident. For surjectivity, let f ∶ Zξ → colim Z ′ . Since Zυ are κ-small relative to I- cell for every υ < ξ, then morphisms f zυξ factorises at Zξυ′ for every υ < ξ. Notice that ∣{ξυ′ ∣υ < ξ}∣ = ∣ξ∣ ≤ ∣ξ∣ < κ and since λ′ is a κ-filtered ordinal. Then, ξ ′ ∶= sup{ξυ′ ∣υ < ξ} < λ′ . Hence, all morphisms f zυξ factorises at Zξ′ ′ as f zυξ = z ′ ξ′ fυ′ . λ′

For every  < υ < ξ, we have z ′ ξ′ f′ = f zξ = f zυξ zυ = z ′ ξ′ fυ′ zυ λ′

λ′

Since z ′ ξ′ is a monomorphism, then f′ = fυ′ zυ every  < υ < ξ. Then, by the universal property of the colimit, there exists a unique morphisms f ′ ∶ colim Ziλξ → Zξ′ ′ . Applying the universal property of λ′

colimits again for the morphisms f zυξ for every υ < ξ yield that f = z ′ ξ′ f ′ , hence f factorises at Zξ′ ′ , and ϑZξ is surjective. Hence, ϑZξ is an isomorphism and Zξ is κ-small relative to I- cell. λ′

Therefore, by the transfinite induction colim Ziλξ is κ-small relative to I- cell for every ξ ≤ λ, in particular, for ξ = λ, colim Z = colim Ziλλ is κ-small relative to I- cell. Therefore, cofibrant objects of C are small relative to the class of all fibrations.  The above lemma is a combination of specialised assertions, customised to fit the interest of the book, for more general treatment of the ideas used here see [Hir03, §10 and §.12]

79

6. Proper Model Categories Definition 6.1. Let C be a model category. Then (1) C is called left proper if the class of its weak equivalences is closed under push-outs along cofibrations. (2) C is called right proper if the class of its weak equivalences is closed under pull-backs along fibrations. (3) C is called proper if it is both left and right proper. Proper model categories facilitate approximation in model categories where it guarantees that finite (co)limits of approximations are corresponding approximations of the finite (co)limits, and enables solving extension-lifting problem on the level of an approximation, among other desired properties. This is particularly useful in the study of localisation of model categories. Therefore, recall the relevant results hereby. Lemma 6.2. Let C be a model category.

RA FT

̃ → Ṽ a cofibration ● Let C be left proper, i ∶ U → V a cofibration and j ∶ U → U ′ a morphism in C, ̃i ∶ U cofibrant approximation of i. Then, the push-out of i along j has a cofibration cofibrant approximation ̃ → U → U ′ ). that is push-out of ̃i along α(U ̂ → Ŷ a fibration ● Let C be right proper, p ∶ X → Y a fibration and q ∶ Y ′ → Y a morphism in C, p̂ ∶ X fibrant approximation of p. Then, the pull-back of p along q has a fibration fibrant approximation that is pull-back of p̂ along δ(Y ′ → Y → Ŷ ). Proof. See [Hir03, 13.3.10].



Lemma 6.3 (Kan). Let C be a model category, i ∶ U → V a cofibration and p ∶ X → Y a fibration in C. ̃ → Ṽ a cofibration cofibrant approximation of i. Then, if p has the RLP ● Let C be left proper, and ̃i ∶ U ̃ with respect to i then it has the RLP with respect to i. ̂ → Ŷ a fibration fibrant approximation of p. Then, if i has the LLP ● Let C be right proper, and p̂ ∶ X with respect to p̂ then it has the LLP with respect to p. Proof. The two statements are dual. Therefore, we prove the first, then the second follows by duality. Assume that has p has the RLP with respect to ̃i, and consider the extension-lifting problem given in E e0

U

D

i

E

 V

e1

/X

p

 / Y.

Then, it can be extended by the cofibration cofibrant approximation square to form the below diagram ∼ qU

̃ U ̃ U

 Ṽ

∼ qV

/U

e0

i

E

 /V

e1

/X p

 / Y.

̃ → U yields the solid commutative diagram Then, considering the pus-out of the diagram Ṽ ← U ̃ U

∼ qU

̃ i

 Ṽ

/U iU

iV

/7 X

e0

h

i

⌟  /W

p

∼

j

qV

' V

e1

 / Y.

Where W ∶= Ṽ ∐ U , and j is the unique morphism induced by the universal property of push-outs that makes ̃ U

the diagram commutes. Since ̃i ∈ {p}- proj by assumption, and {p}- proj is closed under push-outs by lemma 1.20, then iU ∈ {p}- proj. Hence, the dotted lift h ∶ W → X exists, making the above diagram commutative, i.e. hi = e0

, and

ph = e1 j.

80

Since C is left proper, then push-out of a weak equivalence along cofibration is a weak equivalence, hence iV is a weak equivalence in C, and by the two-out-of-three property j is also a weak equivalence in C. Moreover, since cofibrations are closed under push-forward, then iU is a cofibration in C. Consider the category U ↓ C ↓ Y endued with the model structure induced from the model structure of C. It is a simplicial model category with cofibrations, fibration and weak equivalences corresponding to cofibrations, fibration and weak equivalences in C. Therefore, (iU , e1 j) and (i, e1 ) are cofibrant objects in U ↓ C ↓ Y , because i and iU are cofibrations in C. Similarly, we see that (e0 , p) is a fibrant object in U ↓ C ↓ Y , because p is a fibration in C. j ∶ (iU , e1 j) → (i, e1 ) is a weak equivalence in U ↓ C ↓ Y , for being a weak equivalence in C. Since j is a a weak equivalence between cofibrant objects in U ↓ C ↓ Y and (e0 , p) is a fibrant objects in U ↓ C ↓ Y , then [Hir03, Cor.7.7.5] implies that the induced morphism j ∗ ∶ U ↓ C ↓ Y ((i, e1 ), (e0 , p)) Ð→ U ↓ C ↓ Y ((iU , e1 j), (e0 , p)) is a bijection. Since, h ∈ U ↓ C ↓ Y ((iU , e1 j), (e0 , p)) then there exists h′ ∈ U ↓ C ↓ Y ((i, e1 ), (e0 , p)) such that h′ j = h. Since h′ ∈ U ↓ C ↓ Y ((i, e1 ), (e0 , p)), then by the definition of morphisms in U ↓ C ↓ Y one sees that h′ is a lift of E. 

RA FT

Notice that in Top gluing16 of topological is invariant under weak equivalence. That does not hold in general for arbitrary model category. However, that is remedied in left proper model categories, as seen in the below lemma. Lemma 6.4. Let C a model category.

● Let C be left proper, then for every solid commutative diagram

f

g

!

∼

∼

j

/V

i

U

/ V′

′

U′

i

iV

j′

⌟  /Z

iW

h

iV ′

h

!  W′

 / Z ′.

∼

∼

D

 W

⌟

iW ′

with i, i′ cofibrations and diagonal solid morphisms being weak equivalences, the induced morphism Z → Z ′ is a weak equivalence for Z ∶= W ∐U V and Z ′ ∶= W ′ ∐U ′ V ′ . ● Let C be right proper, then for every solid commutative diagram YO o

XO g

f

!

∼

∼

q

p

YO ′ o

p

′

XO ′

πX q′

Zo

πW

⌟

πX ′

W h

h

∼

∼

Z′ o

πW ′

⌟!

W ′.

with i, i′ cofibrations and diagonal solid morphisms being weak equivalences, the induced morphism Z → Z ′ is a weak equivalence for Z ∶= W ∐U V and Z ′ ∶= W ′ ∐U ′ V ′ . Proof. See [ibid., 13.3.10]. 16Push-out along cofibrations, i.e. along monomorphisms for nice topological spaces.



81

7. Simplicial Model Category Model categories may lack to some desired property that facilitate working with them, namely neither left-homotopy nor right-homotopy in a model category is necessary an equivalent relation on morphism between objects that are not cofibrant or fibrant, weak equivalences are usually not easy to detect and in general they are not closed cartesian or monoidal categories, although they precise a Cartesian product. The above issue are better realised in simplicial model categories. 7.1. Simplicial Categories. The category of simplicial sets sSet is a monoidal category with the monoidal product being the cartesian product, and the identity being ∗. Hence, we can consider enriched categories over sSet. sSet is closed monoidal (actually closed cartesian) category, .... Definition 7.1 (Simplicial Categories). Simplicial category C is defined to be sSet-enriched category that is SM6) tensored and powered over sSet, i.e. Tensored) For every object X ∈ C, the functor Map(X, −) ∶ C → sSet has a sSet-enriched left adjoint X ⊗ − ∶ sSet → C. Powered) For every object Y ∈ C, the functor Map(−, Y ) ∶ Cop → sSet has a sSet-enriched left adjoint

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Y − ∶ sSet → Cop . Example 7.2. The category of simplicial set is an enriched category over itself, with the hom-simplicial sets given by Map(X, Y )n = sSet(X × ∆n , Y ), X, Y ∈ sSet, n ≥ 0 as seen in 5. The above mentioned sSet-adjunctions are express by the existence of isomorphisms (26)

MapC (X ⊗ K, Y ) ≅ MapsSet (K, MapC (X, Y )) ≅ MapCop (Y K , X) ≅ MapC (X, Y K ). that are natural in X, Y and K.

The above definitions shows that for every simplicial set K, there exist two functors , and

−K ∶ C → C

D

−⊗K ∶C→C

Lemma 7.3. Let C be a simplicial category, K a simplicial set. Then, − ⊗ K is a sSet-left adjoint of −K . Proof. See [GJ09, Lem.2.2].



The above lemma shows that C is a closed monoidal sSet-enriched category. Hence, it explains the terminology of calling ⊗ tensor product, for being left adjoint for internal hom. Also, based on the above arguments we have the bi-functors − ⊗ − ∶ C × Set → C

, and

−− ∶ C × sSetop → C.

Remark 7.4. Notice that every simplicial category C is a category with its hom-sets given by C(X, Y ) = Map(X, Y )0 , ∀X, Y ∈ C, i.e. hom-sets coincide with the vertices of the hom-simplicial sets of C. We call this category the underlying category of the simplicial category C. Some authors call Map(X, Y ) the function complex, or simplicial mapping spaces, from X to Y , as in [ibid., Def.9.1.2]. Hence, the above adjunction on sSet-enriched categories implies adjunctions of catgeories on the level of vertices, given by isomorphism (27)

C(X ⊗ K, Y ) ≅ sSet(K, MapC (X, Y )) ≅ Cop (Y K , X) ≅ C(X, Y K ).

that are natural in X, Y and K, as in [ibid., Prop.9.1.8]. Notice that in example 7.2, hom-objects are given in terms of hom-sets. That holds in general for arbitrary simplicial category, and it is mainly due to C being sSet- tensored, as in the SM6 axiom, as in the below lemma. Lemma 7.5. Let C be a simplicial category, X, Y ∈ C, n ≥ 0. Then, there is a bijection Map(X, Y )n ≅ C(X ⊗ ∆n , Y ) natural in X, Y and [n] ∈ ∆.

82

Proof. Recall that for every simplicial set K, there is a natural bijection Kn ≅ sSet(∆n , K), hence, we have Map(X, Y )n ≅ sSet(∆n , Map(X, Y )) ≅ C(X ⊗ ∆n , Y ).  Hence, we see easily that 0

X ⊗ ∂∆0 ≅ ∅ , X ∂∆ ≅ ∗ , X ⊗ ∆0 ≅ X

, and

0

X∆ ≅ X

Notice that both the tensor product and the power are compatible with the monoidal structure (cartesian product) of simplicial sets, which is usually referred to of being associative (which is an abuse of the conventional notation of associativity) as seen in the below lemma: Lemma 7.6. Let C be a simplicial category, X, Y ∈ C and K, L be simplicial sets. Then, there exists isomorphism X ⊗ (K × L) ≅ (X ⊗ K) ⊗ L

, and

L

Y (K×L) ≅ (Y K )

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natural in X, Y, K and L. Proof. See [Hir03, Prop.9.1.11].



Recall that every locally small category can be embedded into the category of small sets using Yoneda embedding h− , and its dual h− . Similarly, simplicial categories are embedded into the category of simplicial sets, and that is mainly due to the fact that C(X, Y ) is recovered from Map(X, Y ) for all X, Y in a simplicial category C. Lemma 7.7. Let C be a simplicial category. Then, for every X ∈ C, we have the functor Map(X, −) ∶ C → sSet

D

that sends Y ∈ C to Map(X, Y ), and g ∶ Y → Z in C (taken as a conventional category) to g∗ ∶ Map(X, Y ) → Map(X, Z)

given by post-composition, i.e. by Map(X, Y )

∼

/ ∗ × Map(X, Y )

ιg ×idMap(X,Y )

/ Map(Y, Z) × Map(X, Y )

○X,Y,Z

/ 2 Map(X, Z).

g∗

Also, for every Y ∈ C, we have the functor (pre-sheaf) Map(−, Y ) ∶ Cop → sSet that sends X ∈ C to Map(X, Y ), and f ∶ W → X in C (taken as a conventional category) to f ∗ ∶ Map(X, Y ) → Map(W, Z). given by pre-composition, i.e. by Map(X, Y )

∼

/ Map(X, Y ) × ∗

idMap(X,Y )×ιf

/ Map(X, Y ) × Map(W, X)

○W,X,Y

/ 2 Map(W, Y ).

f∗

Proof. See [ibid., Prop.9.1.3].



Remark 7.8. The above lemma applies to any M-enriched categories, for a monoidal category M, as well as hom-sets can be recovered canonically from the hom-objects.

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7.2. Simplicial Model Categories. Since sSet is equipped with model structures (Quillen, Joyal,...), then when considering simplicial categories, whose underlying category admits a model structure, it is natural to distinguish these simplicial categories with model structures that are compatible with the considered model structure of simplicial sets. Usually, when talking about the model structure of simplicial sets, refers to Quillen classical model structure, described in 3, unless mentioned otherwise. Such simplicial categories are called simplicial model categories and recalled below: Lemma 7.9. Let C be a simplicial category, with its underlying category being a model category. Then, the following conditions are equivalent: ● Let i ∶ U → V be a cofibration and p ∶ X → Y a fibration in C. Then, the induced morphism of simplicial sets i∗ × p∗ ∶ Map(V, X) → Map(U, X) ×Map(U,Y ) Map(V, Y ) is a Kan fibration. Moreover, if either of i or p is a weak equivalence in C, then so is i∗ × p∗ . ● Let i ∶ U → V be a cofibration and j ∶ K → L a cofibration of simplicial sets. Then, the induced morphism of simplicial sets i◻j ∶U ⊗L ∐ V ⊗K →V ⊗L U ⊗K

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is a cofibration in C. Moreover, if either of i or j is a weak equivalence in C or sSet, respectively, then so is i ◻ j. ● Let p ∶ X → Y be a fibration in C and j ∶ K → L a cofibration of simplicial sets. Then, the induced morphism of simplicial sets pj ∶ X L → X K ×Y K Y L is a fibration in C. Moreover, if either of p or j is a weak equivalence in C or sSet, respectively, then so is pj . Proof. See [Hir03, Prop.9.3.7]



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Definition 7.10 (Simplicial Model Categories). Let C be a simplicial category, with its underlying category being a model category. Then, we say that C is a simplicial model category if it satisfies the homotopy lifting extension theorem, i.e. if it satisfies any of the equivalent conditions of the above lemma 7.9, particularly satisfies: SM7 Let i ∶ U → V be a cofibration and p ∶ X → Y a fibration in C. Then, the induced morphism of simplicial sets i∗ × p∗ ∶ Map(V, X) → Map(U, X) ×Map(U,Y ) Map(V, Y ) is a Kan fibration. Moreover, if either of i or p is a weak equivalence in C, then so is i∗ × p∗ . Remark 7.11. Some authors defines simplicial categories to be sSet-enriched categories, and require SM6 to be part of the definition of simplicial model categories, like in [ibid.]. However, since SM6 is independent of the model structure of C, then it makes sense not to be included in this definition. Remark 7.12. Axiom SM7 above, is the homotopy lifting extension theorem. It was proven by Kan for the category of simplicial objects. The below lemma shows the compatibility meant earlier between the model struture of C and this of simplicial sets Lemma 7.13. Let C be a simplicial model category. Then, if W is a cofibrant object, p ∶ X → Y is a (weak) fibration in C, then p∗ ∶ Map(W, X) → Map(W, Y ) is a (weak) Kan fibration. Also, if Z is a fibrant object, i ∶ U → V is a (weak) cofibration, then i∗ ∶ Map(V, Z) → Map(U, Z) is a (weak) Kan fibration. Proof. It is a direct result of SM7. Where U is taken to be the initial object in C in the first case, and Y is taken to be the terminal object in C in the second. 

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Corollary 7.14. Let C be a simplicial model category. Then, if W is a cofibrant object, p ∶ X → Y is a weak equivalence between fibrant objects in C, then p∗ ∶ Map(W, X) → Map(W, Y ) is a weak equivalence of simplicial sets. Also, if Z is a fibrant object, i ∶ U → V is a weak equivalence between cofibrant objects in C, then i∗ ∶ Map(V, Z) → Map(U, Z) is a weak equivalence of simplicial sets. Proof. If W is a cofibrant object, then Map(W, −) ∶ C → sSet sends weak fibrations to weak fibrations, hence in particular Map(W, −) sends weak fibrations between fibrant objects into weak fibrations, particurally weak equivalence. then, by Ken Brown’s lemma 3.5 Map(W, −) sends all weak equivalences between fibrant objects to weak equivaleces. Hence, for a weak equivalence between fibrant objects p ∶ X → Y , we have p∗ ∶ Map(W, X) → Map(W, Y ) a weak equivalence of simplicial sets, actually it is a weak equivalence between fibrant objects Map(W, X), Map(W, Y ). The second statement, holds by duality. 

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. Recall that weak fibrations of simplicial sets are determined by the RLP with respect to the set of inclusions of boundaries into standard simplexes. This can be use to determine when fibrations or cofibrations induce weak equivalences on hom-objects. Lemma 7.15. Let C be a simplicial model category. (1) Let i ∶ U → V be a cofibration, Z a fibrant object in C. The induced morphism i∗ ∶ Map(V, Z) → Map(U, Z) is a weak equivalence if and only if the unique morphism X → has the RLP with respect to the set of canonical morphism Ii = {i ◻ ∂[n] ∶ U ⊗ ∆n

n n ∐ B ⊗ ∂∆ → B ⊗ ∆ ∣n ≥ 0}

A⊗∂∆n

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(2) Let p ∶ X → Y be a fibration, W a cofibrant object in C. The induced morphism p∗ ∶ Map(W, X) → Map(W, Y ) is a weak equivalence if and only if the unique morphism ∅ → X has the LLP with respect to the set of canonical morphism n

n

n

I p = {p∂[n] ∶ X ∆ → Y ∆ ×Y ∂∆n X ∂∆ ∣n ≥ 0}

Proof. We prove the first sentence, and the second holds by ’duality’. Since i is a cofibration and Z is a fibrant object in C , then i∗ is a Kan fibration. Hence, i∗ is a weak equivalence if and only if it is a weak Kan fibration, which is equivalent to having the RLP with respect to the set of generating cofibrations of the standard model structure on simplicial sets, namely I = {∂∆n ↪ ∆n ∣n ≥ 0}. Then, lemma [Hir03, Lem.9.3.6] shows that is equivalent of the unique arrow X → ∗ having the RLP with respect to Ii .  7.3. Detecting Weak Equivalences. Lemma 7.16. Let C be a simplicial model category, f ∶ X → Y in C. Then, f is a weak equivalence if either ● For every fibrant object Z, the induced morphism f ∗ ∶ Map(Y, Z) → Map(X, Z) is a weak equivalence of simplicial sets. ● For every cofibrant object W , the induced morphism f∗ ∶ Map(W, X) → Map(W, Y ) is a weak equivalence of simplicial sets. Proof. See [ibid., Prop.9.7.1] Lemma 7.17. Let C be a simplicial model category.



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(1) A morphism i ∶ U → V in C is a weak equivalence if and only if for every fibrant object Z, the induced morphism ∗ Q(i) ∶ Map(Q(V ), Z) → Map(Q(U ), Z) is a weak equivalence of simplicial sets, where Q is the cofibrant replacement of C. (2) A morphism f ∶ X → Y in C is a weak equivalence if and only if for every cofibrant object W , the induced morphism R(f )∗ ∶ Map(W, R(X)) → Map(W, R(Y )) is a weak equivalence of simplicial sets, where S is the cofibrant replacement of C. Proof. Let i ∶ U → V be a morphism in C, if i is weak equivalence, then Q(i) is so by the two-out-of∗ three property. Since Q(U ), Q(V ) are cofibrant objects, then, by corollary 7.14, Q(i) is a weak equivalence ∗ of simplicial sets for every fibrant object Z. On the other hand, suppose that Q(i) is a weak equivalence of simplicial sets for for every fibrant object Z, then by the previous lemma 7.16, Q(i) is a weak equivalence. Then, by the two-out-of-three property, i is a weak equivalence in C. The other statement holds by duality. 

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Corollary 7.18. Let C be a simplicial model category. (1) If U and V are cofibrant objects in C, then a morphism i ∶ U → V in C is a weak equivalence if and only if for every fibrant object Z, the induced morphism i∗ ∶ Map(V, Z) → Map(U, Z)

is a weak equivalence of simplicial sets. (2) If X and Y are fibrant objects in C, then a morphism f ∶ X → Y in C is a weak equivalence if and only if for every cofibrant object W , the induced morphism f∗ ∶ Map(W, X) → Map(W, Y )

is a weak equivalence of simplicial sets.

Proof. Straight forward result of the previous lemma 7.17.



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7.4. Simplicial Functors. Recall that in categories morphisms are always define to preserve the structure of interest of a give collection of objects, same concept generalises to 2-categories. Hence, a morphism between simplicial categories should be defined to be a functor that preserve the sSet-enrichness of the category. [Hir03, Ex.9.8.6] give an example of a functor that does not respect the sSet-enriched structure. Hence, the distinguishing becomes important. Definition 7.19 (Simplicial Functor). Let C, D be simplicial categories. Then a simplicial functor F ∶ C → D is defined to be a sSet-enriched functor from C to D, i.e. ● A class function F ∶ Ob(C) → Ob(D) that send X ∈ C to F (X) ∈ D. ● It assigns for every pair of objects X, Y ∈ C a morphism of simplicial sets FX,Y ∶ Map(X, Y ) → Map(F (X), F (Y )) that is compatible with the simplicial composition, i.e. makes the following diagram commutative for every X, Y, Z ∈ C Map(Y, Z) × Map(X, Y ) FY,Z ×FX,Y

cX,Y,Z

/ Map(X, Z) FX,Z

  cF (X),F (Y ),F (Z) / Map(F (X), F (Z)) Map(F (Y ), F (Z)) × Map(F (X), F (Y )) A natural question arise here, namely when a given functor between simplicial categories is a simplicial functor. This is addressed in the below lemma. Lemma 7.20. Let F ∶ C → D be a functor between simplicial categories. The, F can be extended to a simplicial functor if and only if for every finite simplicial set K and object X ∈ C there exist a natural transformation σ ∶ F (−) ⊗ − → F (− ⊗ −), where F (−) ⊗ −, F (− ⊗ −) ∶ C × sSet → D are evident such functors, such that satisfy the triangle and pentagon identities, i.e.

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(1) For every object X ∈ C, σX,∆0 is an isomorphism that makes the evident triangle commute σX,∆0

F (X) ⊗ ∆0

/ F (X ⊗ ∆0 )

≅ ≅

%

y

≅

F (X) (2) For every object X ∈ C and finite simplicial sets K and L, the following pentagon commutes ≅ F (X) ⊗ (K × L) / (F (X) ⊗ K) ⊗ L σX,K ⊗idL

σX,K×L

 F (X ⊗ (K × L)) ≅

 F (X ⊗ K) ⊗ L σX⊗K,L

.

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{ $ F ((X ⊗ K) ⊗ L) Proof. See [Hir03, Th.9.8.5].



7.4.1. Simplicial Quillen Adjunction. Working with simplicial model categories, one has the notions of Quillen adjunction and simplicial functors. Then it is natural to consider the case when both concepts are compatible, as in the below definition. Definition 7.21. Let (F, G, ϕ) ∶ C → D be a Quillen adjunction between simplicial model categories. then (F, G, ϕ) is called a Quillen adjunction of simplicial model categories if it is an sSet-enriched adjunction. Example 7.22. Let (C, τ ) be a small site, ∆op PSh(C), ∆op Shvτ (C) be the category of simplicial presheaves on C and sheaves on (C, τ ), respectively, endued with τ -local injection model structure, as in chapter 4. Then, there is the natural Quillen adjunction:

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(−aτ , i, ϕ) ∶ ∆op PSh(C) → ∆op Shvτ (C)

where i is the inclusion functor, −aτ is the functor induced by the sheafification functor with respect to τ . Both ∆op PSh(C) and ∆op Shvτ (C) are simplicial model categories, and as seen in chapter 4, (−aτ , i, ϕ) is a Quillen adjunction of simplicial model categories. 8. Localisation of Model Categories To motivate the argument of this chapter, we start with topology, where homotopical invariants are functors from the category of topological spaces (or at least a nice such category) that are invariant under homotopy type, like homotopy groups, singular homology and cohomology groups, etc. In particular, spaces X and X × I have same such groups, where I is the unit interval. On the other hand in algebraic geometry, in the category of some ’nice’ schemes over a base filed k, there are well-known A1k -invariant, like Chow groups that CH ∗ (X × A1k ) ≅ CH ∗ (X) for any scheme X. Then it is natural to ask if there exists a homotopy theory on schemes over k (or at least some ’nice’ ones) with respect to which Chow groups are homotopical invariant, among other desired properties. This has been realised in A1 -homotopy theory [MV99], where a category of ’nice’ schemes is embedded in the category of their Nisnevich sheaves and then localised with respect to the affine line A1k , i.e. contracting the affine line to a point. Therefore, in order to fully understand A1 -homotopy theory, one needs to understand localisation of model categories. On the other hand, localising topological spaces is an interesting subject in its own, see [BK71]. Since, model categories are motivated by the case of topological spaces, then it is natural to look for a framework of localising objects in model categories. That is done in localisation of objects, section 8.1. Notice that what one is actually wants to localise is the homotopy theories, as in motivic homotopy theory. Since a homotopy category is better understood as a homotopy category of some model category, then it becomes important to understand this localisation on the level of model categories. This is called the localisation of model

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categories. Roughly speaking the localisation of the model category C at a class of morphisms S, if exists, is a new model structure on C with the smallest class of weak equivalences WS that contains both of W , the original class of weak equivalences, and S. Hence, the images of S in the resulting homotopy category, if exists, are isomorphisms; i.e. the resulting homotopy category is a localisation of the original homotopy category with respect to images of S in the original homotopy category, in the sense of localisation of categories, def. 8.1. That should be done in a minimalistic way, i.e. having the localisation C → LocS C to be initial such morphism in the ’category’ of model categories. Each of the notions of morphisms of model categories, namely left and right Quillen functors, gives rise to their corresponding notions of localisation of model categories, namely left and right localisation of model category. Hirschhorn has provided general treatment on the localisation of model categories in [Hir03]. However, we restrict attention to simplicial model categories, that model categories that arises in motivic homotopy theory are simplicial and that simplifies the arguments. In this section we mainly follow [ibid.], [Far96], and [Bou77]. We start by recalling left and right localisations, (co)local objects and (co)local equivalences, localisations of objects, which form the basic topological motivation for localisation of model category. Then we pass to Bousfield localisation, and used the accumulated result, especially from localisations of objects, to prove the existence of Bousfield localisations.

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Throughout, this section C will denote a model category with a model structure (C, F, W, (α, β), (γ, δ)), S a class of morphisms of C, or set of such morphisms if specified.

DR AF

Definition 8.1 (Localisation of Model Category). Let C be a model category, S a class of morphisms of C. Then, (1) A left localisation of C with respect to S, if exists, is a pair (LS C, ηS ), where LS C is a model category, and ηS ∶ C → LS C is a left Quillen functor such that there exists a cofibrant approximation functor Q′ ∶ C → Cc for which (a) the total left derived functor LQ′ ηS ∶ HC → H LS C takes images of S in HC to isomorphisms into H LS C. (b) any left Quillen functor θ ∶ C → D, such that the total left derived functor LQ′ θ ∶ HC → HD takes images of S in HC into isomorphisms in HD, factorises uniquely through ηS , i.e. there exists a unique left Quillen functor θS ∶ LS C → D such that the following diagram of left Quillen functors commutes ηS / LS C . C ↺

θS

"  D (2) A right localisation of C with respect to S, if exists, is a pair (RS C, S ), where RS C is a model category, and S ∶ C → RS C is a right Quillen functor such that there exist a fibrant approximation functor R′ ∶ C → Cf for which (a) the total right derived functor RR′ S ∶ HC → H RS C takes images of S in HC into isomorphisms in H RS C. (b) any right Quillen functor θ ∶ C → D, such that the total right derived functor RR′ θ ∶ HC → HD takes images of S in HC into isomorphisms in HD, factorises uniquely through S , i.e. there exists a unique right Quillen functor θS ∶ RS C → D such that the following diagram of right Quillen functors commutes S / RS C . C θ

↺ θ

θS

"  D

We see later that the choice of the cofibrant or fibrant approximation functors. is not an essential part of the definition. In fact, corollary 8.15 indicates that the above definition is independent of the choice of the above fibrant or cofibrant approximation functors, hence one can always consider the canonical cofibrant or fibrant replacement functors, as done in most categories considered in these notes. However, occasionally in practise one has nicely behaved different cofibrant and fibrant approximation functors that are more practical to work with compared to the cofibrant and fibrant replacement functors, respectively.

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Remark 8.2. Lemma 3.2 shows that we can rephrase the definition of right localisation of C with respect to S, with left Quillen functors S ∶ RS C → C, θ ∶ D → C and θS ∶ D → RS C, inverting the diagram accordingly. Remark 8.3. One observes the similarity between of the above definitions of left and right localisation of a model category C with respect to S and what would have been ’universal arrows’ from C to H, inverting morphisms of S. However, H is not a functor, but rather 2-functor so strictly speaking it does not make sense to talk of universal arrows to H. If a generalisation of ’universal arrows’ is to be considered here, then it should be from (C, S) in the 2-category of pairs of model categories and classes in them to H. Lemma 8.4. Let C be a model category, S a class of morphisms of C. Then, if the left (right) localisation of the model category C with respect to S exists, then it is unique up to isomorphism. Proof. We prove the statement for left localisations, and the proof for right localisation holds by duality. Let (LS C, ηS ), (L′S C, ηS′ ) be left localisations of C with respect to S. Then, by the second axiom of the above definition of left localisation of model category, there exist unique left Quillen functors θS ∶ LS C → L′S C and θS′ ∶ L′S C → LS C such that the following diagrams of left Quillen functor commutes C

/ LS C

ηS

↺

θS

!  L′S C

ηS

/ LS C . O

↺ ′ ηS

′ θS

! L′S C

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′ ηS

C

Applying the second axiom of the definition of left localisation of model category again for θS θS′ and θS′ θS shows that θS θS′ = idLS C , and θS′ θS = idL′S C . Hence, the two left localisations are isomorphic.



Remark 8.5. Notice that Quillen functors do not take care of functorial factorisations. Hence, If two isomorphism left (right) localisation of a model category with respect to a class of morphisms have also the same classes of weak equivalences, cofibrations, and fibrations. That does not imply that they do coincide, because they may have different functorial factorisations.

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To start with consider (C, W ) a category with weak equivalence, and S a class of morphism in C. The above definition of localisation of model category can be generalised evidently to a localisation of categories with weak equivalences. Then a naive idea of forming the localised category with weak equivalence (C, W ) localised by S is to consider the new class of weak equivalences being the union W ⋃ S, because one needs S to be weak equivalences in the localised category with weak equivalences. However, that does not work because, for example, the two-out-of-three property does not necessary hold for the class W ⋃ S. Of course one can take big enough class that contains S and W and satisfies the two-out-of-three property, like M orC. However, W should be enlarged by adjoining S in minimalistic way, as the definition 8.1 is given by (initial universal morphism)-like property. Here one can see the role of model categories; that having this idea in mind, they provide a workaround, that is particularly convenient in simplicial model categories. Where one fixes either of the classes of cofibrations or fibrations. Then, if S ⋃ W is to be a subclass of the new weak equivalences, lemma 7.17 provide a tool to define new fibrant or cofibrant objects, respectively. Then, applying the same lemma again extend S ⋃ W to the minimal possible class of weak equivalences for the previously fixed class of cofibrations or fibrations, respectively. The lifting properties provides then the third class of the new structure, and this is what is refereed to by the left or right Bousfield localisation, respectively. We see below that this gives rise to the desired localisation of simplicial model category. Therefore, we start by recalling S-(co)local objects and S-local equivalences which will be the new (co)fibrant objects and weak equivalences in the new structures, respectively. This is also possible in more general model categories as done in [Hir03]. However, all the model categories of our interest in motivic homotopy theory are simplicial model category, and this restriction simplifies arguments, hence the restriction. We need first to recall the below technical lemma in order to be able to recall the definition of S-(co)local objects and equivalence. Lemma 8.6. Let C be a simplicial model category, S a class of morphisms of C. ● Let Z be a fibrant object of C, i ∶ U → V a morphism of C. Then, having the induced morphism ̃ , Z) ̃i∗ ∶ Map(Ṽ , Z) → Map(U

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weak equivalence of simplicial sets for a cofibrant approximation ̃i of i is independent of the choice of the cofibrant approximation of i. ● Let W be a cofibrant object of C, p ∶ X → Y a morphism of C. Then, having the induced morphism ̂ → Map(W, Ŷ ) p̂∗ ∶ Map(W, X) weak equivalence of simplicial sets for a fibrant approximation p̂ of p is independent of the choice of the fibrant approximation of p. Proof. The two statements are dual. Therefore, we prove the first, then the second follows by duality. Let Z be a fibrant object in C, i ∶ U → V be a morphism in C, ̃i an arbitrary cofibrant approximation of i. Then, applying a cofibrant replacement functor Q′ on the defining square of ̃i yield a commutative diagram Q′ (̃ i)

̃) Q′ (U Q′ (̃ pU ̃) pU ̃

$ Q′ (U )

̃) / Q′ (U Q′ (̃ pṼ )

$ / Q′ (V )

Q′ (i) pṼ

pU

 ̃ U

 / Ṽ

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̃ i

̃U p ̃

$  U

̃Ṽ p

i

pV

$/  V.

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Morphisms of the left and right sides of the diagram are weak equivalences, either by definition of by the fact that cofibrant approximation functors preserve weak equivalences. Since all vertices of the above diagram, apart from U and V are cofibrant objects, then corollary 7.18 implies that Map(pŨ , Z), Map(pṼ , Z), Map(Q′ (̃ pŨ ), Z) and Map(Q′ (̃ pŨ ), Z) are weak equivalences of simplicial sets, then by the two-out-of-three property of simplicial sets Map(̃i, Z) is a weak equivalence of simplicial sets if and only if Map(Q′ (i), Z) is so. Since, both the cofibrant approximation functor Q′ and the cofibrant approximation ̃i of i are independent from each other, then having Map(̃i, Z) a weak equivalence of simplicial sets is independent of the choice of the cofibrant approximation ̃i of i.  Definition 8.7 (S-(co)local Objects). Let C be a simplicial model category, S a class of morphisms of C. Then, ● An object Z ∈ C is called S-local object if Z is a fibrant object such that for every morphism i ∶ U → V in S the induces ̃ , Z) ̃i∗ ∶ Map(Ṽ , Z) → Map(U is a weak equivalence of simplicial sets for a cofibrant approximation ̃i of i (and hence any, by lemma 8.6). If S consists of a single morphisms i, then Z is called i-local object. Moreover, if i ∶ ∅ → V , then Z called V -local or V -null object. ● An object W ∈ C is called S-colocal object if W is a cofibrant object such that for every morphism p ∶ X → Y in S the induces ̂ → Map(W, Ŷ ) p̂∗ ∶ Map(W, X) is a weak equivalence of simplicial sets for a fibrant approximation p̂ of p (and hence any, by lemma 8.6). If S consists of a single morphisms p, then W is called p-colocal object. Moreover, if p ∶ X → ∗, then W called X-colocal object. Definition 8.8 (S-local Equivalences). Let C be a simplicial model category, S a class of morphisms of C. Then, ● A morphism i ∶ U → V is called S-local equivalence if the induced morphism of simplicial sets ̃ , Z) ̃i∗ ∶ Map(Ṽ , Z) → Map(U is a weak equivalence of simplicial sets for every S-local object Z ∈ C, for a cofibrant approximation ̃i of i (and hence any, by lemma 8.6). If S consists of a single morphism j, then i is called j-local equivalence. Moreover, if j ∶ ∅ → W , then i called W -local equivalence.

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● A morphism p ∶ X → Y is called S-colocal equivalence if the induced morphism of simplicial sets ̂ → Map(W, Ŷ ) p̂∗ ∶ Map(W, X) is a weak equivalence of simplicial sets for every S-colocal object W ∈ C, for a cofibrant approximation p̂ of p (and hence any, by lemma 8.6). If S consists of a single morphisms q, then p is called q-colocal equivalence. Moreover, if q ∶ Z → ∗, then p called Z-colocal equivalence. Lemma 8.9. Let C be a simplicial model category, S a class of morphisms of C. Then, ● A morphism i ∶ U → V between cofibrant objects is an S-local equivalence if and only if the induced morphisms i∗ ∶ Map(V, Z) → Map(U, Z) is a weak equivalence of simplicial sets for every S-local object Z ∈ C. ● A morphism p ∶ X → Y between fibrant objects is an S-colocal equivalence if and only if the induced morphisms p∗ ∶ Map(W, X) → Map(W, Y ) is a weak equivalence of simplicial sets for every S-colocal object W ∈ C. Proof. Any morphisms i ∶ U → V and p ∶ X → Y that are between cofibrant objects and fibrant objects, respectively, are cofibrant and fibrant approximation of themselves, respectively. Hence, the statement of the lemma holds. 

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Lemma 8.10. Let C be a simplicial model category, S a class of morphisms of C. Then, ● A fibrant object Z ∈ C is an S-local object in C if and only if the induced morphisms i∗ ∶ Map(V, Z) → Map(U, Z)

is a weak equivalence of simplicial sets for every S-local equivalence i ∶ U → V between cofibrant objects. ● A cofibrant object W ∈ C is an S-colocal object in C if and only if the induced morphisms p∗ ∶ Map(W, X) → Map(W, Y )

is a weak equivalence of simplicial sets for every S-colocal equivalence p ∶ X → Y between fibrant objects.

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Proof. The two statements are dual. Thererfore, we prove the first, then the second follows by duality. Let Z ∈ C be an S-local object in C, then by the previous lemma i∗ is a weak equivalence of simplicial sets for every S-local equivalence i ∶ U → V between cofibrant objects. On the other hand assume that Z ∈ C is a fibrant object in C for which the induced morphisms i∗ ∶ Map(V, Z) → Map(U, Z)

are weak equivalences of simplicial sets for every S-local equivalence i ∶ U → V between cofibrant objects. Any morphism i ∈ S is an S-local equivalence, since cofibrant approximation preserve S-weak equivalences then so is any cofibrant approximation ̃i of i, which is between cofibrant objects. Then, by the assumption ̃i∗ is a weak equivalence of simplicial sets for every i ∈ S and Z ∈ C is an S-local object in C.  The above definitions can be given in more general settings, in terms of homotopy complexes, as in [Hir03, Def.3.1.4]. Lemma 8.11. Let C be a simplicial model category, S a class of morphisms of C. Then, a weak equivalence in C is both S-local equivalence and S-colocal equivalence. Proof. Since S-local objects are fibrant object in C and S-colocal objects are cofibrant object in C, then the above statement is a straightforward result from the above definitions and lemma 7.17.  In other wards, classes of S-local equivalence and S-colocal equivalence contain weak equivalences in C. Also, by their very definition they contain all morphisms of S, hence they contain W ⋃ S. We see later that S-local equivalence serves as the class of weak equivalences of the left localisation of C with respect to S, if exists. Whereas S-colocal equivalence will form the class of weak equivalences of the right localisation of C with respect to S, if exists. Since S-local equivalences or S-colocal equivalences are expected to play the role of weak equivalences in the left or right localised model category, respectively. Therefore, we start by verifying that they do indeed satisfies the model categories axioms that are solely related to weak equivalence, namely the two-out-of-three axiom and closeness under retract. Lemma 8.12. Let C be a simplicial model category, S a class of morphisms of C. Then, both classes of S-local equivalences and S-colocal equivalences satisfy the two-out-of-three axiom.

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Proof. We prove the statement for S-local equivalences, and the proof for S-colocal equivalences holds by duality. Consider the commutative diagram of morphisms in C 5/ W

ji

U (

i

j

V.

Then, since we can use any cofibrant approximation, by lemma 8.6, we apply the cofibrant replacement functor Q of C. Then, we have the diagram of morphisms between cofibrant objects in C Q(j)Q(i)

Q(U ) Q(i)

)

/ Q(W ) . 4 Q(j)

Q(V )

Applying the functor Map(−, Z) ∶ Cop → sSet yield a commutative diagram of simplicial set for every S-local object Z in C Q(i)∗ Q(j)∗

Map(Q(U ), Z) ok Q(i)

∗

Map(Q(V ))

s

Map(Q(W ), Z) . Q(j)

∗

∗

∗

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If two morphism out of i, j, and ji are S-local equivalences, then two morphism out of Q(i) , Q(j) , and ∗ ∗ Q(i) Q(j) are weak equivalences of simplicial sets for every S-local object Z, hence the other morphisms is so. Therefore, the other morphism of i, j, and ji is an S-local equivalence.  Lemma 8.13. Let C be a simplicial model category, S a class of morphisms of C. Then, both classes of S-local equivalences and S-colocal equivalences are closed under retract. Proof. We prove the statement for S-local equivalences, and the proof for S-colocal equivalences holds by duality. Let i′ ∶ U ′ → V ′ be an S-local equivalence in C, and let i ∶ U → V be a retract of i′ , i.e. there exists a commutative diagram of morphisms in C

D

U

i

 V

d0

/ U′

r0

D

i′

R

d1

 / V′

r1

/U

 /V

i

with horizontal composite being identities. Applying the functor Map(Q(−), Z) ∶ Cop → sSet yield a commutative diagram of simplicial set for every S-local object Z in C Map(Q(U ), Z) o O Q(i)∗

Map(Q(V ), Z) o

Q(d0 )∗

Map(Q(U ′ ), Z) o O

Q(r0 )∗

Q(D)∗

Q(i′ )∗

Q(R)∗

Q(d1 )∗

Map(Q(V ′ ), Z) o ∗

Q(r1 )∗

Map(Q(U ), Z) O Q(i)∗

Map(Q(V ), Z) ∗

with horizontal composite being identities. Hence, Q(i) is a retract of Q(i′ ) in sSet. Since i′ is an S-local ∗ equivalence, then Q(i′ ) is a weak equivalence of simplicial sets for every S-local object Z. Weak equivalence ∗ of simplicial sets is closed under retract, hence Q(i) is a weak equivalence of simplicial sets for every S-local object Z, i.e. i is an S-local equivalence.  The construction of left or right localisation of model category C with respect to the class S of morphisms of C involves both a construction of a model category and left or right Quillen functors, ηS or S , whose total derived functors sends images of morphisms of S in HC to isomorphisms of the localised homotopy category. The below theorem shows that S-local objects and equivalences or S-colocal objects and equivalences provide tools to determine whether a total left (right) derived functor for a left (right) Quillen functor inverts the image of the class of morphisms S in HC by examining the original Quillen functor between model categories, before passing to their homotopy categories. Hence, they provide a tool to determine if given ηS or S satisfies the first axiom of the definition of localisation of model categories. Again, this can be done in full generality as in [Hir03].

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Theorem 8.14. Let (F, G, φ) ∶ C → D be a Quillen adjunction of simplicial model categories. (1) Let S be a class of morphisms of C, Q′ a cofibrant approximation functor in C. Then, the following statements are equivalent: (a) The total left derived functor LQ′ F ∶ HC → HD takes images of elements of S in HC into isomorphisms in HD. (b) The functor F takes the Q′ cofibrant approximation of elements of S to weak equivalences in D. (c) The functor G takes fibrant objects in D into S-local objects in C. (d) The functor F takes S-local equivalences between cofibrant objects into weak equivalences in D. (2) Let T be a class of morphisms of D, R′ a fibrant approximation functor in D. Then, the following statements are equivalent: (a) The total right derived functor RR′ G ∶ HD → HC takes images of elements of T in HD into isomorphisms in HC. (b) The functor G takes the R′ fibrant approximation of elements of T to weak equivalences in C. (c) The functor F takes cofibrant objects in C into T -colocal objects in D. (d) The functor G takes T -local equivalences between fibrant objects into weak equivalences in C.

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Proof. The proof of 2 is dual of the proof of 1. Therefore, we prove 1, then 2 holds by duality. (a) ⇐⇒ (b) The equivalence of (a) and (b) is a straightforward result from the definition of the total left derived functor with respect to Q′ . (b) ⇐⇒ (c) (b) means that for every i ∈ S, F (Q′ (i)) is a weak equivalence. Since F is a left Quillen functor then F (Q′ (i)) is morphisms between cofibrant objects in D. Then, corollary 7.18 implies that (b) is equivalent to having the induced morphism ∗

F (Q′ (i)) ∶ Map (F (Q′ (V )), Z) → Map (F (Q′ (U )), Z) a weak equivalence of simplicial sets for every fibrant object Z in D for every i ∈ S. Since (F, G, φ) is a Quillen adjunction of simplicial model categories then that is equivalent to having the induced morphism ∗ Q′ (i) ∶ Map(Q′ (V ), G(Z)) → Map(Q′ (U ), G(Z)) a weak equivalence of simplicial sets for every fibrant object Z in D for every i ∈ S. That in turn is equivalent to having G(Z) an S-local object for for every fibrant object Z in D, i.e. equivalent to (c). (c) ⇐⇒ (d) (c) means that for every fibrant object Z ∈ D, G(Z) is an S-object in C, i.e. the induced morphism

D

i∗ ∶ Map(V, G(Z)) → Map(U, G(Z))

a weak equivalence of simplicial sets for every S-local equivalence i ∶ U → V between cofibrant objects, due to lemma 8.10. Since (F, G, φ) is a Quillen adjunction of simplicial model categories then that is equivalent to having the induced morphism ∗

F (i) ∶ Map(F (V ), Z) → Map(F (U ), Z) a weak equivalence of simplicial sets for every S-local equivalence i ∶ U → V between cofibrant objects and for every fibrant object Z ∈ D. Since F is a left Quillen functor then F (i) is morphisms between cofibrant objects in D. Then, corollary 7.18 implies that the above is equivalent to F taking S-local equivalences between cofibrant objects into weak equivalences in D, i.e. to (d).  Corollary 8.15. Let C be a simplicial model category, S a class of morphisms of C. Then, ● If (LS C, ηS ) is a left localisation of C with respect to S then it satisfies the conditions of definition 8.1 for any cofibrant approximation functor Q′ . ● If (RS C, S ) is a right localisation of C with respect to S then it satisfies the conditions of definition 8.1 for any fibrant approximation functor R′ . Proof. The two statements are dual. Therefore, we prove the first, then the second follows by duality. Assume that (LS C, ηS ) is a left localisation of C with respect to S. Then, it satisfies the conditions of definition 8.1 for a cofibrant approximation functor Q′ . Let Q′′ be another cofibrant approximation functor. Since the conditions are satisfied for Q′ , then ● LQ′ ηS takes images of elements of S in HC into isomorphisms in H LS C. By the above theorem 8.14, that is equivalent to ηS taking S-local equivalences between cofibrant objects into weak equivalences in LS C. then, applying the same theorem again for the cofibrant replacement Q′′ yield that LQ′′ ηS takes images of elements of S in HC into isomorphisms in H LS C.

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● Similarly, using theorem 8.14 shows that having a left Quillen functor θ ∶ C → D such that LQ′ θ taking images of elements of S in HC into isomorphisms in HD is equivalent to LQ′′ θ taking images of elements of S in HC into isomorphisms in HD.  S-(co)local objects and S-(co)local equivalences are the main objects and morphisms of interest in the localised model category, particularly Bousfield localisation with respect to S, if exists, as seen in the above theorem. Therefore, it is important to develop an understanding of the S-(co)local objects and S-(co)local equivalences, and the rest of this section is devoted to present results that provide better understanding of these objects and morphisms in special case of interest. This will prove particularly practical in the prove of the existence of left Bousfield localisation with respect to a set of morphisms in a left proper cellular simplicial model category. The next section will also provide better understanding of these objects and morphisms. Lemma 8.16. Let C be a simplicial model category, S a class of morphisms of C. ● A fibrant object in C is an S-local object if it is weakly equivalent to an S-local object in C. ● A cofibrant object in C is an S-colocal object if it is weakly equivalent to an S-colocal object in C.

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Proof. The two statements are dual. Therefore, we prove the first, then the second follows by duality. One direction is evident, as any object is weakly equivalent to itself. To get started for the other direction let f ∶ X → Y be a weak equivalence between fibrant objects, one of which is an S-local object. Then, for every morphism i ∶ U → V in S, we have the commutative square Map(Q(V ), X)

Q(i)∗

/ Map(Q(U ), X)

f∗

 Map(Q(V ), Y )

Q(i)∗

f∗

 / Map(Q(U ), Y ).

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Since Q(V ) and Q(U ) are cofibrant objects and f is a weak equivalence between fibrant objects, then corollary 7.18 implies that vertical morphisms are weak equivalences. Since weak equivalences of simplicial sets satisfy the two-out-of-three property then upper morphism is a weak equivalence if and only if the lower morphism is so. Therefore, X is an S-local object if and only if Y is an S-local objects. Now, let X and Y be weakly equivalent fibrant objects such that one of them is an S-local object, then there exist a zigg-zagg of weak equivalences of length n < ∞ X = Z0

Z2 #

Z1

~

Zn = Y " z Zn−1

Then, applying a fibrant approximation functor R′ , one has the commutative diagram Z2

X = Z0 %

Z1

& y

 R′ (Z0 )

Zn = Y

Zn−1

x

 R′ (Z2 ) $  z R′ (Z1 )

 R′ (Zn ) %  y R′ (Zn−1 )

with vertical morphisms being weak equivalences. Since fibrant approximation functors preserve weak equivalences, then applying the above result on the evident set of weak equivalences between fibrant objects connecting X and Y in the above diagram proves the statement of the lemma.  Lemma 8.17. Let C be a simplicial model category, S be a class of morphism of C. Then,

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● A cofibration i ∶ U → V between cofibrant objects in C is an S-local equivalence if and only if it has n n the LLP with respect to canonical morphisms ∂[n] ∗ ∶ Z ∆ → Z ∂∆ for every S-local object Z in C and for every integer n ≥ 0. ● A fibration p ∶ X → Y between fibrant objects in C is an S-colocal equivalence if and only if it has the RLP with respect to canonical morphisms idW ⊗ ∂[n] ∶ W ⊗ ∂∆n → W ⊗ ∆n for every S-colocal object W in C and for every integer n ≥ 0. Proof. The two statements are dual. Therefore, we prove the first, then the second follows by duality. Let i ∶ U → V be a cofibration between cofibrant objects. Then, the induced morphism i∗ ∶ Map(V, Z) → Map(U, Z) is a Kan fibration for every fibrant object Z ∈ C be lemma 7.13. Since i ∶ U → V is between cofibrant objects, then it is an S-local equivalence if and only if the induced morphisms i∗ are weak equivalence of simplicial sets for every S-local object Z ∈ C, by lemma ??. Then, that is equivalent to having i∗ weak Kan fibrations for every S-local object Z ∈ C. Which is equivalent to having all commutative squares ∂∆ n_

/ Map(V, Z)

d0

∂[n]

i∗

 / Map(U, Z)

RA FT

 ∆n

d1

admitting lifts, for every Z ∈ C and every integer n ≥ 0. Since, C is a simplicial model category, then the adjunction described in (27) illustrates that above squares are in one-to-one correspondence with squares U

d′0

/ Z ∆n

i

 V



d′0

∂[n] ∗

/ Z ∂∆n

D

For every Z ∈ C and for every integer n ≥ 0. Moreover, the same adjunction shows one-to-one correspondence between lifts of both collection of squares. Hence, i is an S-local equivalence if and only if it has the LLP with respect to all ∂[n] ∗ for every S-local object Z ∈ C and every integer n ≥ 0.  Corollary 8.18. Let C be a simplicial model category, S be a class of morphism of C. Then, ● A morphism i ∶ U → V in C is an S-local equivalence if and only if a cofibration cofibrant approximation ̃i of i has the LLP with respect to canonical morphisms ∂[n] ∗ ∶ Z ∆n → Z ∂∆n for every S-local object Z in C and for every integer n ≥ 0. ● A morphism p ∶ X → Y in C is an S-colocal equivalence if and only if a fibration fibrant approximation p̂ of p has the RLP with respect to canonical morphisms idW ⊗ ∂[n] ∶ W ⊗ ∂∆n → W ⊗ ∆n for every S-colocal object W in C and for every integer n ≥ 0. Proof. The two statements are dual. Therefore, we prove the first, then the second follows by duality. Consider the commutative diagram of the cofibration cofibrant approximation ̃i of i ̃ U  U

̃ i

i

/ Ṽ  / V.

Since vertical morphisms are weak equivalences, and S-local morphisms satisfy the two-out-of-three morphisms then i is an S-local morphism if and only if ̃i is so. Since ̃i is a cofibration between cofibrant objects, then the above lemma proves the statement of the corollary.  Lemma 8.19. Let C be a simplicial model category, S be a class of morphism of C. Then, ● A cofibration i ∶ U → V between cofibrant objects in C is an S-local equivalence if and only if i ⊗ idK ∶ U ⊗ K → V ⊗ K is an S-local equivalence for every simplicial set K. ● A fibration p ∶ X → Y between fibrant objects in C is an S-colocal equivalence if and only if pidL ∶ X L → Y L is an S-local equivalence for every simplicial set L.

95

Proof. The two statements are dual. Therefore, we prove the first, then the second follows by duality. Let i ∶ U → V be a cofibration between cofibrant objects, K a simplicial set, and Z a fibrant object in C. Since C is a model category, then the adjunction (26) induces the below commutative square ≅

Map(V ⊗ K, Z)

/ MapsSet (K, Map(V, Z)) (i∗ )∗

(i⊗idK )∗

 Map(U ⊗ K, Z)

 / MapsSet (K, Map(U, Z))

≅

with horizontal morphisms being the adjunction isomorphisms. Hence, the left morphism is a weak equivalence of simplicial sets if and only if the right morphism is so. Recall that i∗ is a Kan fibration, because Z is a fibrant object and i ∶ U → V is a cofibrant in the simplicial model category C. Since, i is between cofibrant objects, then it is an S-local equivalence if and only if i∗ is a weak equivalence of simplicial sets, hence a weak Kan fibration, for every S-local object Z ∈ C. Then, by corollary 3.11 that is equivalent to having the right morphism (i∗ )∗ a weak Kan fibration for every S-local object Z ∈ C, which is equivalent to having the left ∗ morphism (i ⊗ idK ) being weak Kan fibration for every S-local object Z ∈ C. Since C is simplicial model category and i is cofibration, then so is i ⊗ idK is a cofibration between cofibrant objects, by lemma 7.9. Hence, the above equivalent statements are equivalent to having i ⊗ idK an S-local equivalence. 

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Corollary 8.20. Let C be a simplicial model category, S be a class of morphism of C. Then, ̃ ⊗ K → Ṽ ⊗ K is an ● A morphism i ∶ U → V in C is an S-local equivalence if and only if ̃i ⊗ idK ∶ U S-local equivalence for every simplicial set K and for a cofibration cofibrant approximation ̃i of i. ̂ L → Ŷ L is an S-local ● A morphism p ∶ X → Y in C is an S-colocal equivalence if and only if p̂idL ∶ X equivalence for every simplicial set L and for a fibration fibrant approximation p̂ of p. Proof. The prove is identical to the proof of corollary 8.18, using lemma 8.19 instead of lemma 8.17.



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In the below study of Bousfield localisation in section 8.2, we will restrict attention to left (right) proper cellular simplicial model categories. Therefore, it is important to understand how S-(co)local objects and S-(co)local equivalences behave under such constrains. In fact we are more interested in left proper model categories. However, dual arguments hold for the right proper. The below two lemmas are particularly useful to be use with Joyal Trick show the existence of left Bousfield localisation. Lemma 8.21. Let C be a simplicial model category, S a class of morphisms of C. ● If C is left proper, then the class of cofibrations that are S-local equivalences is closed under push-outs. ● If C is right proper, then the class of fibrations that are S-colocal equivalences is closed under pullbacks. Proof. The two statements are dual. Therefore, we prove the first, then the second follows by duality. Let i ∶ U → V be a cofibration in C that is an S-weak equivalence and j ∶ U → U ′ a morphism in C, and consider the push-out square j

U i

 V

/ U′ i′

j′

⌟  / V′

where V = U ′ ∐ V . Since cofibrations are closed under push-outs, then i′ is cofibration. We will use corollary U

̃ → Ṽ be a cofibration cofibrant approximation of i. 8.18 to show that i′ is also an S-local equivalence. Let ̃i ∶ U Since i is a cofibration and C is left proper then lemma 6.2 indicates that there is exist a cofibration cofibrant ̃′ → V ̃′ of i′ that is the push out of ̃i along α(U ̃ → U → U ′ ). To apply corollary 8.18, let Z approximation ĩ′ ∶ U be an S-local object and n ≥ 0 an integer. Then every commutative square ̃′ U ĩ′

 ̃ V′

d0

/ Z ∆n

D



d1

∂[n] ∗

/ Z ∂∆n

96

extend to the commutative square ̃ j

̃ U

d0

ĩ′

̃ i

 Ṽ

̃′ /U

j̃′

/ Z ∆n ∂[n] ∗

⌟  ̃′ /V

d1

 / Z ∂∆n

n Since i is an S-local equivalence, then corollary 8.18 implies that outer diagram admits a lift h ∶ Ṽ → Z ∆ . n ̃′ → Z ∆ such that Then, the universal property of push-out there exist a unique morphisms h′ ∶ V

h′ i′ = d0

h′ j ′ = h.

, and

then, using the universal property of push-out again for the morphism ∂[n] ∗ and d1 shows they coincide, hence h′ is a lift for the square D. Then, by corollary 8.18, i′ is an S-local equivalence.  Lemma 8.22. Let C be a left proper simplicial model category, S a class of morphisms of C. Then, the class of cofibrations that are S-local equivalences is closed under transfinite composition. Proof. See [Hir03, prop.3.2.11]



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8.1. Localisation of Objects. Weak S-(co)local Whitehead theorem, below, shows that weak equivalences detects S-(co)local equivalence between S-(co)local objects. Therefore, if there exist a coaugmented functor (LS ∶ C → C, jS ) such that LS (U ) is an S-local object for every U ∈ C and jS,U is an S-local equivalence for every U ∈ C. Then, since S-local equivalences satisfy the two-out-of-three property then a morphism i ∶ U → V is an S-local equivalence if and only if jS,X ∶ LS (U ) → LS (V ) is a weak equivalence. The dual argument applies for S-colocal objects. Therefore, we start by recalling weak S-(co)local Whitehead theorem. Theorem 8.23 (Weak S-(co)local Whitehead theorem). Let C be a simplicial model category, S be a class of morphism of C. Then, ● If U and V are S-local objects, then i ∶ U → V is an S-local equivalence if and only if it is a weak equivalence. ● If X and Y are S-colocal objects, then p ∶ X → Y is an S-colocal equivalence if and only if it is a weak equivalence.

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Proof. The two statements are dual. Therefore, we prove the first, then the second follows by duality. Let i ∶ U → V be a morphism in C between S-local objects, lemma 8.11, shows that i is an S-local equivalence if it is a weak equivalence. For the other direction, assume that i is an S-local equivalence. Consider the fibrant cofibrant replacement of i, i.e. consider the commutative diagram Q(U ) Q(i) pU

iQ(U )

# Q(V )

iU

i

R(Q(i))

& / R(Q(V ))

iQ(V ) R(pU )

pV

 U

/ R(Q(U ))

 / R(U )

R(pV ) R(i)

#  V

iV

&  / R(V )

iU , pU , iQ(U ) , iV , pV and iQ(V ) are weak equivalences, then by the two-out-of-three property of weak equivalences, R(pU ) and R(pV ) are so. Lemma 8.16 implies that R(U ) and R(Q(U )) are S-local objects for being fibrant objects that are weakly equivalent to the the S-local object U . Similarly, R(V ) and R(Q(V )) are S-local objects. Since morphisms in the front and back face of the diagram are weak equivalence, i is an S-local equivalence by assumption, and S-local equivalences satisfy the two-out-of-three property, then R(Q(i)) is an S-local equivalence. Moreover, R(Q(U )), R(Q(V )) are also cofibrant objects because Q(U ), Q(V ) are cofibrant objects, iQ(U ) , iQ(V ) are cofibrations, and cofibrations form a subcategory. Having R(Q(i)) an S-local equivalence between S-local objects that are cofibrant objects induces the below two weak equivalences of simplicial sets ∗ R(Q(i)) ∶ Map(R(Q(V )), R(Q(U ))) → Map(R(Q(U )), R(Q(U )))

97 ∗

R(Q(i)) ∶ Map(R(Q(V )), R(Q(V ))) → Map(R(Q(U )), R(Q(V ))). Therefore, in particular we have a bijection of the connected component sets ∗

π0 (R(Q(i)) ) ∶ π0 (Map(R(Q(V )), R(Q(U )))) → π0 (Map(R(Q(U )), R(Q(U )))) ∗

π0 (R(Q(i)) ) ∶ π0 (Map(R(Q(V )), R(Q(V )))) → π0 (Map(R(Q(U )), R(Q(V )))) . Hence, R(Q(i)) is a homotopy equivalence, by the evident simplicial homotopy induced from the above bijections. Since homotopy equivalences between cofibrant-fibrant objects are weak equivalences, by lemma ??, [Hir03, Th.7.8.5], then R(Q(i)) is a weak equivalence. Then, using the two-out-of-three of weak equivalence on the above diagram shows that i is a weak equivalence.  The above argument gives rise to the below definition. In fact, there is an even more natural and intrinsic motivation for S-localisation, that the localisation of model categories in general is motivated by the localising topological spaces homology groups, which are special cases of S-localisations functors.

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Definition 8.24 (S-localisation). Let C be a simplicial model category, S be a class of morphism of C. Then, (1) (a) An S-localisation of an object U ∈ C is an S-local equivalence jU ∶ U → U ′ where U ′ is an S-local object. Moreover, an S-localisation jU of U is called a cofibrant S-localisation of U if jU is also a cofibration. (b) An S-localisation of a morphism i ∶ U → V is a morphism (i, i′ ) ∶ jU → jV in MorC from an S-localisation jU of U to an S-localisation jV of V . (c) A (cofibrant) S-localisation functor is a coaugmented functor (LS ∶ C → C, jS ∶ idC → LS ) such that jS,U is a (cofibrant) S-localisation of U, ∀U ∈ C. (d) Let S ′ be a set of morphisms of C. Then, a relative S- cell S-localisation functor is an S-localisation functor (LS ∶ C → C, jS ∶ idC → LS ) such that jS,U is a relative S- cell complex ∀U ∈ C. (2) (a) An S-colocalisation of an object X ∈ C is an S-colocal equivalence qX ∶ X ′ → X where X ′ is an S-colocal object. Moreover, an S-colocalisation qX of X is called a fibrant S-colocalisation of X if qX is also a fibration. (b) An S-colocalisation of a morphism f ∶ X → Y is a morphism (f ′ , f ) ∶ qX → qY in MorC from an S-colocalisation qX of X to an S-colocalisation qY of Y . (c) A (fibrant) S-colocalisation functor is a augmented functor (coLS ∶ C → C, qS ∶ coLS → idC ) such that qS,X is a (fibrant) S-colocalisation of X, ∀X ∈ C. (d) Let S ′ be a set of morphisms of C. Then, a relative S-cocell S-colocalisation functor is an S-colocalisation functor (coLS ∶ C → C, qS ∶ coLS → idC ) such that qS,X is a relative S-cocell complex ∀X ∈ C. An S-(co)localisation functor encodes all the information about all S-local equivalences, where it allows us to understand S-local equivalences through weak equivalence of the original model category as seen from the below lemma. Lemma 8.25. Let C be a simplicial model category, S be a class of morphism of C. (1) If (i, i′ ) ∶ jU → jV is an S-localisation of i ∶ U → V , then i is an S-local equivalence if and only if i′ is a weak equivalence. (2) If (f ′ , f ) ∶ qX → qX is an S-colocalisation of f ∶ X → Y , then f is an S-colocal equivalence if and only if f i′ is a weak equivalence. Proof. It is enough to prove either of the statements, then the other follows by duality. Let (i, i′ ) ∶ jU → jV be an S-localisation of i ∶ U → V , then we have the commutative diagram U

jU

i

 V

/ U′ i′

jV

 / V′

with horizontal morphisms being S-local equivalence. Since the class of S-local equivalences satisfies the twoout-of-three property, then i is an S-local equivalence if and only if i′ is an S-local equivalence. Since i′ is between S-local objects, then by the weak S-local Whitehead theorem 8.23 it is an S-local equivalence if and only if it is a weak equivalence. 

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T

Therefore, it becomes natural to look for S-(co)localisation functors to gain better understanding of S-(co)local equivalences. Recall that in the definitions of S-(co)local objects and S-(co)local equivalences, S-(co)local objects are thought of as (co)fibrant objects in the localised model category. Hence, for example, if a left localisation LS C exists, it might be natural for one to expect S-local objects to coincide with fibrant objects in LS C. That is not correct in general, as illustrated in the counter example [ibid., Ex.2.1.6], but if C if left proper, then they do coincide, as seen in lemma 8.43. In such case, if there to be LS C with its weak equivalences and fibrant objects being S-local equivalences and S-local objects, respectively, then its fibrant replacement functor (which is induced by the functorial factorisation of LS C as composition of fibrations with weak cofibrations in LS C) is a cofibrant S-localisation. That makes it natural to look for S-localisation functors that are induced from such functorial factorisations, which in most ’nice’ cases are realised as a result of the transfinite small object argument for an appropriate choice of set of morphisms. Hence, one can start by looking for a set of morphisms that induces a cofibrant S-localisation functors through the transfinite small object argument. Denote a generic such desired set of morphism by J S , it should admits the small object argument, J S - cell should consists of cofibrations that are S-local equivalence, so in particular J S ⊂ J S - cell should consists of cofibrations that are S-local equivalence, and for every Z ∈ C, dom βJ S (∗Z ) should be an S-local object for the unique morphism ∗Z ∶ Z → ∗. A fibrant object Z in C is an S-local object if and only if the induced morphisms ̃ , Z) ̃i∗ ∶ Map(Ṽ , Z) → Map(U are weak equivalences of simplicial sets for some cofibrant approximation ̃i of i for every i ∶ U → V in S. In particular if the cofibrant approximations are chosen to be cofibration cofibrant approximations, then all the above induced morphisms are Kan fibrations, by lemma 7.13, hence a fibrant object Z in C is an S-local object if and only if the induced morphisms ̃i∗ are weak Kan fibrations of simplicial sets for some cofibration cofibrant approximation ̃i of i for every i ∶ U → V in S, i.e.if and only if these morphisms have the RLP with respect to the set of canonical inclusions of simplicial sets {∂[n] ∶ ∂∆n ↪ ∆n ∣n ≥ 0}.

for every i ∶ U → V in S. Then, by lemma 7.15, this is equivalent to the unique morphism Z → ∗ having the RLP with respect to class ̃ ⊗ ∆n ∐ Ṽ ⊗ ∂∆n → Ṽ ⊗ ∆n ∣i ∶ U → V ∈ S, n ≥ 0}, Λ(S) ∶= {̃i ◻ ∂[n] ∶ U ̃ ⊗∂∆n U

Where ̃i is some cofibration cofibrant approximation of i. Therefore, the desired set J S has at least to capture the information encoded in the set Λ(S). A dual argument applies for left localisation. Remark 8.26. The above observation will be the base point, based on which, two examples of J S are constructed. The construction is understood in the settings of left cellular simplicial model categories, and that is sufficient in the study of motivic homotopy theory. In a cofibrantly generated simplicial model category (C, I, J), J already encodes all fibrations, hence fibrant objects, therefore J is twisted to form the desired set J S . This goes as follows, starting from a set S of morphisms of C, one forms Λ(S) the set of S-horns, this set encodes enough data to determine if a fibrant object in C is an S-local object, however, it does not have enough data to determine S-local object not knowing already they are fibrant objects in C. Hence, it is enlarged by J to form the set of augmented S-horns. This sets now contains enough data to determine S-local object by itself. Moreover, it admits the small object argument giving that the original cofibrantly generated model category (C, I, J) is nice enough, for example cellular with dom(J) are cofibrant objects, which gives rise to cofibrant S-localisation functor. This set can be also refined further to obtain inclusions of sub-complexes S-localisation functor and that is practically useful because inclusions of sub-complexes enable us to use settheoretical argument in the settings of abstract categories. In particular, in cellular model categories, the have the advantage of being closed under pull-back, push-forward and they give description of pull-backs and pushforwards by the set-theoretical operators of intersection and union. In addition, they provide the notation of boundedness, which will be the key point in proving the existence of left Bousfield localisation. For the above argument to work, S has to be a set, otherwise the small object argument can not be applied. Avoid the confusion, neither of the above sets contains enough information to play the role of generating weak cofibration of the left localised category LS C, if exists, particularly not every cofibration that is an S-weak equivalence is necessary given as a cofibration of any of these sets. Also, notice that once the model category is considered to be cofibrantly generated then duality argument fails,

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in such case left and right localisation should be considered separately, and here we are more interested in left localisation. Definition 8.27 (S-(Co)horns). Let C be a simplicial model category. ● Let i ∶ U → V be a morphism in C. Then a horn on i is defined to be a canonical morphism ̃ ◻ ∂[n] ∶ U ̃ ⊗ ∆n ∐ Ṽ ⊗ ∂∆n → Ṽ ⊗ ∆n Q(i) ̃ ⊗∂∆n U

̃ ̃ → Ṽ is the cofibration cofibrant replacement for i ∶ U → V in C, for some n ≥ 0, where Q(i) ∶ U definition 1.38, with ∂[n] ∶ ∂∆n ↪ ∆n being the canonical embedding. Let be S a class of morphisms in C. Then, Λ(S), the class of S-horns is defined to be the class of all horns, for every n ≥ 0, on all morphisms of S. ● Let p ∶ X → Y be a morphism in C. Then a cohorn on p is defined to be a canonical morphism ̂ ∂[n] ∶ X ̂ ∆n → Ŷ ∆n × ̂ ∂∆n X ̂ ∂∆n R(p) Y ̂ )∶X ̂ → Ŷ is the fibration fibrant replacement for f ∶ X → Y in C. Let S be for some n ≥ 0, where R(f a class of morphisms in C. Then, V(S), the class of S-cohorns is defined to be the class of all cohorns, for every n ≥ 0, on all morphisms of S.

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Actually, one can generalise the above definition to consider any cofibration cofibrant approximation ̃i of i or fibration fibrant approximation p̂ of p. But, that does not seem to be more practical within the scale of these notes. The class of horns is called after the set of generating weak equivalences in spaces (simplicial sets, topological spaces,...), as it ’almost’ generate weak cofibrations in left Bousfield localisation, if exists. Hereafter, we examine if classes of S-(co)horns satisfy the desired properties of the set J S , if it fails to satisfy any of the conditions, the set of S-(co)horns will be altered to remedy the conditions it violates. This procedure will iterate tell we get J S . Lemma 8.28. Let C be a simplicial model category. Then, ● Horns on i ∶ U → V in C are cofibrations between cofibrant objects. ● Cohorns on p ∶ X → Y in C are fibrations between fibrant objects.

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Proof. The two statements are dual. Therefore, we prove the first, then the second follows by duality. ̃ ∶U ̃ → Ṽ is a cofibration between cofibrant objects in C, canonical inclusions ∂[n] ∶ ∂∆n ↪ ∆n are Since Q(i) ̃ ◻ ∂[n] cofibrations of simplicial sets for every n ≥ 0 and all simplicial sets are cofibrant, then by lemma 7.9, Q(i) is a cofibration between cofibrant objects for every n ≥ 0.  Lemma 8.29. Let C be a simplicial model category, S a class of morphisms. ● If C is also left proper, then S-horns are S-local equivalence. ● If C is also right proper, then S-cohorns are S-colocal equivalence. Proof. The two statements are dual. Therefore, we prove the first, then the second follows by duality. Let i ∶ U → V be a morphism in S, n ≥ 0 an integer. Since cofibrant approximations preserve S-weak ̃ ∶U ̃ → Ṽ is a cofibration between cofibrant objects that is S-weak equivalence. Consider equivalences then Q(i) the below diagram induced by the universal property of push-out ̃ ⊗ ∂∆n U

̃ Q(i)⊗id ∂∆n

/ Ṽ ⊗ ∆n

̃ ⊗∂[n] U

 ̃ ⊗ ∆n U

iU ̃ ⊗∆n

⌟

/ ●

̃ ⊗∂[n] U ̃ Q(i)◻∂ [n]

̃ Q(i)⊗id ∆n

%  . Ṽ ⊗ ∆n .

̃ ∶U ̃ → Ṽ is a cofibration Since every simplicial set is a cofibrant object in sSet, ∂[n] is a cofibration in sSet, Q(i) between cofibrant objects in C, then lemma 7.9, implies that the morphisms of the outer diagram are cofibrations between cofibrant objects in C. Since cofibrations are closed under push-outs, in addition to the above lemma ̃ 8.28, then all morphisms of the above diagram are cofibrations between cofibrant objects in C. Since Q(i) is a ̃ ̃ S-weak equivalence, then corollary 8.20 implies that Q(i)⊗id∂∆n and Q(i)⊗id∆n are S-weak equivalences. Since

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C is left proper, then cofibrations that are S-weak equivalences are closed under push-outs along cofibrations, by ̃ ⊗ id∂∆n is a cofibration that is a S-weak equivalence and U ̃ ⊗ ∂[n] is a cofibration, lemma 8.21, in particular Q(i) hence iŨ ⊗∆n is cofibration that is a S-weak equivalence. Since S-weak equivalences satisfy two-out-of-three ̃ ◻ ∂[n] is an S-weak equivalence, and hence S-horns are S-local equivalence. property then Q(i)  The essential part of verification is to examine if S-local objects are distinguished by the RLP with respect to Λ(S). Actually, Λ(S) cannot determine S-local objects by itself, and that is not surprising because it does not take care of J the set of generating weak cofibrations of C, if C is a cofibrantly generated. Hence, it fails to distinguish fibrations of C. However, it contains enough information to distinguish if it is an S-local object knowing that an object is already a fibrant object in C, as seen in the below lemma. Lemma 8.30. Let C be a simplicial model category, S a class of morphisms in C. ● Let Z ∈ C. Z is an S-local object in C if and only if Z is a fibrant object in C and the unique morphism Z → ∗ has the RLP with respect to the class of S-horns. ● Let W ∈ C. Z is an S-colocal object in C if and only if W is a cofibrant object in C and the unique morphism ∅ → Z has the LLP with respect to the class of S-cohorns.

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Proof. The two statements are dual. Therefore, we prove the first, then the second follows by duality. Assume that Z ∈ C is an S-local object in C. Then, by definition, it is a fibrant object in C. Also, the ̃ ∗ are weak equivalences of simplicial sets for every i ∈ S, where Q ̃ are the cofibration induced morphisms Q(i) ̃ ̃ ∗ are cofibrant replacement of C. Since Q(i) are cofibrations for every i ∈ S and Z is a fibrant object then Q(i) weak Kan fibrations of simplicial sets for every i ∈ S. Weak Kan fibrations of simplicial sets are precisely those morphisms that have the RLP with respect to canonical inclusions ∂[n] ∶ ∂∆n → ∆n for every n ≥ 0. Then, ̃ ∗ weak Kan fibrations of simplicial sets for every i ∈ S is equivalent to the unique by lemma 7.15, having Q(i) morphism Z → ∗ having the RLP with respect to the class of S-horns. On the other hand, assume that Z is a fibrant object in C and the unique morphism Z → ∗ has the RLP ̃ ∗ have the RLP with respect to all with respect to the class of S-horns. Then, by lemma 7.15, morphisms Q(i) ̃ ∗ are weak Kan fibrations for canonical inclusions ∂ n ∶ ∂∆n → ∆n for every n ≥ 0, for every i ∈ S. Hence, Q(i) every i ∈ S, so in particular weak equivalences of simplicial sets for every i ∈ S, hence Z is an S-local object in C. 

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Since, in cofibrantly generated model categories, the set of generating weak cofibrations determines fibrations in general, particularly fibrant objects, then it is natural to examine the case of cofibrantly generated simplicial model categories and consider the union of the class Λ(S) and the set J in cofibrantly generated simplicial model category as a candidate to be the desired set of generating weak cofibrations JS for LS C. Definition 8.31. Let (C, I, J) be a cofibrantly generated simplicial model category, S a class of morphisms. Then the class of augmented S-horns is defined to be the class of morphisms Λ(S) = Λ(S) ⋃ J. Lemma 8.32. Let (C, I, J) be a left proper cofibrantly generated simplicial model category, S a class of morphisms. Then, the class of augmented S-horns consists cofibrations that are S-weak equivalences. Proof. Since morphisms of Λ(S) are cofibrations that are S-weak equivalences by lemmas 8.28 and 8.29, and morphisms of J are weak cofibrations in C, hence cofibrations that are S-weak equivalences. Then, the class of augmented S-horns consists of cofibrations that are S-weak equivalences.  The class of augmented S-horns provides an appropriate solution to determine S-local objects, as seen in the below lemma. Lemma 8.33. Let (C, I, J) be a left proper cofibrantly generated simplicial model category, S a class of morphisms in C. An object Z ∈ C is an S-local object in C if and only if the unique morphism Z → ∗ has the RLP with respect to the class of augmented S-horns. Proof. By lemma 8.30, having Z ∈ C to be an S-local object is equivalent to having Z to be a fibrant object in C and the unique morphism Z → ∗ having the RLP with respect to the class Λ(S). Since (C, I, J) is cofibrantly generated, then having Z to be a fibrant object is equivalent to the unique morphism Z → ∗ having the RLP with respect to the set J. Hence, having Z ∈ C to be an S-local object is equivalent to unique morphism Z → ∗ having the RLP with respect to both set J and class Λ(S), i.e. having the RLP with respect to the class Λ(S) = λ(S) ⋃ J of augmented S-horns. 

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Recall that we are looking for a set J S , in order to hope it admits the small object argument, hence it is natural to restrict S to be a set, in which case Λ(S) is also a set. Then, all what is left to be able to construct a cofibrant S-localisation coming from a functorial factorisation is to show that J S admits the small object argument and show that morphisms of J S - cell are both cofibrations and S-local equivalence. Then, we can construct a cofibrant S-localisation of U through applying the deduced functorial factorisation to the unique morphism U → ∗. Lemma 8.34. Let (C, I, J) be a left proper cellular simplicial model category, such that dom(J) consists of cofibrant objects, S a set of morphisms in C. Then, Λ(S) admits the transfinite small object argument. Proof. This proof is based on the convenient property of cellular model categorises, that every cofibrant object in cellular model category is small relative to the class of its cofibrations, as in theorem 5.17. dom Λ(S) = dom Λ(S) ⋃ dom J dom J is given to consist of cofibrant objects, hence objects in dom J are small relative to the class of I- cof. On ̃ is cofibrant object for every i ∶ U → V in S, all simplicial sets are cofibrant objects in the other hand, since U ̃ ⊗ ∂∆n is cofibrant object for every n ≥ 0 and i ∶ U → V in S. Also, by sSet, then by lemma 7.9, in particular U lemma 7.9, the canonical morphisms ̃ ⊗ ∂∆n → U ̃ ⊗ ∆n id ̃ ◻ ∂ n ∶ U U

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̃ ◻ id∂∆n ∶ U ̃ ⊗ ∂∆n → Ṽ ⊗ ∂∆n Q(i) are cofibration for every n ≥ 0 and i ∶ U → V in S. Since cofibrations are given by I- cof, and I- cof is closed under push-outs then the push-outs injections ̃ ⊗ ∆n → U ̃ ⊗ ∆n ∐ Ṽ ⊗ ∂∆n U ̃ ⊗∂∆n U

are cofibration for every n ≥ 0 and i ∶ U → V in S. Since composition of I- cof are so, then the unique morphisms ̃ ⊗ ∆n ∐ Ṽ ⊗ ∂∆n ∅→U ̃ ⊗∂∆n U

̃ ⊗ ∆n are cofibration for every n ≥ 0 and i ∶ U → V in S, hence U

n ∐ Ṽ ⊗ ∂∆ are cofibrant objects for every ̃ ⊗∂∆n U

n ≥ 0 and i ∶ U → V in S. Therefore, objects in dom Λ(S) are small relative to the class of I- cof. Since morphisms of Λ(S) are cofibrations, i.e. in I- cof, and that I- cof is closed under pus-outs and transfinite composition, by lemma 4.27, then Λ(S)- cell ⊂ I- cof. Therefore, objects in dom Λ(S) are small relative to the class of Λ(S)- cell, i.e. Λ(S) admits the small object argument.  Lemma 8.35. Let (C, I, J) be a left proper cellular simplicial model category, such that dom(J) consists of cofibrant objects, S a set of morphisms in C. Then, morphisms of Λ(S)- cell are both cofibrations and S-local equivalences. Proof. Since C is left proper cellular simplicial model category, then morphisms of Λ(S) are cofibrations that are S-local equivalences. Since, C is left proper, then lemmas 8.21 and 8.22 imply that the class cofibrations that are S-local equivalences is closed under push-out and transfinite composition, i.e. closed under forming its relative cell complexes, hence morphisms of Λ(S)- cell are both cofibrations and S-local equivalences.  Remark 8.36. Let (C, I, J) be a left proper cellular simplicial model category, such that dom(J) consists of cofibrant objects, S a set of morphisms in C. Still, not every cofibration that S-local equivalence belongs to Λ(S)- cof, for example see [Hir03, Ex.2.1.6]. Remark 8.37. Let (C, I, J) be a left proper cellular simplicial model category, such that dom(J) consists of cofibrant objects, S a set of morphisms in C. Then, the above lemma 8.35 and the transfinite small object argument implies the existence of a functorial factorisation (αΛ(S) , βΛ(S) ) on C with αΛ(S) (f ) ∈ Λ(S)- cell and βΛ(S) (f ) ∈ Λ(S)- inj. Then, in particular, it gives rise to a coaugmented functor (LS , jS ) on C that sends each object U ∈ C to dom βΛ(S) (U → ∗) with jS,U = αΛ(S) (U → ∗). For every U ∈ C, LS (U ) is an S-local object and jS,U is a cofibration that is S-local equivalence, by lemmas 8.33 and 8.35, hence (LS , jS ) is a cofibrant S-localisation functor. Moreover, jS,U is a relative Λ(S)- cell complex for every U ∈ C. Applying lemma 8.25 above to the cofibrant S-localisation functor

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(LS , jS ) we see that a morphism i ∶ U → V is an S-local equivalence if and only if LS (i) is a weak equivalence in C. Also using 8.16, one sees that a fibrant object U is an S-local object if and only if it is weakly equivalent to its localisation LS (U ). Therefore, it is useful to find fibrant S-localisation that preserve inclusions of sub-complexes. In deed the cofibrant S-localisation of remark ?? preserve inclusions of sub-complexes. Lemma 8.38. Let (C, I, J) be a left proper cellular simplicial model category, such that dom(J) consists of cofibrant objects, S a set of morphisms in C, (LS , jS ) be the cofibrant localisation functor of remark ??. then, if i ∶ U → V is an inclusion of I- cell sub-complexes, the so is LS (i) ∶ LS (U ) → LS (V ). Proof. [Hir03, Prop.4.4.1 using 12.4.7].



8.2. Bousfield Localisation. Having a model category C with a class S of morphisms of C, Bousfield provided a canonical method, now is called Bousfield localisation, how to construct the localised model category with respect to S, providing that the model category satisfy some commonly occurring condition. These conditions varies, and we are interested in localising left proper cellular simplicial model category.

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Definition 8.39. Let (C, (W, C, F )) be a simplicial model category, S a class of morphisms. Then, (1) The class of S-local equivalences in C is called the S-local weak equivalences, denoted by WS . (2) The class of cofibrations in C is called the S-local cofibrations. (3) The class of morphisms in C that have the RLP with respect to the class of morphisms that are both S-local cofibrations and S-local weak equivalences is called the S-local fibrations, denoted by FS . If the unique morphism Z → ∗ is an S-local fibrations for Z ∈ C, then Z is called S-local fibrant. (4) The class of S-colocal equivalences in C is called the S-colocal weak equivalences, denoted by W S . (5) The class of fibrations in C is called the S-colocal fibrations. (6) The class of morphisms in C that have the LLP with respect to the class of morphisms that are both S-colocal fibrations and S-colocal weak equivalences is called the S-colocal cofibrations, denoted by C S . If the unique morphism ∅ → W is an S-colocal cofibrations for W ∈ C, then W is called S-colocal fibrant.

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Definition 8.40 (Bousfield Localisations). Let C be a simplicial model category, S a class of morphisms. (1) A left Bousfield localisation of C with respect to S, if exists, is a model category LS C on the underlying category of C such that its weak equivalences, cofibrations, and fibrations are the classes of S-local weak equivalences, S-local cofibrations, and S-local fibrations, respectively. (2) A right Bousfield localisation of C with respect to S, if exists, is a model category RS C on the underlying category of C such that its weak equivalences, cofibrations, and fibrations are the classes of S-colocal weak equivalences, S-colocal cofibrations, and S-colocal fibrations, respectively. Remark 8.41. Notice that given a category C and a class of morphisms S if a left (right) Bousfield localisation exist, it is not necessary unique that they might have different functorial factorisations, especially if the functorial factorisations are to be obtained by the transfinite small object argument. Lemma 8.42. Let C be a simplicial model category, S a class of morphisms of C. (1) Suppose that a left Bousfield localisation LS C exists, then the class of S-local fibrations is subclass of fibrations of C. Whereas the class of S-local weak fibrations coincides with the class of weak fibrations of C. Moreover, fibrant objects of LS C are S-local objects. (2) Suppose that a right Bousfield localisation RS C exists, then the class of S-colocal cofibrations is subclass of cofibrations of C. Whereas the class of S-colocal weak cofibrations coincides with the class of weak cofibrations of C. Moreover, cofibrant objects of RS C are S-colocal objects. Proof. The two statements are dual. Therefore, we prove the first, then the second follows by duality. Recall that in model categories fibrations are given by the RLP with respect to weak cofibrations, and weak fibrations are given by the RLP with respect to cofibrations. If LS C exists, then S-local fibrations are those morphisms of C that have the RLP with respect to the class of S-local weak cofibrations, so in particular they that the RLP with respect to the subclass of weak cofibrations of C, hence S-local fibrations are fibrations in C. Also, S-local weak fibrations are precisely those morphisms of C that have the RLP with respect to the class of S-local cofibrations, i.e. to the class of cofibrations in C. Hence, S-local weak fibrations are precisely weak fibrations in C. Let Z be a fibrant object in LS C, then it is a fibrant object in C because fibrations in LS C are fibrations in C. Since every i ∈ S is an S-local equivalence, i.e. a weak equivalence in LS C and

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cofibrant approximations in C are cofibrant approximations in LS C then lemma 7.17 implies that the induced morphisms ̃ , Z) ̃i∗ ∶ Map(Ṽ , Z) → Map(U are weak equivalences of simplicial sets for a cofibrant approximation ̃i of i in C, hence a cofibrant approximation of i in LS C, for every i ∈ S. Therefore, Z is an S-local object in C.  Avoid the confusion that might arise here. Although (co)fibrant objects of localised category with respect to S, if exists, are S-(co)local objects in C, the converse is not true in general. Still, if the original model category C is nice enough, for example, left (right) proper, then S-(co)local objects coincide with (co)fibrant objects in the localised category with respect to S, as seen in the below lemma. Lemma 8.43. Let C be a simplicial model category, S a class of morphisms of C. ● Suppose that C is left proper, and that a left Bousfield localisation LS C exists. Then, Z is a fibrant objects in LS C if and only if it is an S-local objects in C. ● Suppose that C is right proper, and that a right Bousfield localisation RS C exists. Then, W is a cofibrant objects in RS C if and only if it is an S-colocal objects in C.

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Proof. We prove the first statement, then the proof of the second statement is obtained by duality. Lemma 8.42 above shoes that if Z is a fibrant object in LS C then it is an S-local object in C. For the other direction, assume that Z ∈ C is an S-local object in C. Then, in particular Z is a fibrant object in C. In order to show that Z is a fibrant object in LS C, we need to show that the unique morphism Z → ∗ has the RLP with respect to all weak cofibrations in LS C, i.e. with respect to S-local weak cofibrations in C. Since Z is fibrant in C, then the unique morphism Z → ∗ is a fibration in C. Let i ∶ U → V an S-local weak cofibrations in C. Since C is a left proper Z → ∗ is a fibration in C and i ∶ U → V is a cofibration in C then lemma 6.3 states that it is enough to show that Z → ∗ has the RLP with respect to a cofibration cofibrant approximation of i in order ̃ → Ṽ be a cofibration cofibrant approximation of to show that Z → ∗ has the RLP with respect to i. Let ̃i ∶ U i. Since i is an S-local weak equivalence, and Z is an S-local object in C then 8.18 implies that ̃i has the LLP with respect to canonical morphisms n n id[n] ∗ ∶ Z ∆ → Z ∂∆ for every integer n ≥ 0. So in particular, for n = 0 we have Z ∆ = Z and Z ∂∆ = Z ∅ = ∗. Hence, ̃i has the LLP with respect to the unique morphism Z → ∗.  0

0

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Based on the lemma 8.42, one can see readily that that if left (right) Bousfield localisation exist, then the identity adjunction provides a Quillen adjunction between the original model category and the Bousfield localisation. Corollary 8.44. Let C be a simplicial model category, S a class of morphisms of C. (1) Suppose that a left Bousfield localisation LS C exists, then (idC , idC , ididC ) ∶ C → LS C is a Quillen adjunction. (2) Suppose that a right Bousfield localisation RS C exists, then (idC , idC , ididC ) ∶ RS C → C is a Quillen adjunction. As the name suggests, left (right) Bousfield localisations are indeed left (right) localisation of model categories as asserted in the below lemma. Theorem 8.45. Let C be a simplicial model category, S a class of morphisms of C. (1) Suppose that a left Bousfield localisation LS C exists, then (LS C, idC ) is a left localisation of C with respect to S. (2) Suppose that a right Bousfield localisation RS C exists, then (RS C, idC ) is a right localisation of C with respect to S. Proof. The two statements are dual. Therefore, we prove the first, then the second follows by duality. Suppose that the left Bousfield localisation LS C exists. By the previous corollary idC ∶ C → LS C is a left Quillen functor. Since the left idC ∶ C → LS C takes takes S-local equivalences between cofibrant objects in C into themselves, in particular into S-local weak equivalences, i.e. weak equivalences of LS C, then thereon 8.14 asserts that the left derived functor L idC takes images of morphisms of S in HC into isomorphisms in H LS C. Let (F, Q, ϕ) ∶ C → D be a Quillen adjunction such that the left derived functor L F takes images of morphisms of S in HC into isomorphisms in HD. Since the underlying categories of C and LS C coincides, then (F, Q, ϕ) ∶ LS C → D is an adjuction. We need to show that it is a Quillen adjunction. Using lemma 3.9

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it is sufficient to show that F preserves cofibrations and G preserves fibrations between fibrant objects. Since F ∶ C → D is a left Quillen functor, then it takes cofibrations in C into cofibrations in D. Cofibrations in LS C are the same of cofibrations in C, hence F ∶ LS C → D takes cofibrations in LS C into cofibrations in D. Let p ∶ X → Y be a fibration between fibrant objects in D, then by the assumption that L F takes images of morphisms of S in HC into isomorphisms in HD and theorem 8.14 one sees that G(X) and G(Y ) are S-local objects in C. Since G ∶ D → C is a right Quillen functor then G(p) is fibration in C. All what is left to show is that G(p) is a fibration in LS C. Since G(p) is a fibration between S-local objects in C, then lemma ?? implies that it is a fibration in LS C  It is easy to see that axioms CM1 to CM3 and one part of each of CM4 and CM5 are satisfied for the classes of S-(co)local weak equivalence, S-(co)local cofibrations and S-(co)local fibrations, and that will be recalled later on. However, other parts of axiom CM5 does not necessary hold in general.

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So far the notions of left and right localisation have been dual. However, most model categories that we are interested in localising are cofibrantly generated. That both facilitate the existence of Bousfield localisations and breaks the duality, especially that we will be making explicit use of the sets of generating cofibrations and generating weak cofibrations. Therefore, they are treated separately with more emphasis put on the left localisation.

8.3. Left Bousfield localisation. The existence of left Bousfield localisation depends on the properties the model category satisfy, and the ’size’ of the class being localised with respect to. In motivic homotopy theory the model categories of interest are all left proper cellular simplicial model categories, and that is good enough for the left Bousfield localisation to exist, given that the class of morphisms one is localising with respect to is a small set. It worth noticing that one does not have to be that restrictive for the left Bousfield localisation to exist. The reader interested in a more general theory is referred to [Hir03].

D

Left Bousfield localisation of a left proper cellular simplicial model category (C, I, J) with respect to a set of morphism S of C is obtained as a cofibrantly generated model through ’twisting’ the set of generating weak cofibrations by S, keeping the set of generating cofibrations fixed, hence fixing cofibrations. Therefore, we start by recalling how to ’twisting’ the set of generating weak cofibrations by S, and use Kan Criteria 5.6 to prove the existence of LS C. Since one wants cofibrations and S-local equivalences in C to serve as the classes of cofibrations and weak equivalences in LS C, and if LS C is to form a cofibrantly generated model category (C, I, JS ), then the class of S-local weak cofibrations should be refined to a subclass JS that is both small enough to be a small set, and encodes enough information to give the class of all S-local weak cofibrations as JS - cof. If left Bousfield localisation LS C exists, then the class of all S-local weak cofibrations is precisely the class of morphisms that have the LLP with respect to the class of S-local fibrations. Therefore, it is enough to find a set of S-local weak cofibrations JS such that the class of S-local fibrations equals JS - inj. In this pursuit, one notices at first that, in cofibrantly generated model categories, every (weak) cofibration is a retraction of (weak) relative I- cell complex. So if we let J ′ be the class of S-local weak relative I- cell complexes, then on the one hand J ′ - inj contains the class of S-local fibrations. On the other hand, when LS C exist and is a cofibrantly model category, all S-local weak cofibrations are retracts of S-local weak relative I- cell complexes, then by lemma 1.18, J ′ - inj coincides with the the class of S-local fibrations. The class of S-local weak relative I- cell complexes is usually a proper class, and the rest of this section basically shows that there exist refinement of this class that proves the existence of left Bousfield localisation, under some mild assumptions. Inclusions of I- cell sub-complexes play essential role in Bousfield localisation, and that is mainly due to their set-theoretical-like nature in cellular model categories, as seen in lemmas 4.23, 4.26 and 4.25. Therefore, we start by showing that when looking for JS attention can be restricted to the class of S-local weak I- cell sub-complexes as seen in the below lemma. Lemma 8.46. Let (C, I, J) a left proper cofibrantly generated simplicial model category, S set of morphisms of C, p ∶ X → Y be a fibration in C. If p has the RLP with respect to the class of all S-local weak inclusions of I- cell sub-complexes, then it has the RLP with respect to the class of all S-local weak cofibrations.

105

Proof. Let i ∶ U → V be a S-local weak cofibration. Applying the functorial factorisation (αI , βI ) for the unique morphisms ∅ → U , then to the composition iβI (∅ → U ) yield the commutative square QI (U )

∼ qU

Q′I (i)

 Q′I (V )

∼ qV

/U  /V

i

RA FT

Where QI (U ) and QI (V ) are I- cell complex, particularly QI (U ) is an I- cell sub-complexes of QI (V ), Q′I (i) an inclusion of I- cell sub-complexes, and horizontal morphisms the evident weak fibrations, induced by the transfinite small object argument. Since i ∶ U → V is S-local weak equivalence, horizontal morphisms are S-local weak equivalences for being weak equivalences in C, and that the class of S-local weak equivalences satisfies the two-out-of-three property then Q′I (i) is an S-local weak inclusion of I- cell sub-complexes. Hence, p has the RLP with respect to Q′I (i). Since Q′I (i) is a particular cofibration cofibrant approximation of i, and C is left proper then having p has the RLP with respect to Q′I (i) implies that p has the RLP with respect to i, by Kan lemma in (left) proper model categories 6.3. Therefore, p has the RLP with respect to the class of all S-local weak cofibrations.  Corollary 8.47. Let (C, I, J) a left proper cofibrantly generated model simplicial model category, S set of morphisms of C, J ′ be the class of of all S-local weak inclusions of I- cell sub-complexes. Then, J ′ - inj equals the class of S-local fibrations. Proof. J ′ is a subclass of S-local weak cofibrations (I- cof), since S-local fibrations are given by the RLP with respect to S-local weak cofibrations, then we have J ′ - inj ⊇ (C ⋂ WS )- inj = FS

The other inclusion is a direct implication of the above lemma 8.46.



D

In general the class of all S-local weak inclusions of I- cell sub-complexes is not necessary a small set, hence it needs to be refined further. We see below that this issue can be solved though refining the class further to a subclass JS,κ of representatives of isomorphism classes of all S-local weak inclusions of I- cell sub-complexes of size less than a given cardinal κ. This subclass doesn’t necessary satisfy the desired lifting properties for arbitrary cardinal κ. However, it is a set for every given cardinal κ, and there exist a cardinal κ for which JS,κ - inj equals S-local fibrations. One direction of the equality is clear since JS,κ is a subclass of S-local weak cofibrations, morphisms that has the RLP with respect to the class of S-local weak cofibrations belong to JS,κ - inj for every cardinal κ. Therefore, attention can be restricted to proving the existence of a cardinal κ for which morphisms of JS,κ - inj have the the RLP with respect to the class of S-local weak cofibrations. That is covered over the rest of this section following the Bousfield-Smith Cardinality Argument (see [Hir03, §4.5]). The order of our presentation here differs from [ibid.]. To gain a better and more straightforward understanding of the argument the author preferred to present the reduction of the above problem into relatively simpler problems first, that sets a natural direction toward the supplementary constrictions needed to solve the problem. This presentation is motivated by the author’s understanding of [ibid.] and [Jar87]. To start with we show that JS,κ is a set for every cardinal κ. Lemma 8.48. Let Let (C, I, J) be left proper cellular simplicial model category, S a set of morphisms of C, κ a cardinal. Then, the class of all isomorphism classes of S-local weak inclusions of I- cell sub-complexes of size at most κ is a set. Proof. Isomorphism classes of S-local weak inclusions of I- cell sub-complexes of size at most κ are κ determined uniquely by its set of cells of S of size at most κ. Since, there is at most ∣S∣ sets of cells of S of size at most κ, hence isomorphism classes of S-local weak inclusions of I- cell sub-complexes of size at most κ κ is a set bounded by the cardinality ∣S∣ .  Corollary 8.49. Let Let (C, I, J) be left proper cellular simplicial model category, S a set of morphisms of C, κ a cardinal. Then, JS,κ , the class of representative of isomorphism classes of S-local weak inclusions of I- cell sub-complexes of size at most κ, is a set.

106

Lemma 5.12 reduces the above problem (which is in essence a collection of right lifting problems of JS,κ - inj with respect to I ′ ) to right I ′ -partial extension problems of JS,κ - inj, for the class I ′ of S-local weak inclusions of I- cell sub-complexes. The resulting I ′ -partial extensions problems are reduced by theorem 5.15 to κ-Bounded sub-complexes S-condition, definition 5.14, where W ′ is being the class of S-local weak equivalences (the name is a modification of the bounded cofibrations condition, due to Jardine [ibid.], where it was presented for simplicial (pre-)sheaves, whereas the current formalisation is due to Hirschhorn [Hir03]). Therefore, the above problem is deduced to showing there exist a cardinal κ for which I ′ satisfies the κ-Bounded sub-complexes S-condition, providing the technical construction of cardinal κ at the end of the section. Theorem 8.50 (Bounded sub-complexes S-condition). Let (C, I, J) be left proper cellular simplicial model category, S a set of morphisms of C. Then, there exist a cardinal κ for which the class of S-local weak inclusions of I- cell sub-complexes satisfies the κ-Bounded sub-complexes S-condition, definition 5.14. Proof. See [Hir03, 4.4.5].



Definition 8.51. Let C be a left proper cellular simplicial model category, S a set of morphisms in C. Then, we define the set JS to be JS,κ the set (lemma 8.48) of all S-local weak inclusions of I- cell sub-complexes of size at most κ, for the minimal κ that proves the bounded sub-complexes S-condition, theorem ??.

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Corollary 8.52. Let (C, I, J) be left proper cellular simplicial model category, S a set of morphisms of C. Then, JS - inj coincide with the class of S-local fibrations. Proof. Since JS consists of S-local weak equivalences, then one direction is evident. For the other direction, let p be a morphism that has the RLP with respect to JS = JS,κ . Theorem 8.50 implies that S-local weak inclusions of I- cell sub-complexes satisfies the κ-Bounded sub-complexes S-condition for κ in definition 8.51, then theorem 5.15 implies that p has the RLP with respect to the class of S-local weak inclusions of I- cell sub-complexes. Then, lemma lemma 8.46 implies that p has the RLP with respect to the class of S-local weak cofibrations, i.e. p is an S-local fibration.  Corollary 8.53. Let (C, I, J) be left proper cellular simplicial model category, S a set of morphisms of C. Then, JS - cof coincide with the class of S-local weak equivalences.

D

Proof. Since JS consists of S-local weak equivalences, then one direction is evident. For the other direction, let i ∶ U → V be an S-local weak equivalence. Domains of elements of JS are I- cell sub-complexes, since C is cellular, then theorem 5.17 implies that domains of JS are small relative to cofibrations in C, in particular small relative to JS - cell, because JS is a set of cofibrations and cofibrations are closed under the formation of relative cell complexes. Therefore, JS admits the small object argument. Hence, there is a factorisation of i as i = βJS (i)αJS (i)

where βJS (i) ∈ JS - cell and αJS (i) ∈ JS - inj. Corollary 8.52 above implies that αJS (i) is an S-local fibration, hence it has the RLP with respect to i by the definition of S-local fibrations. Then, the retract argument theorem 1.17 implies that i is retract of βJS (i) ∈ JS - cell. Since JS - cell ⊆ JS - cof, and JS - cof is closed under retract, then i ∈ JS - cof.  Hereby, we recall the main theorem in this section. It proves the existence of left Bousfield localisation for left proper cellular simplicial model category with respect to set of its morphisms. Where we follow [ibid.] and use Kan criteria theorem 5.6 to prove that LS C exists as a cofibrantly generated model category, generated by I and JS . Theorem 8.54. Let C be a left proper cellular simplicial model category, S a set of morphisms in C. Then, the left Bousfield localisation LS C exists as a cofibrantly generated model category, generated by I and JS . Moreover, LS C is a left proper cellular simplicial model category. Moreover, LS C is left proper cellular simplicial model category. Proof. The category C is complete and cocomplete for being a model category. We will examine if the classes (WS , C, FS ) and the sets of morphisms I, JS satisfy Kan criteria theorem 5.6. (1) The class of S-local weak equivalence satisfies the two-out-of-three property, and is closed under retract by lemmas 8.12 and 8.13. (2) The set I admits the small objects argument because (C, I, J) is a cellular model category. Corollary 8.49 implies that JS is a set. Domains of elements of JS are I- cell sub-complexes, since C is cellular, then theorem 5.17 implies that domains of JS are small relative to cofibrations in C, in particular small relative to JS - cell, because JS is a set of cofibrations and cofibrations are closed under the formation of relative cell complexes. Therefore, JS admits the small object argument.

107

(3) Corollary 8.53 above implies that WS ⋂ I- cof = J- cof. (4) Since JS consists of cofibrations, i.e. I- cof, and - cof is a stable operator, then JS - cof ⊆ I- cof. A straightforward verification shows that having JS - cof ⊆ I- cof implies I- inj ⊆ JS - inj. on other hand, since (C, (W, C, F )) is cofibrantly generated we have I- inj ⊆ W ⋂ J- inj, then in particular morphisms of I- inj are weak equivalences, hence S-local weak equivalence. Therefore, I- inj ⊆ WS ⋂ JS - inj. Since classes (WS , C, FS ) and the sets of morphisms I, JS satisfy Kan criteria theorem 5.6. Then, LS C left Bousfield localisation of C with respect to the set S exists. Moreover, by Kan criteria, we see that LS C is cofibrantly generated with the set I of generating cofibration and JS of generating S-local weak cofibrations. Actually it is cellular, than (1) Domains and codomains of morphisms of I are compact relative to I, because (C, I, J) is cellular. (2) Domains of of morphisms of JS are I- cell complexes, as seen above because they are cofibrant objects and C is cellular. (3) Cofibrations of LS C coincide with cofibrations of C, hence they are effective monomorphisms.  8.4. Examples of Localised Model Structures.

RA FT

8.5. The commutativity of Localisation Functors. Quillen adjunction occur naturally in many context, and it is natural to ask if they preserve localisation Lemma 8.55. Let (F, G, ϕ) ∶ (C, I, J) → (D, F (I), F (J)) be a Quillen adjunction of left proper cellular ̃ simplicial model categories. Then, each set S of morphisms of S and relative Λ(F (S))- cell F (S)-localisation ̃ functor (LF (S) , jS ) in D induce an relative Λ(S)- cell S-localisation functor (G LF (S) F, ϕ−,LF (S) (F (−)) (jS,F (−) )) in C. ̃ Proof. For every X ∈ C, F (X) ∈ D. Since (LF (S) , jS ) is a relative Λ(F (S))- cell F (S)-localisation ̃ functor in D. Then, ∀X ∈ C, j is an F (S)-local equivalence that is relative Λ(F (S))- cell complex F (X)

jS,F (X) ∶ F (X) → LF (S) (F (X)).

Then, by the adjunction (F, G, ϕ), there exist a morphism

′ ∶= ϕX,LF (S) (F (X)) (jF (X) ) ∶ X → (G LF (S) F ) (X). jS,X

D

̃ cell S-localisation functor in C, we will show that ∀X ∈ C, To show that (G LF (S) F, jS′ ) is an relative Λ(S)′ ̃ cell complex, hence by lemma ?? (G LF (S) F ) (X) is an S-local object in C, and that jS,X is a relative Λ(S)an S-local equivalence. Since LF (S) (F (X)) is an F (S)-local object in D, then the induced morphisms ∗

F (i) ∶ Map (F (V ), LF (S) (F (X))) → Map (F (U ), LF (S) (F (X))) are weak equivalences of simplicial set for every i ∶ U → V in S. The Quillen adjunction of simplicial model categories implies the commutativity of the below diagram F (i)

Map (V, (G LF (S) F ) (X))

∗

Map (F (U ), LF (S) (F (X)))

≃

≃

Map (F (V ), LF (S) (F (X)))

i∗

Map (U, (G LF (S) F ) (X))

with vertical morphisms being isomorphisms of simplicial sets induced by the adjunction of simplicial model categories. Therefore, the below induced morphisms i∗ ∶ Map (V, (G LF (S) F ) (X)) → Map (U, (G LF (S) F ) (X)) are weak equivalences of simplicial set for every i ∶ U → V in S, hence (G LF (S) F ) (X) is an S-local object in C. ̃ Since j ∶ F (X) → L (F (X)) is a relative Λ(F (S))- cell complex, i.e. a transfinite composition S,F (X)

F (S)

of push-out squares of morphisms of F (S), hence there exist an ordinal λ and a λ-sequence Z ∶ λ → F (X) ↓ D of push-outs of morphisms of F (S) such that jS,F (X) isomorphic to the transfinite composition Z0 → colim Z. Since F commutes with colimits for being a left adjoint, then Z factorises through the evident λ-sequence

108

Z ′ ∶ λ → X ↓ C of push-outs of morphisms of S such that Z = F Z ′ . Moreover, the commutativity of F with colimits implies that the evident morphism ≃

LF (S) (F (X)) ≅ colim Z = colim F Z ′ → F (colim Z ′ ) is an isomorphism. Hence, applying the adjunction on the inverse of the previous isomorphism yield an isomorphism ≃ colim Z ′ → G(colim Z) ≅ (G LF (S) F ) (X) ̃ cell complex. Then, hence j ′ ∶ X → (G L F ) (X) is a transfinite composition of Z ′ , hence a relative Λ(S)S,X

F (S)

D

RA FT

̃ cell S-localisation functor in by lemma ??, we see that (G LF (S) F, ϕ−,LF (S) (F (−)) (jS,F (−) )) is an relative Λ(S)C. 

CHAPTER 3

The Model Structure on sSet In this section we recall the canonical Quillen’s model structure on the category of simplicial sets, where we will follow [GJ09],[Hov99], and [PB13a]. We will recall the three classes of morphisms of simplicial sets Kan fibrations, weak equivalence and cofibrations, and study some of its properties. These classes will form the canonical model structure on the category of simplicial sets, Th. 4.1. We distinguish two classes of canonical injection of simplicial sets, that play a central role in defining the desired classes of fibrations and cofibrations of the canonical model structure on the category of simplicial sets. Hereby, we fix the notation in this section for:

T

I ∶= {∂[n] ∶ ∂∆n ↪ ∆n ∣n ∈ Z≥0 } . Λ ∶= {λnr ∶ Λnr ↪ ∆n ∣n ∈ Z>0 , 0 ≤ r ≤ n}

DR AF

These classes will be the building blocks for (weak) cofibrations and fibrations using class operations -inj and -inj. 1. Weak Equivalences

Definition 1.1 (Weak Equivalence). Let f ∶ X → Y be a morphism of simplicial sets. Then, f is called a weak equivalence if the realisation ∣f ∣ ∶ ∣X∣ → ∣Y ∣ is a weak equivalence of topological spaces, i.e. π0 (∣f ∣) ∶ π0 (∣X∣) → π0 (∣Y ∣)

is a bijection of sets, and for every n ∈ Z≥1

πn (∣f ∣, x) ∶ πn (∣X∣, x) → πn (∣Y ∣, ∣f ∣(x))

is a a group isomorphism for x ∈ ∣X∣.

Remark 1.2. Notice that the condition on π0 could be formulated using the topological homotopy pointed sets (following definition 10.2), which makes the statement of the condition more consistent, but more redundant. So, it is a matter of taste which condition to choose on π0 . We prefer to use the consistent notations in formulating statements, and, of course, use the more efficient condition in proofs. Remark 1.3. Notice that by the construction of the geometric realisation, every path connected component of ∣X∣ contains at least a vertex of X. Since we have an isomorphism πn (∣X∣, x) ≅ πn (∣X∣, x′ ) for every pair of points x, x′ ∈ X that lies in the same path connected component, then the above definition can be given more efficiently by restricting x to the vertices of X. Actuality, it is enough to restrict the choice of x to one point in each path connected components. However, in practice, computing the path connected components of a given simplicial sets may require much more effort than considering all the vertices. Weak equivalence can be defined equivalently using the base-free homotopy sets, as in the below lemma Lemma 1.4. Let f ∶ X → Y be a morphism of simplicial sets. Then, the following statements are equivalent ● f is a weak equivalence. ● ∀n ≥ 0, the induced square, constructed in (104), is a cartesian square. πntop (X) pX

 X0

f∗ ↺

f0

/ π top (Y ) n pY

 / Y0

Proof. A direct result of the above remark and corollary 10.5. 109



110

2. Cofibrations Definition 2.1 (Cofibrations of sSet). A morphism of simplicial sets i ∶ X → Y is saidto be a cofibration if it is in I-cof. Lemma 2.2. A morphisms i ∶ X → Y of simplicial sets is a cofibrations if and only if it is an inclusion (a monomorphism of simplicial sets). Hence, all simplicial sets are cofibrant. Proof. Joe 2.4.4  Lemma 2.3. Cofibrations of simplical sets are relative I-cell complexe, i.e. I-cof⊂ I-cell - that the other inclusion is always satisfied for any class of morphisms, by ??((c)). 

Proof. 2.1. Anodyne Extension.

Definition 2.4 (Anodyne Extension). A morphism of simplicial sets i ∶ X → Y is saidto be a anodyne extension if it is in J-cof. Lemma 2.5. A morphism of simplicial sets is a weak cofibrations iff it is an anodyne extensions.

T

Proof. TBC.



Remark 2.6. Anodyne extensions are cofibrations, but not necessarily weak equivalences.

DR AF

2.1.1. Product and Anodyne Extensions.

Proposition 2.7. Let i ∶ K → L be an inclusion of simplicial sets, and j ∶ U → V be an anodyne extension. Then, j ◻ i is an anodyne extension. Proof. Joe 2.4.19, b is a result of (a), the fact that J-inj=(J-cof)-inj and the previous lemma.



Corollary 2.8. Let i ∶ K → L be an inclusion of simplicial sets, j ∶ U → V be an Anodyne extension. Then: (1) j × idL ∶ U × L → V × L is an anodyne extension. (2) idV × i ∶ V × K → V × L is an anodyne extension. Proof. The unique morphisms i′ ∶ ∅ → L, and j ′ ∶ ∅ → V are inclusions of simplicial sets. Notice that U × ∅ = V × ∅ = ∅ × K = ∅ × L = ∅. Hence, (1) j ∐ i′ = ∅ ∐∅ (U × L) = U × L. Thus, j × idL coincide with j ◻ i′ , and by proposition 2.7, j × idL is an anodyne extension. (2) j ′ ∐ i = (V × K) ∐∅ ∅ = V × K. Thus, idV × i coincide with j ′ ◻ i, and by proposition 2.7, idV × i is an anodyne extension.  3. Kan Fibrations Definition 3.1 (Kan Fibrations). A morphism of simplicial sets p ∶ X → Y is saidto be a fibration (Kan fibration) if it has the RLP with respect to all canonical injections Λnr ↪ ∆n for n ∈ Z>0 , 0 ≤ r ≤ n, i.e. the class of Kan fibrations is J-inj. Moreover, we call a simplicial set X Kan fibrant (Kan complex) if the unique morphism X → ∗ is a Kan fibration. the condition on a simplicial set X to be a Kan fibrant used to be called the extension condition, because it means that every morphism of simplicial sets Λnr → X can be extended to morphism h ∶ ∆n → X, that the below soliddiagram is always commutative. /X Λnr > _ λn r

h

  /∗ ∆n The extension condition is used to be given by combinatorial relations, we recall the equivalence between the two definitions in the below lemma: Lemma 3.2 (Extension condition). Let p ∶ X → Y be a morphism of simplicial sets, then the below two statements are equivalents:

111

(1) p is a Kan fibration. (2) ∀n ∈ Z>0 , 0 ≤ r ≤ n: r , ..., xn ) of X such that dn−1 (xj ̂ For every n-tuple of (n − 1)-simplices (x0n−1 , x1n−1 , ..., x n−1 i n−1 n−1 ) = n−1 i n i dj−1 (xn−1 ) for r ≠ i < j ≠ r , and for every n-simplex yn ∈ Yn such that di (yn ) = p(xn−1 ), i ≠ r, there exists xn ∈ Xn such that dni (xn ) = xin−1 , and pn (xn ) = yn . Wherêindicates a deleted entry. Proof. Since Λnr is generated by, n many of (n − 1)-simplex, ∂ni , 0 ≤ i ≠ r ≤ n, that satisfy the set minimal simplicial relations (97), then by lemma 2.10 we see that giving a morphism Λnr → X is equivalent to giving n−1 i r , ..., xn ) of X such that dn−1 (xj ̂ n-tuple of (n−1)-simplices (x0n−1 , x1n−1 , ..., x n−1 i n−1 n−1 ) = dj−1 (xn−1 ) for r ≠ i < j ≠ r. Also, since ∆n is generated by the n-simplex id[n] and the empty set of minimal simplicial relations, then giving a morphism ∆n → Y is equivalent to giving an n-simplex yn ∈ Yn . In the rest of the proof, 0 ≤ i ≠ r ≤ n, and 0 ≤ j ≠ r ≤ n: 1⇒2 Assume that p is a Kan fibration, and assume that the hypothesis of (2) holds, then there is a soliddiagram e0 /X . Λnr > _ λn r

h e1

p

 /Y

RA FT

 ∆n

e0,n−1 (∂ni )

xin−1 ,

where = and e1,n (id[n] ) = yn . By 2.10, this diagram is commutative iff it is commutative on generators. Since dni (yn ) = pn−1 (xin−1 ), then dni (e1,n (id[n] )) = pn−1 (e0,n−1 (∂ni )). e1 is a morphism of simplicial sets, then dni e1,n = e1,n−1 dn−1 , hence: i pn−1 (e0,n−1 (∂ni )) = e1,n−1 (dn−1 (id[n] ) = e1,n−1 (∂ni ) = e1,n−1 (λnr,n−1 (∂ni )) i i.e, pe0 = e1 λnr , and the diagram is commutative. Then, there exists a lift h ∶ ∆n → X, let xn = h(id[n] ), then the commutativity of the whole diagram implies that pn (xn ) = yn , and dni (xn )

= dni (hn (id[n] )) = hn−1 (dni (id[n] )) = hn−1 (∂ni ) = hn−1 (λnr,n−1 (∂ni )) = e0,n−1 (∂ni ) = xin−1 .

D

2⇒1 Assume (2), and let e0 and e1 be morphisms of simplicial sets that makes the following soliddiagram commutative: e0 /X . Λnr > _ λn r

 ∆n

h e1

p

 /Y

The morphism of simplicial sets e0 implies the existence n-tuple of (n − 1)-simplices ̂ (e0,n−1 (∂ 0 ), e0,n−1 (∂ 1 ), ..., e0,n−1 (∂ r ), ..., e0,n−1 (∂ n )) n n−1 di (e0,n−1 (∂nj ))

n n n−1 i dj−1 (e0,n−1 (∂n )) for

n

of X such that = i < j. The morphism of simplicial sets e1 distinguishes the n-simplex yn = e1,n (id[n] ) ∈ Yn . Whereas the commutativity of the soliddiagram implies that dni (yn )

= dni (e1,n (id[n] )) = e1,n−1 (dni (id[n] )) = e1,n−1 (∂ni ) = e1,n−1 (λnr,n−1 (∂ni )) = pn−1 (e0,n−1 (∂ni )) = pn−1 (xin−1 ).

Hence, there exists xn ∈ Xn such that dni (xn ) = e0,n−1 (∂ni ), and pn (xn ) = yn . Then, since ∆n is generated by id[n] and the empty set of simplicial relations, we define h ∶ ∆n → X to be the unique morphism of simplicial sets such that hn (id[n] ) = xn . We need to see that h makes the whole diagram commutative. Since dni (xn ) = e0,n−1 (∂ni ), then: e0,n−1 (∂ni ) = dni (hn (id[n] )) = hn−1 (dni (id[n] )) = hn−1 (∂ni ) = hn−1 (λnr,n−1 (∂ni )) i.e. e0 = hλnr . Since pn (xn ) = yn , then e1,n (id[n] ) = yn = pn (xn ) = pn (hn (id[n] )), i.e. e1 = ph, and the whole diagram is commutative.  We recall below a classical example of Kan fibrants, that links Kan fibrants to Serre fibrants, but before we need to recall the following topological fact: Lemma 3.3. Let n ∈ Z>0 , 0 ≤ r ≤ n, then Λnr,Top is a retract of ∆ntop .

112

Proof. Define:

rnr ∶ ∆ntop t

Ð→ Λnr,Top ⎧ ti − min {ti } i ≠ r ⎪ ⎪ 0≤j≠r≤n ⎪ ⎪ ⎪ ui = ⎨ ⎪ ⎪ 1 − ∑ ui i = r. ⎪ ⎪ ⎪ 0≤j≠r≤n ⎩ ⊂ ∆ntop , ∃0 ≤ j ≠ r ≤ n, such that ti = 0, i.e.

z→ u,

rnr is a well-defined continuous map. For t ∈ Λnr,Top and

rnr (t)

min {ti } = 0,

0≤j≠r≤n

= t.



Example 3.4. Let X be a topological space, then S(X), the singular simplicial set of X, is a Kan fibrant. i Proof. Let fn−1 ∶ ∆n−1 top → X for 0 ≤ j ≠ r ≤ n be n (n − 1)-simplices of S(X), such that j i dn−1 (fn−1 ) = dn−1 i j−1 (fn−1 ) for r ≠ i ≤ j ≠ r.

Define fn ∶

∆ntop t

Ð→ X i r r r z→ fn−1 ((rnr (t))0 , (rnr (t))1 , ..., (r̂ n (t))i , ..., (rn (t))n ) where (rn (t))i = 0.

T

i ’s implies that fn is well-defined. Also, it is continuous, i.e. fn ∈ S(X)n . It is easy The above conditions on fn−1 to see that the Kan extension condition is satisfied for the choice of fn , and that S(X) is a Kan fibrant. 

Kan fibrations are linked to Serre fibration through the following lemma and Quillen’s Theorem 3.40.

DR AF

Lemma 3.5. Let p ∶ X → Y be a continuous map of topological spaces. Then, p is a Serre fibration iff S(p) ∶ S(X) → S(Y ) is a Kan fibration. Proof. It is a direct result of corolloray ??, for C = sSet, D = Top, (L, R, φ) = (∣−∣, S, ϕ) the adjundtion between geometric realisation and the singular functor, I = J, and J = {p}.  .

Example 3.6 (Counter Example). Consider the solidcommutative square: 0

2

Λ20 _

1

_

 0

2



/ ∂∆ 2 > _

h

λ20

∆

1 _

3

/

   ∆2

0

2

∆∂3 ∣∂∆2

 / ∆3

3 ∂3

 0

2

3

1

1 2

2

This diagram does not admit a lift h ∶ ∆ → ∂∆ , that if we assume for the sake of contradiction that it does then the commutativity of the whole diagram implies that ∂3

3

∆23 (id[2] ) = ∆∂3 ∣∂∆2 2 (h2 (id[2] )) i.e. ∂33 = ∂33 h2 (id[2] ) in ∆. Since ∂33 is a monomorphism, in ∆, we have h2 (id[2] ) = id[2] ∉ ∂∆2 . We got a 3 contradition, and such h does not exist, i.e. ∆∂3 ∣∂∆2 is not a Kan fibration. Remark 3.7. Moreover, all the morphisms of I and J are not Kan fibration, that we have the commutative squares idλnr , id∂[n] ∈ M or(sSet)∀n ∈ Z≥0 , 0 ≤ r ≤ n. One can see easily that neither of these squares has a lift, that the existance of such lift requires id[n] ∈ Λnr , id[n] ∈ ∂∆n , respectively, which is not the case, and elements of I and J are not Kan fibration. Also, notice that ∀n ∈ Z≥0 , 0 ≤ r ≤ n, ∆n is a Kan fibrant, whereas Λnr , and ∂∆n are not.

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Example 3.8. Let X be a simplicial set. Then, we have a unique morphism ∅ → X. Since Λ00 ↪ ∆0 is not in J, we do do not need to consider whether a lift exist for the commutative square ∅ = Λ00

/∅

 ∗ = ∆0

 / X.

ιx0

In fact, such lift does not exist. Actually, the condition of ∅ → X being a Kan fibration is satisfied automatically. On the other hand, one can see that ∅ → X is Kan fibration as a result of the functorial factorisation as a fibration and weak cofibration and the fact that ∅ is only weak equivelant to itself. Remark 3.9. Notice that the fibres of Kan fibration are Kan fibrant that, by Lemma ????. Since the aim of this section is to recall the canonical model structure on the category of simplicial sets. It is of a particular interest to study Kan fibrations, which are also weak equivalences, called weak fibrations. However, the study of weak fibrations is not easy, and the method we adopt here requires familiarity with function complexes, simplicial homotopy and minimal fibrations. Therefore, we devote the rest of this section to recall these concept and eventually classify weak fibrations.

T

3.1. Function Complexes and Kan Fibrations.

DR AF

Proposition 3.10. Let i ∶ K → L be an inclusion of simplicial sets, and p ∶ X → Y be a Kan fibration. Then, (i∗ , p∗ ) is a Kan fibration. Proof. Joe 2.4.19, b is a result of (a), the fact that J-inj=(J-cof)-inj and the previous lemma.



Corollary 3.11. Let i ∶ K → L be an inclusion of simplicial sets, p ∶ X → Y be a Kan fibration, and X ′ be a Kan fibrant. Then: (1) p∗ ∶ Hom(K, X) → Hom(K, Y ) is a Kan fibration. (2) i∗ ∶ Hom(L, X ′ ) → Hom(K, X ′ ) is a Kan fibration. Proof. The unique morphism p′ ∶ X ′ → ∗ is a Kan fibration. Notice that Hom(∅, X) = Hom(∅, Y ) = Hom(L, ∗) = Hom(K, ∗) = ∗. (1) i′∗ ∏ p∗ = ∗ ×∗ Hom(L, Y ) = Hom(L, Y ). Thus, p∗ coincide with (i′∗ , p∗ ), and by proposition 3.10, p∗ is a Kan fibration. (2) i∗ ∏ p′∗ = Hom(K, X) ×∗ ∗ = Hom(K, X). Thus, i∗ coincide with (i∗ , p′∗ ), and by proposition 3.10, i∗ is a Kan fibration.  3.2. Homotopy of Simplicial Sets. We saw before that simplicial homotopy is not an equivalence relation on the class of morphisms of simplicial sets. However, the below lemma provides the framework where the simplicial homotopy is an equivalence relation. 3.3. Simplicial Homotopy Groups. Below, we recall the simplicial homotopy groups that are constructed in an analogue way of constructing the topological homotopy groups. Where, first we distinguish morphisms of simplicial sets from the standard simplexes to a simplicial sets X, that are constant on the boundary. Then, we define then the simplicial homotopy groups of X to be the equivalences classes of the simplicial homotopy of such morphisms. However, as we have seen before, simplicial homotopy is not an equivalence relation in general. Therefore, the definition of the simplicial homotopy groups will be restricted to the case were X is a Kan fibrant, that the below lemma guarantees that the simplicial homotopy of such morphisms forms an equivalence relation. Then, we recall the relation that one is expecting between the simplicial homotopy groups of X, and the the homotopy groups of its geometric realisation. Lemma 3.12. Let X be a Kan fibrant, and i ∶ K → L be an inclusion of simplicial sets. Then, (1) The simplicial homotopy of morphisms L → X is an equivalence relation. (2) The simplicial homotopy of morphisms L → X (rel K) is an equivalence relation. Moreover, the graded equivalence relation defined on X, by identifying n1−simplices of X with morphisms ∆n → X, respects faces and degenerations. Proof. TBC.



114

Lemma 3.13. Let p ∶ X → Y be a Kan fibration, and i ∶ K → L be an inclusion of simplicial sets. Then, (1) The simplicial homotopy of morphisms L → X with respect to p is an equivalence relation. (2) The simplicial homotopy of morphisms L → X (rel K) with respect to p is an equivalence relation. Moreover, the graded equivalence relation defined on X, by identifying n1−simplices of X with morphisms ∆n → X, respects faces and degenerations. Proof. TBC.



In topological homotopy groups we consider homotopy classes at fixed basepoints, below we recall how to make sense of these concepts for simplicial homotopy. Definition 3.14 (Loops at vertex). Let X be a simplicial set, x0 ∈ X0 be a vertex of X. Let xn ∈ Xn , we say that xn is a loop at x0 if its boundary being pure degeneracy of x0 , i.e. if ιxn being constant to x0 on the boundary ∂∆n , which is equivalent of requiring the below diagram to commute: / ∆0 = ∗

∂∆ n_

ιx0

∂∆n

 /X

ιxn

We use the same notation to say that ιxn is a loop at x0 .

T

 ∆n

DR AF

Lemma 3.15. Let X be a simplicial set, x0 ∈ X0 , and xn ∈ Xn . Then, xn is a loop at x0 iff ∀0 ≤ i ≤ n, dni (xn ) = ιx0 ,n−1 (σn−1 ). Hence, ∂(xn ) = (ιx0 ,n−1 (σn−1 ), ιx0 ,n−1 (σn−1 ), ..., ιx0 ,n−1 (σn−1 )). Proof. The commutativity of the below diagram: ∆nn

∆n ∂i

n

ιx0 ,n



/ Xn 

dn i

/ Xn−1

∆nn−1 ιx

0 ,n−1

implies that

dni (xn ) = ιxn ,n−1 (∂ni )  / ∆n Since ∂∆n is generated by ∂ni ’s, then dni (xn ) ∈ im( ∂∆nn−1 On the one hand, if xn is a loop at x0 , then im( ∂∆nn−1 hence dni (xn ) = ιx0 ,n−1 (σn−1 ).



/ ∆n

ιxn

/ X ).

ιxn

/ X ) = {Xσ (x0 ) = ιx ,n−1 (σn−1 )}, n−1 0

 / ∆n ιxn / X equals On the other hand, having dni (xn ) = ιx0 ,n−1 (σn−1 ), ∀0 ≤ i ≤ n implies that ∂∆nn−1 the constant morphisms on ∂∆n to x0 , on generators, hence they coincide, and xn is a loop at x0 .  Definition 3.16. Let X be a Kan fibrant, x0 ∈ X0 be a vertex of X. Then, for n ∈ Z≥0 πn (X, x0 ) is defined to be the set of homotopy classes of loops ∆n → X at x0 relative to the boundary the ∂∆n , i.e. the set of homotopy classes of morphisms f ∶ ∆n → X (rel ∂∆n ) for which the below diagram commutes: ∆O n ∂∆ n

? ∂∆n

f

/X O ιx0

/ ∆0 = ∗

Moreover, X is saidto be connected if π0 (X) consists of one element, the change of notation of dropping the vertex from π0 is explain in remark 3.18, below. Remark 3.17. Since f is a loop at x0 , then the requirement of the simplicial homotopy in the above definition to be relative to ∂∆n is to say that all the element of the desired equivalent class are also loops at x0 . If we think in terms of geometric realisation, it is the analogue of finding a homotopy classes of loops going through some fixed point of the topological space, in order to construct the topological fundamental group.

115

Remark 3.18. Notice that for n = 0, ∂∆0 = ∅, hence any morphism f ∶ ∆0 → X satisfy the above condition on f automatically, and π0 (X, x0 ) is then the set of homotopy classes of vertices of X, and it is independent of the choice of x0 , therefore we denote it by π0 (X). Lemma 3.19. Let X be a Kan fibration, x0 ∼ x′0 ∈ X0 , then ∀n ∈ Z≥0 , there is a bijection πn (X, x0 ) ≅ πn (X, x′0 ). Proof. TBC.



Hence, if X is connected then we can drop the vertex of the notation, and write π(X). in an analogue of the topological homotopy groups, one would like to equip πn (X, x0 ) with a group structure. Definition 3.20. Let X be a Kan fibration, x0 ∈ X0 , and xn , x′n ∈ Xn are loops at x0 . Then, consider the ̂ n+1 ) of n-simplices of X, where: (n + 1)-tuple (x0n , x1n , ..., xnn , x n ⎧ ι (σ ) ⎪ ⎪ ⎪ x0 ,n n xin = ⎨ xn ⎪ ′ ⎪ ⎪ ⎩ xn

0≤i≤n−2 i=n−1 i=n

RA FT

Notice that lemma 3.15 and remark 2.3 imply that dni (xjn ) = ιx0 ,n−1 (σn−1 ) for 0 ≤ i ≤ n and 0 ≤ j ≤ n. Then in particular dni (xjn ) = dnj−1 (xin ) for n + 1 ≠ i < j ≠ n + 1. Since, X is a Kan fibrant, then there exists an ′ (n + 1)-simplex xn+1 ∈ Xn+1 such that dn+1 (xn+1 ) = xin for 0 ≤ j ≤ n. We denote dn+1 n+1 (xn+1 ) by h(xn , xn ), for i such xn+1 . . Notice that such xn+1 is not necessary unique for given xn and x′n . However, the below lemmas show that the h(xn , x′n ) ∈ Xn is a loop at x0 for every such xn+1 , and that the equivalence class [ιh(xn ,x′n ) ](rel ∂∆n ) is independent of the choice of the representatives xn and x′n of the equivalence classes [ιxn ] and [ιx′n ] (rel ∂∆n ), it is also independent of the choice of such xn+1 introduced in the above definition. Then, we show that this definition gives rise to a group structure on πn (X, x0 ). Lemma 3.21. Let X be a simplicial set, x0 ∈ X0 , then h(xn , x′n ), defined above, is a loop at x0 .

D

Proof. Using lemma 3.15, it is enough to show that ∀0 ≤ i ≤ n, dni (h(xn , x′n )) = ιx0 ,n−1 (σn−1 ) in order to show that h(xn , x′n ) is a loop at x0 . ∀0 ≤ i ≤ n: n n+1 dni (h(xn , x′n )) = dni (dn+1 (xn+1 )) = dnn (xin ) = ιx0 ,n−1 (σn−1 ). n+1 (xn+1 )) = dn (di

Hence, h(xn , x′n ) goes though x0 .



Lemma 3.22. Let X be a simplicial set, x0 ∈ X0 , then [ιh(xn ,x′n ) ] (rel ∂∆n ) is independent of the choice of representative xn and x′n in the homotopy classes [ιxn ] and [ιx′n ]. Moreover, it is independent of the choice of xn+1 in definition 3.20. Proof. TBC.



Lemma 3.23. Let X be a Kan fibration, x0 ∈ X0 , then ∀n ∈ Z≥1 , πn (X, x0 ), have a canonical group structure given by: ∀[ιxn ], [ιx′n ] ∈ πn (X, x0 ) [ιxn ] ⋅ [ιx′n ] = [h(xn , x′n )].

Moreover, for n ≥ 0, the resulting group is abelian. Proof. TBC.



Lemma 3.24. Let X be a Kan fibration, x0 ∈ X0 , then ∀n ∈ Z≥0 , there is a canonical bijection πn (X, x0 ) ≅ πn (∣X∣, x0 ), which is a group isomorphism for n ≥ 1. Proof. See [Hov99, 3.6.3].



Proposition 3.25. Let X be a non-empty Kan fibrant with no non-trivial simplicial homotopy groups. Then the unique morphism X → ∗ is in I-inj. Proof. See [Hov99, 3.4.7].



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3.4. Minimal Fibrations. Having the above isomorphism in mind, one might wonder if weak fibration coincide with Kan fibration that induces isomorphisms on the level of simplicial homotopy groups. However, the domain and codomain of Kan fibration are not necessary Kan fibrant. Hence, simplicial homotopy groups are not necessarily defined for its domain and codomain. Therefore, minimal fibrations have been introduced to overcome this difficulty. We recall the definition and main properties below. Roughly speaking, the idea of minimal fibration is to distinguish Kan fibrations that induces an isomorphism fibre-wise on level of simplicial homotopy groups. Since, πn (∗, ∗) is trivial, so it makes sense to define minimal fibrations to be Kan fibrations with trivial simplicial group fibre-wise. Minimal fibrations will also provide a sufficient criteria for morphisms to have the RLP with respect to I, i.e. being in I-inj. Definition 3.26. Let q ∶ X → Y be a Kan fibration, we say that q is a minimal fibration if every fibre-wise homotopic simplices of X (rel ∂∆n ) with respect to q coincide, i.e. ∀xn , x′n ∈ Xn such that xn ∼q x′n (rel ∂∆n ), then one has xn = x′n . Lemma 3.27. Let q ∶ X → Y be a fibration. Then, the following are equivalent: (1) q is a minimal fibration. (2) For every fibre-wise homotopy H ∈ Hom(∆n , X)1 (rel ∂∆n ) with respect to q, the below diagram commutes: 1

n

∆ ×∆

0

idL ×∆∂1

// ∆ n × ∆ 1

/ X.

DR AF

(3) ...Path components. Proof. TBC.

H

T

0 idL ×∆∂1



Definition 3.28. A Kan fibrant X is called a minimal fibrant if the unique morphism X → ∗ is a minimal fibration. Counter Example 3.29. One can see easily that the fibrant ∆n is not a minimal fibrant. One can think of minimal fibration as simplicial equivalent to covering spaces, where there is no path between points of the fibres at a given point. However, in the case of minimal fibrations, the fibre can be empty. Lemma 3.30. Minimal fibrations are closed under pullback, and retract. Proof. TBC.



Definition 3.31 (fibre-wise trivial). Let p ∶ X → Y be a fibration. We say that p is fibre-wise trivial iff pr1 ∀n ∈ Z≥0 , yn ∈ Yn , the pull back fibration ∆n ×Y X → ∆n is isomorphic to a product fibration ∆n × F → ∆n over ∆n , i.e. there is an isomorphism of simplicial sets ∆n × F → ∆n ×Y X that makes the below diagram commute: / ∆n ×Y X ∆n × F pr1

$

∆n

z

Lemma 3.32. Let q ∶ X → Y be a minimal fibration. Then, q is fibre-wise trivial. Proof. TBC.



One knows from topological homotopy theory that deformation retraction are homotopy of a topological space with respect to a subspace that is a retraction of the original space. This notion can be extended to simplicial sets in the below manner: Definition 3.33 (Strong Deformation Retract). Let i ∶ K → L be an inclusion of simplicial sets. K is saidto be a strong deformation retract of L, if there exist a retraction r ∶ L → K, i.e. ri = idK , and a homotopy H ∶ L × ∆1 → L relative to K from idL to ir. Definition 3.34 (Strong Fibre-wise Deformation Retract). Let p ∶ X → P be a Kan fibration. A morphism of simplicial sets q ∶ Z → Y is saidto be a strong fibre-wise deformation retract of p, if there exist an inclusion of simplicial sets i ∶ Z → X, and retraction r ∶ X → Z that makes q a retract of p, and a homotopy H ∶ X × ∆1 → X with respect to p relative to Z from idX to ir. Proposition 3.35. Let p ∶ X → Y be a Kan fibration. Then, p has a strong fibre-wise deformation retract q ∶ Z → Y which is a minimal fibration.

117

Proof. TBC.



Lemma 3.36. Let p ∶ X → Y be a Kan fibration, xn , x′n ∈ Xn degenerate such that xn ∼p x′n . Then, xn = x′n . Proof. See [Hov99, 3.5.8].



i.e. minimal fibration imposes its condition only on non-degenerate simplices. ′

p r / / Y , with p′ being X′ Theorem 3.37. Let p ∶ X → Y be a Kan fibration. Then, p factorise as X ′ a minimal fibration, and r a retraction of the simplicial sub-set i ∶ X ↪ X such that r ∈ I-inj.

Proof. See [Hov99, 3.5.9].



The below corollary provides a sufficient criteria for fibrations to be in I−inj Corollary 3.38. Let p ∶ X → Y be a Kan fibration having a non-empty fibres with no non-trivial simplicial homotopy groups ∀y0 ∈ Y0 . Then, p ∈ I−inj. Proof. See [Hov99, 3.5.10].



Proof. TBC.

RA FT

Theorem 3.39 (Gabriel-Zisman). Let q ∶ X → Y be a minimal fibration. Then, the realisation ∣q∣ ∶ ∣X∣ → ∣Y ∣ is a Serre fibration. 

Corollary 3.40 (Quillen’s Theorem). Let p ∶ X → Y be a morphism of simplicial sets. Then, p is a Kan fibration iff ∣f ∣ ∶ ∣X∣ → ∣Y ∣ is a Serre fibration. Proof. See [Hov99, 3.6.2].



then lemma before becomes a corrolry is a result of the fact that all topological spaces are fibrant.

D

Corollary 3.41. Let p ∶ X → Y be a morphism of simplicial sets. Then, p is a weak fibration iff it is in I-inj. following Joe 2.4.60/2.4.7, Hovey 3.6.4. Suppose that p is a weak fibration. Using corollary 3.38, it is sufficient to show that fibres of p are non-empty and have no non-trivial simplicial homotopy groups. ∀y0 ∈ Y0 , consider the pullback /X Xy 0

 ∆0

p

ιy0

 /Y

Then Xy0 is a fibrant, by 1.20. Since, the geometrical realisation functor preserves pullbacks, lemma ??, we have the following pullback square / ∣X∣ ∣Xy ∣ 0

 ∣∆0 ∣

∣p∣

∣ιy0 ∣

 / ∣Y ∣

then, by corollary 3.40, ∣p∣ is a Serre fibration. ∣Xy0 ∣ is the fibre of ∣p∣ over ∣ιy0 ∣. Since ∣p∣ is a weak equivalence, then weak Serre fibrant ∣Xy0 ∣ is non-empty and has no non-trivial homotopy groups. Then, Xy0 is non-empty, and by lemma 3.24 it has no non-trivial simplicial homotopy group. Hence, by corollary 3.38, we see that p ∈ I-inj.  Since every fibration p factorises as p = qr where r ∈ I-inj and q a minimal fibration, and I-inj are weak fibrations. Then, the information ida fibration is a weak fibration is encoded in its minimal fibration factor.

118

4. The Canonical Model Structure Theorem 4.1 (Organising Theorem). The category of simplicial sets sSet, together with the classes of Kan fibrations, cofibrations, and weak equivalences, studied above, forms a model category. Proof. We will use theorem ??. We examine below that the five axioms of model category, Def. ??, are satisfied: CM1 Since the category of small sets Set is complete and cocomplete, then by lemma ??, sSet is complete and cocomplete. Furthermore, for any small diagram F ∶ J → sSet, we have (lim F ) = lim Fn , and (colim F ) = colim Fn J

n

J

J

n

J

for every n ∈ Z≥0 , and the axiom CM1 holds. CM2 (2 out of 3) For any commutative diagram of simplicial sets 6/ Z .

gf

X (

f

Y

g

RA FT

Using the realisation functor, we get the commutative diagram of topological spaces ∣g∣∣f ∣

∣X∣

∣f ∣

(

∣Y ∣

/6 ∣Z∣ .

∣g∣

D

If any two of f, g or gf are weak equivalences of simplicial sets, then by definition two maps of ∣f ∣, ∣g∣ and ∣g∣∣f ∣ are weak equivalences of topological spaces. Using the model structure on Top, the third map is a weak equivalence of topological spaces. Hence, the third corresponding morphism is a weak equivalence of simplicial sets, and the axiom CM2 holds. CM3 (Stability under retract) Let f ∶ X → Y, and g ∶ X ′ → Y ′ be morphisms of simplicial sets such that f is a retract of g. Then, we have the commutative diagram: X f

 Y

d0

/ X′

r0

D

g

R

d1

 / Y′

r1

/X  /Y

f

with the horizontal composite being the identities. Then: (a) Suppose that g is a cofibration, then g ∈ I-cof, then by lemma ?? f , a retract of g, is in I-cof, i.e. f is a cofibration. (b) Suppose that g is a Kan fibration, then g ∈ J-inj, then by lemma ?? f , a retract of g is in J-inj, i.e. f is a fibration. Notice that, here, the cofibration and fibration is quite formal, does not depend on nature of morphisms of simplicial sets, but rather on being defined as the morphisms with LLP or RLP with respect to some classes of morphisms. (c) Suppose that g is a weak equivalence, applying the realisation, we have the commutative diagram in Top, with the horizontal composite being the identities: ∣X∣ ∣f ∣

 ∣Y ∣

∣d0 ∣

/ ∣X ′ ∣

∣r0 ∣

∣D∣

∣d0 ∣

∣R∣

∣d1 ∣

 / ∣Y ′ ∣

∣r1 ∣

/ ∣X∣  /Y

∣f ∣

Since, g is a weak equivalence of simplicial sets, then ∣g∣ is a a weak equivalence of topological spaces. Since weak equivalence of topological spaces is closed under retract, then so is ∣f ∣. Hence, f is a weak equivalence of simplicial sets, and CM3 holds.

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CM4 (Lifting) Let E be a commutative diagram of morphisms of simplicial sets A i

 B

/X

e0

E

 /Y

e1

p

Where i is a cofibration, p is a fibration, and either of i or p is a weak equivalence. Then we need to show that E admits a lift h ∶ B → X. On the one hand, if p is a weak equivalence then p ∈ I-inj, whereas i ∈ I-cof= (I-inj)-proj. Hence, E admits a lift. On the other hand, if i is a weak equivalence. Then, by CM5 proved above, i factorise as i = qj where q is a fibration and j is an anodyne extension, hence a weak cofibration. Then, by the (2 out of 3) property proved above q weak fibration. Then, using the above argument for the diagram A i

q

E

 /B

RA FT

 B

/ A′

j

′

idB

′

we have a lift h ∶ B → A . Then, i is a retract of j, so i is an anodyne extension. Hence, it has Since anodyne extensions have the LLP with respect to all fibrations, hence E admits a lift. CM5 (factorisation) Let f ∶ X → Y be a morphism of simplicial sets. Since every simplicial sets is small, then in particular the domains of I are small-relative to I-cell. Then, by the small object argument, f factorise as: f = pi, p ∈ I-inj, i ∈ I-cell. Since I-cell⊂ I-cof. Then, i is a cofibration. Also, I-inj are weak fibrations, hence p is weak fibration. The domains of J are small-relative to J-cell, hence f also factorise as: f = qj, q ∈ J-inj, j ∈ J-cell.

D

J-inj are Kan fibrations, hence q is a fibration. Also, J-cell⊂ J-cof, i.e. j is anodyne extension, hence a weak cofibration.  4.1. Cofibrations. As seen above the proof of the factorisation axiom CM5 is based on using transfinite small object argument on the classes of morphisms I and J. Therefore, we recall hereby that I and J admit transfinite small object argument in the below lemma. Lemma 4.2. Let κ be an infinite cardinal. Then, κ-bounded simplicial sets are small relative to cofibrations. Proof. Let A be a κ-bounded simplicial set. We need to show that for any κ- filtered ordinal λ, and any λ-sequence F ∶ λ → sSet of cofibrations, the canonical morphism (22) ϑ ∶ colim sSet(A, F− ) → sSet(A, colim F ) λ

λ

is a bijection. Since colimits of sets are quotients of disjoint unions, then to show that ϑ is surjective, we need to show that for any morphism of simplicial set f ∶ A → colim F there exist β < λ and morphism of simplicial λ

set A → Fβ , for which we have the commutative diagram A f



Fβ

...

' / colim F λ

where Fβ → colim F is the evident transfinite composition. λ

Since colimits in sSet are given object-wise, then ∀n ∈ Z≥0 , f induces maps of sets fn ∶ An → colim(F− )n λ

120

where (F− )n ∶ λ → Set is the evident λ-sequences of injections of sets. The transfinite composition of cofibrations is so as we see later in lemma ??, also the same holds for transfinite composition of injections of sets as in lemma ??. Hence, all evident transfinite compositions Fβ ↪ colim F , and (Fβ )n ↪ colim (F− )n ′ ′ β

β

′

are cofibrations and injections, respectively, ∀β < β ≤ λ. Then, by the definition of colimits of sets ∀xn ∈ colim(F− )n there exist βxn < λ such that xn ∈ (Fβn )n ⊆ colim(F− )n . In particular, ∀an ∈ An , there exists λ

λ

βan < λ such that fn (an ) ∈ (Fβan )n . Then, setting βn = sup{βan ∣an ∈ An } we notice that βn < λ because λ is a κ-filtered ordinal, βan < λ and ∣An ∣ < κ, recall definition ??. Then, fn (an ) ∈ (Fβan )n ⊂ (Fβn )n . Hence, maps fn factorise through (Fβn )n . Now, put β = sup{βn ∣n ∈ Z≥0 }. Since ∣Z≥0 ∣ = ℵ0 ≤ κ, and βn < λ, ∀n ∈ Z≥0 , then β < λ. Hence, all maps fn factorise through gn ∶ A → (Fβ )n . Now, in order to show that f itself factorises through Fβ , it is enough to show that gn ’s give rise to a morphism of simplicial set, i.e. they should be natural in their argument, which is evident from the commutativity of the below diagram, and the fact that (Fβ )m ↪ colim(F− )m is an injection of sets (i.e. a monomorphism, cancels

RA FT

λ

from the left), for any µ ∶ [m] → [n] in ∆

.

An

fn

gn

"  (Fβ )n

Aµ

 Am

...

) / colim(F− )n λ

(Fβ )µ

colim(F− )µ λ

fm

D

gm

"   (Fβ )m

...

)  / colim(F− )m λ

Therefore, there exists g ∶ A → Fβ , given object-wise above, such that f = ϑ([g]), where [g] is the class of g in colim sSet(A, F− ). λ

On the other hand, to show that ϑ is injective, let g ∶ A → Fβ , g ′ ∶ A → Fβ ′ such that ϑ([g]) = ϑ([g]), then we need to show that [g] = [g ′ ]. By the definition of ϑ cλβ ○ g = ϑ(g) = ϑ(g ′ ) = cλβ′ ○ g ′ where cλβ ∶ Fβ ↪ colim F , cλβ′ ∶ Fβ ′ ↪ colim F are the evident transfinite compositions. Without loose of λ

λ

generality, let β ≤ β ′ , since cλβ = cλβ′ ○ cββ , and cλβ′ cancels from the left for being a cofibration (a monomorphism ′

in sSet), then we have cββ ○ g = g ′ . But, [cββ ○ g] = [g], hence [g] = [g ′ ]. ′

′



Notice that the converse of the above statement is not true as shown in the below example. Corollary 4.3. The classes I and J defined above admits the transfinite small object argument in sPSh(C).

Proof. Direct result of lemma 4.2, above. That, I and J are classes of cofibrations, and for κ infinite, ∣∂∆n ∣ < κ, and ∣Λnr ∣ < κ, for every n ∈ Z≥0 , 0 ≤ r ≤ n.  5. Kan Extension Ex∞ Kan developed a method of calculating the fibrant replacements in Quillen’s model category of simplicial sets. It is given by Kan Ex∞ functor. For a simplicial set X, Ex∞ (X) is the colimit of an injective system X

jX

/ Ex(X)

jEx(X)

/ Ex2 (X)

jEx2 (X)

/ Ex3 (X)

jEx3 (X)

/ ...

121

of inclusions, where Ex is defied using the subdivision functor sd, by Ex(X)n = sSet(∆n , Ex(X)) = sSet(sd ∆n , X). with faces and degenerations given by pre-composition with cofaces and codegenerations of sd ∆n ’s. Recall that sd ∆n = N nd ∆n , where N is the nerve functor, and the nd − is the poset of nondegenerate simplices. It is worth noticing that Kan extension of Kan fibrants is a fibrant that is more involved and does not coincide with the original fibrant, even for very ’simple’ simplicial sets, as seen in the below example. Example 5.1. ∆1 is a Kan fibrant. Hereby we just calculate sk1 (Ex(∆1 )), and using the injectivity j’s above, deduce that ∆1 ≠ Ex∞ (∆1 ). For n = 0 id[0]

nd ∆0 = D ●

Therefore, sd ∆0 = N nd ∆0 = ∗, hence Ex(∆1 )0 = sSet(∗, ∆1 ) = ∆10 = {∂10 , ∂11 }. Then, for n = 1 ∂10

∂11

id

/ ●[1] o D

nd ∆1 = D ●

D● .

Then,

RA FT

(sd ∆1 )0 = {∂10 , id[1] , ∂11 }. (sd ∆1 )1 = {id∂10 , (∂10 , id[1] ), idid[1] , (∂11 , id[1] ), id∂11 }.

(sd ∆1 )2 = {(id∂10 , id∂10 ), (id∂10 , (∂10 , id[1] )), ((∂10 , id[1] ), idid[1] ), (idid[1] , idid[1] ), ((∂11 , id[1] ), idid[1] ), (id∂11 , (∂11 , id[1] )), (id∂11 , id∂11 ))} . ⋮

To understand a simplicial set is basically equivalent to understanding its faces and degenerations. Hence, the faces and degenerations of sd ∆1 are highlighted below, up to 2-simplicies. s00 maps the 0-simplicies to their identity maps, i.e. it maps ∂10 , id[1] , ∂11 to id∂10 , idid[1] , id∂11 , respectively. Hence, there are only two non-degenerate 1-simplicies in sd ∆1 , namely (∂10 , id[1] ), (∂11 , id[1] ). d10 , d11 maps 1-simplicies to their codomain, domain respectively. s10 , s11 maps 1-simplicies to the 1-simplicies obtained by insetting identity morphisms at the domain, codomain, respectively. In particular, s10 maps

D

id∂10 , (∂10 , id[1] ), idid[1] , (∂11 , id[1] ), id∂11 ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶ ↧ (id∂10 , id∂10 ), (id∂10 , (∂10 , id[1] )), (idid[1] , idid[1] ), (id∂11 , (∂11 , id[1] )), (id∂11 , id∂11 )

and s11 maps id∂10 , (∂10 , id[1] ), idid[1] , (∂11 , id[1] ), id∂11 ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶ ↧ (id∂10 , id∂10 ), ((∂10 , id[1] ), idid[1] ), (idid[1] , idid[1] ), ((∂11 , id[1] ), idid[1] ), (id∂11 , id∂11 ). Then, all 2-simplicies of sd ∆1 are degenerate. Actually that can been seen from the facts that any pair of decomposable morphisms of nd ∆1 contains an identity morphism, and by the construction of degenerations of nerves, all pairs that contains identity morphism are degenerations. Hence, all n-simplicies of sd ∆1 are degenerate for n ≥ 2. d20 , d22 maps 2-simplicies to their codomain, domain respectively, whereas d21 maps 2-simplicies to their composition. Hence sd ∆1 has only two non-degenerate simplicies, in addition to its three vertices with d10 ((∂10 , id[1] )) = id[1] , d10 ((∂11 , id[1] )) = id[1] , and d11 ((∂10 , id[1] )) = ∂10 , d11 ((∂11 , id[1] )) = ∂11 Hence, it has the geometric realisation ∂10

●

id[1] (∂10 ,id[1] )

●

(∂11 ,id[1] )

∂11

● .

Since im d10 ⋃ im d11 = (sd ∆1 )0 , then morphisms of simplicial sets from sd ∆1 are determined by the image of the two 1-simplices (∂10 , id[1] ) and (∂11 , id[1] ). ∆1 has three 1-simplices ∆11 = {∂10 σ00 , id[1] , ∂11 σ00 } Then a straightforward verification shows that there are exactly five morphisms of simplicial sets from sd ∆1 to ∆1 , given below by a visualisation of their images

122

B

A

C ∂01

id[1] ∂01

id[1]

∂11

∂01

D

id[1]

∂11

id[1] ∂11

E ∂11

∂01

id[1]

∂11

∂01

where upper positions represent being mapped to ∂10 σ00 , diagonals to id[ 1], and lower positions to ∂11 σ00 . Since faces and degenerations of Ex(∆1 ) is given by pre-composition with cofaces and codegenerations of sd ∆n ’s, then sk 1 (Ex(∆1 )) has the geometric realisation visualised below: ∂10

C

●

∂10

●

B

D

Then ∆1 ⫋ sk1 (Ex(∆1 )) ⊂ Ex(∆1 ). Since Ex∞ (∆1 ) is the colimit if injective system of inclusions then, ∆1 ⫋ Ex∞ (∆1 ). Remark 5.2. When calculating Kan extension for standard simplexes ∆n . Calculation can be simplified due to the fact that the nerve functor N is faithfully full and that N[n] = ∆n . Then, Ex(∆n )n = Set(sd ∆1 , ∆1 ) = Set(N nd ∆1 , N[n]) ≅ PoS(nd ∆1 , [n])

D

RA FT

as has been used in example 1.42.

CHAPTER 4

Simplicial Pre-sheaves The main aim of this chapter is to recall the category of simplicial (pre-)sheaf on a small Grothendieck site, and recall that it admits a model structure, in fact several different model structures. Hereby, we mainly follow [Jar87], and [PB13b]. op Throughout this chapter, Cτ denotes a small Grothendieck site, PSh(C) ∶= SetC the category of pre-sheaves of sets on the underlying category of the site C, and Shvτ (C) the category of sheaves of sets on Cτ . 1. The Category of Simplicial Pre-sheaves

RA FT

Definition 1.1. [Simplicial Pre-sheaf ] A simplicial pre-sheaf on C is a simplicial object in PSh(C), i.e. a contra-variant functor X ∶ ∆op → PSh(C). The category of simplicial pre-sheaves on C and their natural transformations is denoted by sPSh(C). Then, a simplicial pre-sheaf is given by functors Xn ∶ Cop → Set, ∀n ≥ 0, and natural transformations Xµ ∶ Xn → Xm , ∀µ ∶ [m] → [n] in ∆. Hence, X is given by ● sets Xn (U ) for all n ≥ 0, U ∈ C. ● maps Xn (ϕ) ∶ Xn (U ) → Xn (V ) for n ≥ 0, ϕ ∶ V → U in C. ● maps (Xµ )(U ) ∶ Xn (U ) → Xm (U ) for µ ∶ [m] → [n] in ∆, U ∈ C. The composition Xm (ϕ)Xµ (U ) = Xµ (V )Xn (ϕ) will be denoted by Xµ (ϕ).

D

When it is unambiguous we might write ϕ∗ instead of Xn (ϕ), µ∗ instead of (Xµ )(U ), and for xn,U ∈ Xn (U ) we denote µ∗ ϕ∗ (xn,U ) by xm,V . In this chapter by simplicial pre-sheaves we mean a simplicial pre-sheaf on C unless mentioned otherwise, and are denoted by calligraphic letters X, Y, Z, ..., and their morphisms with the small Latin letters f, g, h, i, j, p, q, .... Also, since Grothendieck site is a generalisation of topological spaces, the objects of C are denoted by the capital Latin letters U, V, W, ..., and its morphisms by ϕ, ψ, .... We keep the notation used in chapter B for the simplicial sets and simplicial category unchanged. A morphism of simplicial pre-sheaves f ∶ X → Y is a natural transformation, then it is given by its components fn , n ≥ 0n ≥ 0, i.e. morphisms in PSh(C), which are in tune natural transformations, given by their components fn,U ∶= (fn )U , U ∈ C. The above definition of simplicial pre-sheaves is coherent with the definition of simplicial objects in general. However, the fact that Cat is a Cartesian closed category, with the exponential given by the functor category , provides an alternative way of thinking of simplicial pre-sheaves. That, there exist natural isomorphisms (28)

≅

≅

Cat(∆op , PSh(C)) → Cat(Cop × ∆op , Set) → Cat(Cop , sSet)

Then, there exist categories isomorphisms: (29)

− ∶ sPSh(C) → PShsSet (C)

given on objects by X(U )n = Xn (U ), X(U )µ = Xµ,U , and X(ϕ)n = Xn (ϕ) for µ ∶ [m] → [n] in ∆, ϕ ∶ U → V in C and X ∈ sPSh(C), and on morphism by (f U )n = (fn )U for f ∶ X → Y. and (30)

− ∶ sPSh(C) → PSh(∆ × C) 123

124

given on objects by X([n], U ) = Xn (U ), , and X(µ, ϕ) = Xµ,ϕ for µ ∶ [m] → [n] in ∆, ϕ ∶ U → V in C and X ∈ sPSh(C), and on morphism by f n (U ) = (fn )U for f ∶ X → Y. Categories isomorphism are exact and coexact, commutes with limits and colimits. Hence, some authors might abuse notation, and use the term simplicial pre-sheaves to refer to an object in either of the three categories in (28). In this notes, we also abuse the notation, and use the notation interchangeably. Definition 1.2 (Simplicial Sheaves). A simplicial sheaf on Cτ is a simplicial object in Shvτ (C). The category of simplicial sheaves on C and their natural transformations is denoted by sShvτ (C). Notice that the embedding ι ∶ Shvτ (C) ↪ PSh(C) induces an embedding ι∗ ∶

sShvτ (C)

X f

↪ ↦ ↦

sPSh(C)

ιX idι ⋅ f

.

Also, the left adjoint −# ∶ PSh(C) → Shvτ (C) induces the functor sPSh(C)

X f

→ ↦ ↦

sShvτ (C)

−# X id−# ⋅ f

.

T

−#∗ ∶

DR AF

Then, using ??, one sees that −#∗ is a left adjoint for ι∗ . 1.0.1. Embeddings.

Remark 1.3. Let E be a set. Then, the constant functor E ∶ ∆op → PSh(C) given by E n (U ) = E

defines a simplicial sheaf on Cτ . Also, let X be a simplicial set. Then, the constant functor X ∶ Cop → sSet given by X(U ) = X defines a simplicial pre-sheaf on Cτ . In particular, for n ≥ 0, ∆n denote the simplicial pre-sheave given by the constant functor Cop → sSet with values ∆n . Lemma 1.4. There is a fully faithful embedding < − >sSet ∶ sSetsPSh(C). Proof. TBC.



In order to stay consistence with the previous notation, then for f ∶ X → Y in C, we denote −sSet (f ) by f ∶X →Y. Remark 1.5. On the other hand, for any pre-sheaf F ∶ Cop → Set, the constant functor F ∶ ∆op → PSh(C) defines a simplicial pre-sheaf, given by F n = F. In particular, Yoneda embedding hU = C(−, U ) ∶ Cop → Set for an object U ∈ C defines a simplicial pre-sheaf U ∶ ∆op → PSh(C) given by U n = hU . Lemma 1.6. There is a fully faithful embedding < − >PSh(C) ∶ PSh(C)sPSh(C). Proof. TBC.



Lemma 1.7. Yoneda embedding induces a fully faithful embedding of C in sPSh(C). Proof. Define −C ∶ (31)

C ↪ sPSh(C) U ↦ U, ϕ ↦ ϕ∗ ,

U n = hU given component-wise by the composition with ϕ.

It is easy to see that −C is a well-defined fully faithful embedding.



125

In order to stay consistence with the previous notation, then for ϕ → U in C, we denote −C (f ) by ϕ ∶ V → U . Hence, we notice that the category of simplicial pre-sheaves encapsulate information about both the underlying site and the category of simplicial sets. Example 1.8. Let C = 0, the category with single object and morphism. Then, C admits a unique Grothendieck topology τ . Moreover, sShvτ (C) = sPSh(C) = sSet. Therefore, it is natural when looking for a model structure on sPSh(C) to seek a model structures induced the canonical model structure on sSet for C = 0. 1.0.2. Restriction. Let X be a simplicial pre-sheaves, U ∈ C. We define the restriction X∣U to be X∣U ∶

(C ↓ U )op ϕ∶V →U V ϕ

ψ

  U

Ð→ sSet ↦ X(V )

/ V′

↦

X(ψ).

ϕ′

RA FT

Then, one can see that X∣U is a simplicial pre-sheaf on C ↓ U - in the sense of the above abuse of notation (29). Also, for f ∶ X → Y a morphism of simplicial pre-sheaves, there exists the induced morphism f ∣U ∶ X∣U → Y∣U

in sSet(C↓U )

op

given by f ∣U ϕ∶V →U = fV . Then, one can see easily that −∣U ∶ sPSh(C) → sSet(C↓U )

op

given above on objects and morphisms defines a functor. 1.0.3. Pointed Restriction. One would like to have a pointed version of the above restriction functor, that will be used later to define an analogue of homotopy groups of simplicial sets in the case of simplicial presheaves, called the homotopy sheaves. Then, one can start with the the naive construction of considering global objects, as in Set∗ , Top∗ and sSet∗ . So, let sPSh(C)∗ be the category whose objects are morphisms of simplicial pre-sheaves of the form ∗ → X, where X is a simplicial pre-sheaf, with evident morphisms. Then, for every object in f ∶ ∗ → X ∈ sPSh(C)∗ , and for U ∈ C, we define define the pointed restriction of f to be

D

f ∣∗U ∶

(C ↓ U )op ϕ∶V →U

V ϕ

ψ

  U

Ð→ sSet∗ ↦ (X(V ), f0,V (∗))

/ V′

↦

X(ψ).

ϕ′

which is easily seen to be well-defined that f0,V (∗) ∈ X0 (V ) and X0 (ψ)(f0,V ′ (∗)) = f0,V (x0,V ). Also, it is easy to see that it defines a pre-sheaf of pointed simplicial sets on C ↓ U . Then, for the morphism XW

g

f

/ X′ F f′

∗ in sPSh(C)∗ , we define

h∣∗U ∶ f ∣∗U → f ′ ∣∗U

given by (g∣∗U )ϕ∶V →U = gV . It is easy to see that (g∣∗U )ϕ∶V →U is a morphism of pointed simplicial sets, and that g∣∗U ) is a natural transformation from f ∣∗U to f ′ ∣∗U . Then, one can see easily that −∣∗U ∶ sPSh(C)∗ → PShsSet∗ (C ↓ U ) given above on objects and morphisms defines a functor. If no confusion is caused then we might abuse notation and write X∣∗U , g∗ instead of f ∣∗U , g∣∗U , respectively; Especially it make it easier to distinguish between the notation of objects and morphisms.

126

However, the above definition is a bit redundant, that different morphisms ∗ → X can give rise to the same pointed restriction when there are U, V ∈ C with C(U, V ) = C(V, U ) = ∅. Therefore, we prefer to use a more efficient version of the above definition. The below version is being used implicitly in the literature without actually being emphasises, and we prefer to adopt to this version. to avoid the above issue, one can simply single out sections of X0 (U ), as follows: For U ∈ C, let sPSh(C)U be the category whose objects are pairs (X, x0,U ), where X is a simplicial presheaf, and x0,U ∈ X0 (U ) is a vertex of X0 (U ), and whose morphisms are morphisms of simplicial pre-sheaves preserving the distinguish U -vertex. Then, we define the pointed restriction to U as (32)

(−∣∗U , ∗) ∶ sPSh(C)U → PShsSet∗ (C ↓ U )

given on objects for (X, x0,U ) ∈ sPSh(C)U by (C ↓ U )op ϕ∶V →U

(X∣∗U , x0,U ) ∶

V ϕ

ψ

  U

Ð→ sSet∗ ↦ (X(V ), ϕ∗ (x0,U ))

/ V′

X(ψ).

↦

ϕ′

∗

RA FT

which is easily seen to be well-defined that ϕ∗ (x0,U ) ∈ X0 (V ) and X0 (ψ)(ϕ′ (x0,U )) = ϕ∗ (x0,U ). Also, it is easy to see that it defines a pre-sheaf of pointed simplicial sets on C ↓ U . Whereas, it is given morphisms for f ∶ (X, x0,U ) → (Y, f0,U (x0,U )) by

(f ∣∗U , x0,U ) ∶ (X∣∗U , x0,U ) → (Y∣∗U , f0,U (x0,U ))

D

such that (f ∣U , x0,U )ϕ∶V →U = fϕ . It is easy to see that (f ∣∗U , x0,U )ϕ is a morphism of pointed simplicial sets, and that (f ∣∗U , x0,U ) is a natural transformation from (X∣∗U , x0,U ) to (Y∣∗U , f0,U (x0,U )). If no confusion is caused then we might abuse notation and write f∗ instead of (f ∣∗U , x0,U ). Also, it is convenient to abuse notation and shorten ϕ∗ (x0,U ) = x0,V , if no confusion may arise. 1.0.4. Pointed Restriction of Pairs. It some occasions, it will convenient to work with pairs of simplicial sets in similar manner to topological spaces. Hence, below we generalise the above definition, which will be used later to define sheaves of relative homotopy groups in a similar manner of the case in topology, see [Swi75, §.3]. For U ∈ C, let sPSh2 (C)U denote the category whose objects are pairs (i ∶ K ↪ L, k0,U ), where i is an injective cofibration, and k0,U ∈ K0 (U ), and its morphism are morphism preserving the distinguish U -vertex, of the form f ∶ L → L′ such that the below diagram commutes  i / K L f ∣K

   K′

f

i′

 / L′ .

We call each object of sPSh2 (C)U a pointed pair at U . Then, similar to (32), pointed restriction is generalised to pointed restriction of pointed pairs at U as follows (33)

2

(−∣∗U , −∣∗U , ∗) ∶ sPSh2 (C)U → PShsSet∗ (C ↓ U )

given on objects for (i ∶ K ↪ L, k0,U ) ∈ sPSh2 (C)U by (i∣∗U , k0,U ) ∶

(C ↓ U )op ϕ∶V →U V ϕ

ψ

  U

/ V′

Ð→ sSet2∗ ↦ (L(V ), K(V ), ϕ∗ (k0,U )) ↦

L(ψ)

ϕ′

where sSet2∗ is the category of pointed pairs of simplicial sets, see ??. It is easy to see that (i∣∗U , k0,U ) well∗ defined that i is an injective cofibration, ϕ∗ (k0,U ) ∈ K0 (V ) and L0 (ψ)(ϕ′ (i0,U (k0,U ))) = i0,U (ϕ∗ (k0,U )). Also, it is easy to see that it defines a pre-sheaf of pairs of pointed simplicial sets on C ↓ U .

127

Whereas, it is given morphisms for f ∶ (i ∶ K ↪ L, k0,U ) → (i′ ∶ K′ ↪ L′ , f0,U (k0,U )) by (f ∣U , k0,U ) ∶ (i∣∗U , k0,U ) → (i′ ∣∗U , f0,U (k0,U )) such that (f ∣U , k0,U )ϕ∶V →U = fϕ . It is easy to see that (f ∣∗U , k0,U )ϕ is a morphism of pointed pair of simplicial sets, and that (f ∣U , k0,U ) is a natural transformation from (i∣∗U , x0,U ) to (i′ ∣∗U , f0,U (k0,U )). If no confusion is caused then we might abuse notation and write f∗ instead of (f ∣∗U , x0,U ), x0,V instead of ϕ∗ (x0,U ), and (L, K, k0,U ), (L∗U , K∗U , k0,U ) if i is to be understood. 1.0.5. Morphisms. Natural transformations are monomorphisms, epimorphisms, or isomorphisms if they are monomorphisms, epimorphisms, or isomorphisms object-wise, respectively. Hence, a monomorphisms X ↪ Y is denoted by X ⊂ Y, and X is called a simplicial sub-pre-sheaf of Y. If X ⊆ Y then Xµ (ϕ) = Yµ (ϕ)∣Xn (U ) for µ ∶ [m] → [n] in ∆, and ϕ ∶ V → U in C. 1.1. Limits and Colimits. Since the category Set is a complete and cocomplete category, then by lemma ?? the category of simplicial pre-sheaves is a complete and cocomplete category with limits and colimits given object-wise. I.e. for a digaram functor F ∶ J → sPSh(C), we have: (lim F )n (U ) = lim Fn (U )text, and (colim F )n (U ) = colim Fn (U ) J

J

J

J

Where Fn (U ) = (−)n (U )F for

RA FT

(−)n (U ) ∶ sPSh(C) → Sets

with (X)n (U ) = Xn (U ) and (f )n (U ) = fn,U .

Whereas, maps (lim F )n,ϕ , (lim F )µ,U , (colim F )n,ϕ , and J

J

J

(colim F )µ,U are induced by universal property of limit and colimit. J

1.1.1. Initial and Terminal Objects. Let X, Y be a simplicial pre-sheaf such that Xn (U ) = ∅, Yn (U ) = ∗ f oralln ≥ 0, U ∈ C, with the evident maps. Then, X is called the empty simplicial pre-sheaf, and it is denoted by ∅, and Y is denoted by ∗. One can readily see that ∅, and ∗ are the initial and terminal objects of sPSh(C), respectively. 1.1.2. Finite Limits and Colimits. We recall here the description of the building blocks of finite (co)limits, namely (co)product, (co)equaliser: Products : Let X, Y be simplicial pre-sheaves, then the product of X and Y is given by

D

(X × Y)n (U ) = Xn (U ) × Yn (U ) , (X × Y)µ,ϕ = Xµ,ϕ × Yµ,ϕ

Hence, elements of (X × Y)n (U ) are ordered pairs (xn,U , yn,U ) where xn,U ∈ Xn (U ) and yn,U ∈ Yn (U ). Faces and degeneracy are given component-wise by: dn,U (xn,U , yn,U ) = (dn,U (xn,U ), dn,U (yn,U ))text, and sn,U (xn,U , yn,U ) = (sn,U (xn,U ), sn,U (yn,U )) i i i i i i More generally, (X × Y)µ,ϕ (xn,U , yn,U ) = (Xµ,ϕ (xn,U ), Yµ,ϕ (yn,U )) Equaliser : Let f, g ∶ X → Y be morphisms of simplicial pre-sheaves, then eq(f, g), the equaliser of f and g is the simplicial sub-pre-sheaf of X given by: eq(f, g)n (U ) = {xn,U ∈ Xn (U )∣(fn,U )(xn,U ) = (gn,U )(xn,U )} ⊂ Xn (U ). One can see easily that eq(f, g)n (U ) defines a simplicial sub-pre-sheaf of X, and that it is the equaliser object of f and g, with the equaliser morphism given by the inclusion eq(f, g) ⊂ X. Fibre Product: Let f ∶ X → Y, f ′ ∶ X′ → Y be morphisms of simplicial pre-sheaves. Then, the fibre product of X ×Y X′ is given the equaliser of f pr1 ∶ X × X′ → Y and f ′ pr2 ∶ X × X′ → Y. Intersection, Union and Difference: Lemma 1.9. Let i ∶ X ↪ Y, i′ ∶ X′ ↪ Y be inclusions of simplicial pre-sheaves. Then, (X ×Y X′ )n (U ) ≅ Xn (U ) ⋂ X′n (U ). Proof. (X ×Y X′ )n (U ) = {(xn,U , x′n,U ) ∈ (X × X′ )n (U )∣(in,U )(xn,U ) = i′n,U (x′n,U )} ≅ Xn (U ) ⋂ X′n (U ).  Lemma 1.10. Let i ∶ X ↪ Y, i′ ∶ X′ ↪ Y be inclusions of simplicial pre-sheaves. Then, (X ×X ⋂ X′ X′ )n (U ) ≅ Xn (U ) ⋃ X′n (U ).

128

Proof. (X×X ⋂ X′ X′ )n (U ) = (X ⊔ X′ )n (U )/(X ⋂ X′ )n (U ) = (Xn (U ) ⊔ X′n (U ))/(Xn (U ) ⋂ X′n (U )) ≅ Xn (U ) ⋃ X′n (U ).  The above lemmas motivate the following definitions Definition 1.11 (Intersection). Let i ∶ X ↪ Y, i′ ∶ X′ ↪ Y be monomorphisms of simplicial pre-sheaves. Then, the intersection X ⋂ X′ is defined to be the fibre product X ×Y X′ . Hence, (X ⋂ X′ )n (U ) ∶= Xn (U ) ⋂ X′n (U ) for every n ≥ 0, U ∈ C. Since pull-back of injection is so, it is easy to see that X ⋂ X′ ⊂ X. Definition 1.12 (Union). Let i ∶ X ↪ Y, i′ ∶ X′ ↪ Y be monomorphisms of simplicial pre-sheaves. Then, the union X ⋃ X′ is defined to be the push-out X ∐X ⋂ X′ X′ . Hence, (X ⋃ X′ )n (U ) ∶= Xn (U ) ⋃ X′n (U )

for every n ≥ 0, U ∈ C.

RA FT

By lemma 2.60 the push-outs of monomorphisms of simplicial pre-sheaves (injective cofibrations) is so, it is easy to see that X ⊂ X ⋃ X′ . Definition 1.13 (Difference).

1.2. Representable Simplicial Pre-sheaves. Representable simplicial pre-sheaves plays an important role in the study of simplicial pre-sheaves, similar to the role of standard simplexes in the study of simplicial sets. Therefore, it is recalled below: Definition 1.14. For any U ∈ C, and n ≥ 0, consider the representable pre-sheaf: ∆ × C(−, ([n], U )) ∶ (∆ × C)op → Set.

Since (∆ × C)op = ∆op × Cop , then (28) gives rise to the simplicial pre-sheaves on C given by:

D

∆nU ∶ ∆op → PSh(C)

(∆nU )m =

hom∆×C (([m], −), ([n], U )) ≅ ∆nm × C(−, U )

((∆nU )µ )− = µ∗ × (idC(−,U ) )∗

for m ≥ 0 for µ ∶ [l] → [m] in ∆

Since these simplicial pre-sheaves are given rise from representable pre-sheaf, then we abuse notation and call ∆nU a representable simplicial pre-sheaf. In fact, for each m ≥ 0 (∆nU )m (−) = ∆nm ×C(−, U ) is representable pre-sheaf. Then, we have

(∆nU )m (V ) ≅ ∆nm × C(V, U )text, and (∆nU )ν,ψ = ν ∗ × ψ ∗ for ν ∶ [l] → [m] in ∆ and ψ ∶ W → V in C.

Moreover, for every morphism (µ, ϕ) ∶ ([m], V ) → ([n], U ) in ∆×C, we have a morphism of simplicial pre-sheaves n ∆µϕ ∶ ∆m V → ∆U

given by (∆µϕ )l,W = µ∗ × ϕ∗ , for ([l], W ) ∈ ∆ × C. Remark 1.15. If the site Cτ has a terminal object U = ∗, then (∆n∗ )m (V ) ≅ ∆nm × ∗ ≅ ∆nm . Then, it is natural to abuse notation and denote the simplicial pre-sheave ∆n∗ by ∆n . Also, for the unique site Cτ with the underlying category C = ∗, PSh(C) = Set. Hence, sPSh(C) = sSet, and ∆n = ∆n∗ . Lemma 1.16. ∆−− , defined on objects and morphisms above, gives a the fully faithful embedding ∆−− ∶ ∆ × C → sPSh(C).

129

Proof. On the one hand, for every (µ, ϕ), (µ′ , ϕ′ ) ∶ ([m], V ) → ([n], U ) in ∆ × C such that ∆µϕ = ∆µϕ′ , then ′

′

(∆µϕ )l,W = (∆µϕ′ )l,W , for ([l], W ) ∈ ∆ × C, particularly for l = m, W = V , then applying the above equality on (id[m] , idV ), we find that (µ × ϕ) = (µ′ , ϕ′ ), hence ∆−− is faithful. n On the other hand, for every f ∶ ∆m V → ∆U , we need to show that there exists (µ, ϕ) ∶ ([m], V ) → ([n], U ) in ∆ × C such that f = ∆µϕ .

Let (µ, ϕ) = fm,V (id[m] , idV ) ∈ ∆nm × C(V, U ), i.e. (µ, ϕ) ∶ ([m], V ) → ([n], U ) in ∆ × C. Then, for every (ν, ψ) ∶ ([l], W ) → ([m], V ) in ∆ × C, we have the below commutative diagram: fm,V

(∆m V )m (V )

(id[m] , idV ) ∈

U

(∆m V )ν,ψ

U

= fl,W (id[m] ν, idV ψ) = fl,W ((id[m] , idV )(ν, ψ)) = fl,W ((∆m V )ν,ψ (id[m] , idV )) = (∆nU )ν,ψ (fm,V (id[m] , idV )) = (∆nU )ν,ψ (µ, ϕ) = (µ, ϕ)(ν, ψ) = (∆µϕ )l,W (ν, ψ).

DR AF

fl,W (ν, ψ)

fl,W

 / (∆n )l,W .

T

Hence, we have:

(∆n U )ν,ψ

↺

 (∆m V )l,W

(ν, ψ) ∈

/ (∆n )m (V )

Hence, f = ∆µϕ , and ∆−− is full.



Remark 1.17. We notice that ∆nU plays in simplicial pre-sheaves similar role to ∆n in simplicial sets. That, Yoneda’s lemma implies that for any simplicial pre-sheaf X, there is a natural isomorphism (34)

⋅

ι ∶ X−,− ≅ Set(∆×C) (∆−− , X) op

i.e. there exist isomorphisms, for n ≥ 0, U ∈ U

≅

ιn,U ∶ Xn (U ) → Set(∆×C) (∆nU , X). op

Hence, a morphism ∆nU → X can be identified with an U -element of an n-simplex of X. We denote ιxn,U ∶ ιn,U (xn,U ), which is the unique morphism that satisfies ιxn,U ,n,U (id[n] , idU ) = xn,U . Then we have the below commutative square for every (µ, ϕ) ∶ ([m], V ) → ([n], U ) in ∆ × C: (35)

xn,U ∈

Xn (U )

ιn,U

op

µ Xϕ

 Xm (V )

/ Set(∆×C) (∆n , X) U

↺



∗ (∆µ ϕ)

/ Set(∆×C) (∆m , X) V op

ιm,V

Definition 1.18 (The category of elements of X, (∆−− ↓ X)). Let X be a simplicial pre-sheaf, the category of elements of X is define to be the comma category (∆−− ↓ X), with objects being the morphisms of simplicial pre-sheaves of the form ∆nU → X, i.e. ιxn,U for all xn,U ∈ Xn (U ), n ≥ 0, U ∈ C, and morphisms being commutative diagrams:

ιxm,V

/ ∆n . U

f

∆m V !

D

X

ιxn,U

~

Since ∆−− is fully faithful, we found in the proof of lemma 1.16 that f = ∆µϕ for a unique morphism (µ, ϕ) ∶ ([m], V ) → ([n], U ) in ∆ × C. Then, we define the evident forgetful functor (36)

UX ∶ (∆−− ↓ X) ιxn,U D

→ ↦ ↦

sPSh(C)

∆nU f

.

130

Similar to the case of simplicial sets, the below lemma shows that all simplicial pre-sheaves are just colimits of diagrams of ∆nU ’s. Hence, it is important to work with functors on sPSh(C) that preserves colimits, because in that case everything can be reduced again to working with ∆nU ’s. Lemma 1.19. Let X be a simplicial pre-sheaf, then X = colim UX . − (∆− ↓X)

Proof. Consider the diagonal functor, ?? ˆ ∶ sPSh(C) → sPSh(C)(∆− ↓X) − −

that sends each simplicial pre-sheaf to its corresponding constant functor. Then, based on the definition of ⋅ ˆ given (∆−− ↓ X) there is a cocone of UX with vertex X, that there is a natural transformation ηX ∶ UX → X n component-wise by ηX,ιn,U = ιn,U ∶ ∆U → X. Then, we need to show that (X, ηX ) is initial universal morphism ˆ. from UX to − ⋅ ˆ is a natural transformation. Hence, for every D ∶ ιx Let (Y, ηY ) be a cocone of UX , i.e. ηY ∶ UX → Y → ιxn,U m,V − in (∆− ↓ X), induced by (µ, ϕ) ∶ in ∆ × C we have the commutative square (∆µ ϕ )l,W

(∆m V )l,W

/ (∆n )l,W U

RA FT

(37)

(ηY,ιx

m,V

↺

)l,W

(ηY,ιx

# { Yl,W

n,U

)l,W

Then, define the collection of maps for n ≥ 0, U ∈ C

ξn,U ∶ Xn (U ) → Yn (U )

D

by ξn,U (xn,U ) = (ηY,ιxn,U )n,U (id[n] , idU ) for xn,U ∈ Xn (U ). This collection of maps makes the following diagrams commute for every µ ∶ [m] → [n] in ∆, ϕ ∶ V → U in C Xn (U ) Xµ,ϕ

 Xm (V )

ξn,U

↺

/ Yn (U )

Yµ,ϕ

ξm,V

 / Ym (V )

that ∀xn,U ∈ Xn (U ), we have: (35)

(ξm,V Xµ,ϕ ) (xn,U ) = ξm,V (Xµ,ϕ (xn,U )) = (ηY,ιXµ,ϕ (xn,U ) )m,V (id[m] , idV ) = (ηY,(ι (37)

= (ηY,ιxn,U ) = (ηY,ιxn,U ) ∗

m,V

m,V

((∆µϕ )m,V (id[m] , idV )) = (ηY,ιxn,U )

m,V

(id[n] µ, idU ϕ) = (ηY,ιxn,U )

m,V

xn,U

∆µ ϕ)

) m,V

(id[m] , idV )

(µ, ϕ)

((∆nU )µ,ϕ (id[n] , idU ))

= Yµ,ϕ ((ηY,ιxn,U )n,U (id[n] , idU )) = (Yµ,ϕ ξn,U ) (xn,U ). (∗) is due to fact that ηY,ιxn,U ’s are natural transformations. Hence, the above diagram commutes, and the collection of ξm,V defines a natural transformation ξ∶X→Y given component-wise above. Notice that ξ makes the following diagram commutative UX ηX

 ˆ X

ηY

ξˆ

ˆ /Y ?

131

that = ξm,V (ιxn,U ,m,V (µ, ϕ)) = ξm,V (ιxn,U ,m,V ((∆nU )µ,ϕ (id[n],idU ))) = ξm,V (Xµ,ϕ (ιxn,U ,n,U (id[n],idU ))) = ξm,V (Xµ,ϕ (xn,U )) = Yµ,ϕ (ξn,U (xn,U )) = Yµ,ϕ ((ηY,ιxn,U )n,U (id[n] , idU )) = (ηY,ιxn,U )m,V ((∆nU )µ, ϕ(id[n] , idU )) = (ηY,ιxn,U )m,V (µ, ϕ)

(ξm,V ηX,ιxn,U ,m,V ) (µ, ϕ)

In order to show the uniqueness of ξ, let ξ ′ ∶ X → Y such that ηY = ξ ′ ηX , then ′ ξn,U (xn,U )

′ ′ (ιxn,U ,n,U (id[n] , idU )) = ξn,U = ξn,U (ηX,ιxn,U ,n,U (id[n] , idU )) = ηY,ιxn,U ,n,U (id[n] , idU ) = ξn,U (xn,U )

UX . Hence, ξ ′ = ξ. Therefore, X = colim −



(∆− ↓X)

In general C might not satisfy some desired properties that sPSh(C) satisfy, hence it is useful to view C as being embedded in sPSh(C), as in the below lemma. Lemma 1.20. The functor ∆0− , given below, is a fully faithful embedding. ∆0− ∶

C → U ↦ ϕ ↦

sPSh(C)

∆0U ∆0ϕ

RA FT

(38) is a fully faithful embedding

Proof. Notice that ∆([l], [0]) is a singleton for every l ≥ 0. Then, we can see that the proof is a simplified version of the proof of lemma 1.16.  1.3. Boundedness. The concept of κ-boundedness is essititial for the proofs of the lemmas leading to the proof of 2.57. CM5. Therefore, in this subsection we recall κ-bounded simplicial pre-sheaves and their properties. Definition 1.21 (κ-bounded Pre-sheaves). Let κ be an infinite cardinal, F ∈ PSh(C), we say that F is κ-bounded iff ∣F(U )∣ < κ, ∀U ∈ C.

D

Definition 1.22 (κ-bounded Simplicial Pre-sheaves). Let κ be an infinite cardinal, X ∈ sPSh(C), we say that X is κ-bounded iff Xn is κ-bounded, ∀n ≥ 0, i.e. iff ∣Xn (U )∣ < κ, ∀n ≥ 0, U ∈ C.

Example 1.23. Let κ be an infinite cardinal, such that κ > 2∣M or(C)∣ , then ∆nU is κ-bounded, ∀n ≥ 0, U ∈ C. That, ∀m ≥ 0, V ∈ C ∣(∆nU )m (V )∣ = ∣∆([m], [n]) × C(V, U )∣. ∣∆([m], [n])∣ is always finite, whereas we distinguish two cases for ∣C(V, U )∣, if it is finite then ∣∆([m], [n]) × C(V, U )∣ is finite, otherwise ∣∆([m], [n]) × C(V, U )∣=∣C(V, U )∣ In both cases ∣(∆nU )m (V )∣ = ∣∆([m], [n]) × C(V, U )∣ < κ. Hence, ∆nU is κ-bounded. Remark 1.24. Notice that if X is a κ-bounded simplicial pre-sheaf for a cardinal κ, and there exists an isomorphism either f ∶ X → Y or g ∶ Y → X, then Y is κ-bounded. Then, the previous example and lemma 1.19 shows that, for an infinite cardinal κ > 2∣M or(C)∣ , any simplicial pre-sheaf X is a colimit of κ-bounded simplicial pre-sheaves, namely im ιxn,U for xn,U ∈ Xn (U ), n ≥ 0, U ∈ C, actually we see below that it is a filtered colimit as in corollary ??. The below two lemmas indicated implicitly in [Jar87], and presented explicitly in [PB13b]. Lemma 1.25. Let κ an infinite cardinal, C, I be small categories, such that ∣Ob(I)∣ < κ, and let F ∶ I → PSh(C) be a functor of κ-bounded pre-sheaves. Then colim F is κ-bounded.

Proof. Colimit of pre-sheaves of sets is given object-wise, hence (colim F )(U ) = colim FU , U ∈ C, where FU ∶ I → Set given by i ↦ Fi (U ) And, colimit of sets is given as a quotient of a disjoint union, i.e. (colim F )(U ) = colim FU = ⋃ Fi (U )/ ∼ i∈I

hence ∣(colim F )(U )∣ < ∣⋃ Fi (U )∣ < κ ⋅ ∣I∣ = max(κ, ∣I∣) = κ i∈I

i.e. colim F is κ-bounded pre-sheaf.



Lemma 1.26. Let κ be an infinite cardinal such that κ > 2∣M or(C)∣ . Then, every pre-sheaf on C is a filtered colimit of its κ-bounded sub-objects.

132

Proof. Let F be a pre-sheaf on C, IF be set of κ-bounded sub-objects of F , with a partial order relation induced by monomorphisms, the we see that I is filtered. Define, the evident functor (the faithful embedding) ϑF ∶ IF → PSh(C) then we have a canonical morphism iF ∶ colimϑF → F . Notice that iF is a monomorphism for being a colimit of monomorphisms. Then, we only need to show that it is epimorphism. Then, let U ∈ C, γ ∈ F (U ). Then, using Yoneda’s lemma, there is a morphism or pre-sheaves ιγ ∶ C(−, U ) → F given by ιγ (ϕ) = F (ϕ)(γ) for ϕ ∶ V → U in C, in particular ιγ (idU ) = γ. ∀V ∈ C, ∣C(V, U )∣ < ∣M or(C) < κ∣, hence C(−, U ) is κ-bounded, hence it is image is κ-bounded sub-object of F , that contains γ, as seen above. Then, im ιγ ⊂ colimϑF , hence γ ∈ (im ιγ )(U ) ⊂ colimϑF (U ), and (iF )U is surjective, hence iF is epimorphism. Therefore, F ≅ colimϑF .  Corollary 1.27. Let κ be an infinite cardinal such that κ > 2∣M or(C)∣ . Then, every simplicial pre-sheaf on C is a filtered colimit of its κ-bounded sub-objects. Proof. This is a direct result of the previous lemma, applied on the category ∆ × C.



2. The Model Structures on Simplicial Pre-sheaves

DR AF

T

In the study of pre-sheaves on a Grothendieck site, it is convenient to adopt the terminology used in the theory of sheaves on topological spaces, which is the motivating example. Therefore, by the local theory of pre-sheaves one means the study of pre-sheaves and their morphisms stock-wise, if the site have enough points; Whereas global theory means the study object-wise (section-wise). It is well-known that the category of simplicial pre-sheaves admits several model structure. Below we recall the local/global injective/projective model structures on the full category sPSh(C). Notice that, assuming the site have enough points, then these structures gives rise to the same homotopy theory. In fact the global model categories are Quillen equivalent to each other, and local ones are so. Moreover, local model categories are left Bousfield localisation of the global ones. These theories were developed mainly by Brown, Joyal and Jardine. All these model structures are based on the canonical model structure of simplicial set. Therefore, throughout this chapter the model structure of simplicial sets refers to the canonical model structure of simplicial sets, studied in chapter 3. A natural way of defining the model structure on sPSh(C) is object-wise, using the model structure of sSet, as in the below definitions. However, the resulting classes of object-wise weak equivalence, cofibration and fibration do not form a model structure. Still, that can be easily remedied by alternating the classes of fibration, or cofibration as done afterwards. Definition 2.1 (Object-wise Morphisms). A morphism of simplicial pre-sheaves f ∶ X → Y is said to be object-wise (section-wise) weak equivalence, cofibration, fibration, weak cofibration, or weak fibration iff fU ∶ X(U ) → Y(U ) is a weak equivalence, cofibration, fibration, weak cofibration, or weak fibration of simplicial sets ∀U ∈ C, respectively. An object-wise weak equivalence is called a global weak equivalence, and is being used in both injective and global projective structures, object-wise cofibrations are called injective cofibrations and they give rise to the injective structures, whereas object-wise fibrations are called projective fibrations and they give rise to the projective structures. Remark 2.2. The terminology used here is different in some cases from the original terminology used in referenced sources. For example what is called here local weak equivalence, local injective fibrations and the local injective model structure, Jardine calls topological weak equivalence , global fibrations, and global structure, respectively. 2.0.1. Global Structures. Definition 2.3 (Global Injective Structure). The global injective structure on sPSh(C) consists of the classes of object-wise cofibrations, global weak equivalences, and global injective fibrations. Where global injective fibrations are the morphisms having the RLP with respect to all object-wise weak cofibration. This is also called the Reedy model structure.

133

Theorem 2.4. The category sPSh(C) endued with the global injective structure forms a model category. Proof. See Heller.



Definition 2.5 (Global Projective Structure). The global projective structure on sPSh(C) consists of the classes of global projective cofibrations, global weak equivalences, and object-wise fibrations. Where projective cofibrations are the morphisms having the LLP with respect to all object-wise weak fibration. This is also called Bousfield-Kan model structure. Theorem 2.6. The category sPSh(C) endued with the global projective structure forms a model category. Proof. See [BK72, §.XI.8] and [Dug01].



Lemma 2.7. The identity functors is a Quillen equivalence between injective and global projective model categories on sPSh(C). Proof. See [DHI04].



Remark 2.8. Notice that the global model structures are defined for the underlying category C, and do not depend on the site topology.

RA FT

Lemma 2.9. The identity functors is a Quillen equivalence between global injective and global projective model categories on sPSh(C). Proof. See [DHI04].



2.0.2. Local Structures. The essential difference between local and global structures lies in weak equivalences. We recall local weak equivalence below in 2.1. However, if the given site have enough points, then, roughly speaking, local weak equivalences are those morphisms that induce weak equivalences of simplicial sets stalk-wise, which also will be made precise in ??.

D

Definition 2.10 (Local Injective Structure). The local injective structure on sPSh(C) consists of the classes of object-wise cofibrations, local weak equivalences, and local injective fibrations. Where local injective fibrations are the morphisms having the RLP with respect to all morphisms that are both local weak equivalence and object-wise cofibrations. Theorem 2.11. The category sPSh(C) endued with the local injective structure forms a model category. Proof. It is recalled in 2.2.



Definition 2.12 (Local Projective Structure). The local projective structure on sPSh(C) consists of the classes of local projective cofibrations, local weak equivalences, and object-wise fibrations. Where local projective cofibrations are the morphisms having the RLP with respect to all morphisms that are both local weak equivalence and object-wise fibrations. Theorem 2.13. The category sPSh(C) endued with the local projective structure forms a model category. Proof. See [?].



Remark 2.14. Although there is an essential difference between global and local weak-equivalences, where the first depends only on the underlying category of the site, whereas the later depends also on its topology. It turns out that if the site is ’nice enough’, having enough points, both structures induce equivalent homotopy categories. 2.0.3. Comparison and Intermediate Structures. Notice in global and local settings that the classes of injective/projective fibrations/cofibrations are contained in one another, as the below lemma shows: Lemma 2.15. projective cofibrations is contained in the class of injective cofibration and injective fibrations is contained in the class of projective cofibration. In the injective structures all objects are cofibrant, whereas in the projective all objects are fibrants. The Bousfield localization of the global model category structure to the local one In between the injective and the projective model structures there are many other model structures obtained by varying the class of generating global cofibrations.

134

2.1. Local Weak Equivalences. Local weak equivalence is the core concept in the ’standard’ local structures. The idea of local weak equivalences is that they should be given by the topology of the site Cτ , such that when considering the trivial topology they coincide with global weak equivalences [Jar15] and [Voe10]. It can be defined in different ways as seen later on. Both definitions due to Jardine [Jar87, Jar07, Jar07, Jar15] and Morel and Voevodsky [MV99] depend on the functors of homotopy sheaves. Therefore, we start by recalling these functors and study the properties of local weak equivalences. In global model structure the canonical morphisms X → LX are not weak weak equivalences, and they do not induce isomorphisms in the homotopy category. However, it is natural to look to for a model structure where such morphism induces isomorphism in the homotopy category. We see that model structures using local weak equivalences do so, but not in an efficient way for this sole purpose. The explanation of the naturality of the below definitions comes from topology, as seen in [Jar15] and [DI04]. And, [Voe10, p.1391] shows that the idea behind local weak equivalences is that they should arise from the topology. Also, [DHI04] [Isa05] show that topological local weak equivalences are given as localisation at hypercovers. Sheaf of Path Connected Components. Viewing the simplicial pre-sheaves as pre-sheaves of simplicial sets on C, and using path connected components of simplicial sets given in (100), one can define the sheaf of path connected components of simplicial pre-sheaves using post-composision, as follows:

given by

RA FT

Definition 2.16. The, the functor of the pre-sheaves of path connected components is defined to be the functor π0pre (−) ∶ sPSh(C) → PSh (C) π0pre (−) = π0Top (−)∗

where π0Top (−)∗ is the post-composition with π0Top (−) given in (100), i.e. π0pre (X)(U ) = π0Top (X(U )) .

For X ∈ sPSh(C) and U ∈ C. Then, the sheaves of path connected components is defined to be the functor π0pre (−) ∶ sPSh(C) → Shvτ (C)

D

given by the sheafification of π0pre (−)

π0top (−) = −a ○ π0pre (−)

We abuse notation and denote π0top (f ) and π0pre (f ) by f∗ for a morphism of simplicial pre-sheaves f ∶ X → Y, if no confusion arise. Lemma 2.17. Sheaves of path connected components commute with colimits. Proof. This is a direct result of the facts that geometric realisation commutes with colimits for being a left adjoint, and topological path connected components commutes with colimits, by lemma ??.  2.1.1. Joyal’s Homotopy Sheaves. For U ∈ C, we saw in ?? that Cτ ↓ U is a site, which is small when C is. Definition 2.18 (Homotopy Sheaves). Let n ≥ 0, U ∈ C. Then, the functor of n-homotopy pre-sheaf at U is defined to be the pre-sheaves (39)

πnpre (−∣U , ∗) ∶ sPSh(C)U → PShGrp (C ↓ U )

given by πnpre (−∣U , ∗) = πnTop (−, ∗)∗ ○ (−∣U , ∗) where πnTop (−, ∗)∗ is the composition with the topological homotopy groups of simplicial sets, and (−∣U , ∗) is the pointed restriction given in (32). Then, the functor of n-homotopy sheaves at U is defined to be the functor πntop (−∣U , ∗) ∶ sPSh(C)U → ShvGrp τ ↓U (C ↓ U ) given by the sheafification of πnpre (−∣U , ∗) πntop (−∣U , ∗) = −# ○ πnpre (−∣∗U , ∗).

135

Hence, for every non-empty simplicial pre-sheaf X, and for every vertex x0,U ∈ X0 (U ), we have the sheaf of pointed groups πntop (X∣U , x0,U ) = (πnTop (−, ∗) ○ (X∣∗U , x0,U ))# ∶ (C ↓ U )op → Grp i.e. the sheafification with respect to Cτ ↓ U of the the n-homotopy pre-sheaf given by πnpre (X∣U , x0,U ) ∶

(C ↓ U )op ϕ∶V →U V ϕ

ψ

  U

/ V′

Ð→ Grp ↦ πn (∣X(V )∣, x0,V ) ↦

πn (∣X(ψ)∣)

.

ϕ′

Having a morphism f ∶ (X, x0,U ) → (Y, f0,U (x0,U ), it is convenient to abuse notation and write f∗ rather than πntop (f ∣U , x0,U ) or πnpre (f ∣U , x0,U ), if no confusion can arise.

RA FT

Remark 2.19. One may want to define πnTop (−, x0,U ) directly on the original site and for all simplicial pre-sheaves. However, in order to define the homotopy groups, we need to fix a base point first. Doing so by choosing x0,U ∈ X0 (U ) one needs to be able to map this vertex fuctorially to a vertex in Y0 (V ). That is the main reason of these restriction. That, when there is a morphism V → U , the choice of the base point in functorial by the functoriality of X, and morphisms X → Y that preserves the distinguished vertex. However, if such morphisms do not exist, then we can not functorially choose a base point in ∣Y(V )∣ to calculate homotopy groups at. Therefore, it is natural to restrict the definition of the functor of sheaves as above. On the other hand, (C ↓ U ) has the advantage of having a terminal object, namely idU , hence it is filtered and all colimit on it are filtered colimits. Definition 2.20 (Local Weak Equivalences). Let f ∶ X → Y be a morphism of simplicial pre-sheaves. Then, f is called a local weak equivalence iff ● π0top (f ) is an isomorphism of sheaves. ● ∀n ≥ 1, U ∈ C, πntop (f ∣U , x0,U ) is an isomorphism of sheaves for all x0,U ∈ X0 (U ).

D

Remark 2.21. Following [Jar15], 2.1.4 explains the naturality of the definition of weak equivalences, and the naturality of the above generalisation of homotopy groups of simplicial sets. Notice that the local weak equivalences depends on the topology of the site. Also, one can see easily that the definition is a bit redundant, and a more effective version of it is obtained by applying the below lemma Lemma 2.22. Let f ∶ X → Y be a morphism of simplicial pre-sheaves, n ≥ 0, U ∈ C, ϕ ∶ V → U in C, x0,U ∈ X0 (U ). If πntop (f ∣V , x0,V )idV is an bijection, where x0,V = X0 (ϕ)(x0,U ), then πntop (f ∣U , x0,U )ϕ is a bijection. Proof. It is a straightforward application of lemma ?? for X ∶= X0 , xU ∶= x0,U , FU,xU = πnpre (X∣U , x0,U ), GU,xU = πnpre (Y∣U , f0,U (x0,U )), and fU,xU = πnpre (f ∣U , x0,U ). That, (fU,xU )ϕ = πnpre (f ∣U , x0,U )ϕ = πn (∣fV ∣, x0,V ) = πnpre (f ∣V , x0,V )idV = (fV,xV )idV . Then, since ((fV,xV )aτ↓V )id = πntop (f ∣V , x0,V )idV is a bijection, lemma ?? implies that πnpre (f ∣U , x0,U )ϕ = V ((fU,xU )aτ↓U )ϕ is a bijection.  Corollary 2.23. Let f ∶ X → Y be a morphism of simplicial pre-sheaves, n ≥ 0. Then, πntop (f ∣U , x0,U )’s are isomorphism of sheaves ∀U ∈ C, x0,U ∈ X0 (U ) iff πntop (f ∣U , x0,U )idU ’s are bijection of sets ∀U ∈ C, x0,U ∈ X0 (U ). Proof. One direction is immediate, whereas the other is a direct result of the previous lemma 2.22.



Notice that the previous corollary give an effective method of working with local weak equivalences. In order to be eligible to call a class of morphisms weak equivalences, it ought to contain all isomorphisms, and satisfy the 2-out-of-3 property. It is evident that local weak equivalences contain all isomorphisms of simplicial pre-sheaves, and the below lemma shows that they satisfy the 2-out-of-3 property as well. Lemma 2.24. The class of local weak equivalences of simplicial pre-sheaves satisfies the 2-out-of-3 property.

136

Proof. Let f ∶ X → Y and g ∶ Y → Z be morphisms of simplicial pre-sheaves. Suppose that two of f, g, or gf are local weak equivalences, we need to show that the third is so. Consider the commutative triangle of simplicial pre-sheaves /6 Z .

gf

X (

f

Y

g

Applying π0top to the above diagram induces the commutative triangle of sheaves

f∗

/ π top (Z) . 80

g ∗ f∗

π0top (X)

(40)

&

g∗

π0top (Y)

Also, ∀n ≥ 1, U ∈ C, applying the functors πntop (−∣U , ∗) induces the commutative triangles of sheaves / πntop (Z∣U , g0,U f0,U (x0,U )) . 3

g∗ f∗

πntop (X∣U , x0,U )

(41)

+

f∗

g∗

πntop (Y∣U , f0,U (x0,U ))

RA FT

for every x0,U ∈ X0 (U ). ● If g and either f or gf are local weak equivalences, then all g∗ and either all f∗ or all (gf )∗ in (40) and (41) are isomorphisms of sheaves, respectively. Hence, all (gf )∗ or f∗ are isomorphisms, respectively. Then, gf or f are local weak equivalences, respectively. ● If f and gf are local weak equivalences, then all g∗ are also isomorphisms in (40) and (41). However, that is not enough to show that g is a local weak equivalence, that we need to show that g∗ ∶ πntop (Y∣U , y0,U ) → πntop (Z∣U , g0,U (y0,U ))

for all y0,U ∈ Y0 (U ), not only those in im f0,U . Let ∀U ∈ C, n ≥ 1, y0,U ∈ Y0 (U ), then by corollary 2.23 to prove that πntop (g∣U , y0,U )’s are isomorphisms of sheaves, it is sufficient to show that πntop (g∣U , y0,U )idU ’s are bijections of sets. Let γ be the path connected component of ∣Y(U )∣ that contains ∣y0,U ∣ . Then, ∣y0,U ∣ ∈ γ ∈ π0pre (Y)(U ). Since, f∗ in (40) is an isomorphism and π0pre is separable, then be lemma ??

D

f∗ ∶ π0pre (X) → π0pre (Y)

is a local isomorphism, so in particular there exists a τ -covering {ϕi ∶ Ui → U }i∈I such that there exists δi ∈ π0pre (X)(Ui ) with (f∗ )Ui (δi ) = ϕ∗i (γ). π0pre (X)(U )

(f∗ )U

ϕ∗i

∣x0,U ∣ ∈ δi ∈

 π0pre (X)(Ui )

/ π pre (Y)(U ) 0

∋ γ ∋ ∣y0,U ∣

ϕ∗i

(f∗ )Ui

 / π pre (Y)(Ui ) 0

∋ ϕ∗i (γ) ∋ ∣y0,Ui ∣.

Put y0,Ui = Y(ϕi )(y0,Ui ), since ϕ∗i is induced from ∣Y(ϕi )∣ on path connected components, we have ∣y0,Ui ∣ ∈ ϕ∗i γ. Also, since the path connected components of realisations of simplicial sets are represented by vertices, as in lemma ??, then there exists x0,Ui ∈ X0 (Ui ) such that ∣x0,Ui ∣ ∈ δi . Since, (f∗ )Ui maps δi to ϕ∗i γ, then ∣f0,Ui (x0,Ui )∣ = ∣fUi ∣(∣x0,Ui ∣) ∈ ϕ∗i γ. Hence, the points ∣fUi ∣(∣x0,Ui ∣) and ∣y0,Ui ∣ of ∣Y(Ui )∣ are path connected. Therefore, πntop (Y∣Ui , y0,Ui ) ≅ πntop (Y∣Ui , f0,Ui (x0,Ui )).

(42)

Applying ∣gUi ∣ to the above argument yield an isomorphism of sheaves πntop (Z∣Ui , g0,Ui (y0,Ui )) ≅ πntop (Z∣Ui , (gf )0,Ui (x0,Ui )).

(43)

So, in particular, we have a commutative diagram of sets (44)

πntop (Y∣Ui , y0,Ui )(ididUi ) (g∗ )idU

i



πntop (Z∣Ui , g0,Ui (y0,Ui ))(idUi )

(f∗ )idU

≅

≅

/

/ π top (Y∣U , f0,U (x0,U ))(idid ) o n i i i Ui 

(g∗ )idU

i

s

i

((gf )∗ )idU

πntop (Z∣Ui , (gf )0,Ui (x0,Ui ))(idUi ).

πntop (X∣Ui , x0,Ui )(idUi ) i

137

Since (f∗ )idUi and ((gf )∗ )idUi are bijections, then so are the right (g∗ )idUi ’s. The horizontal maps are bijections, then so are the left (g∗ )idUi ’s, i.e. πntop (g∣Ui , y0,Ui )idUi ’s are bijections, then by lemma 2.22, we have πntop (g∣U , y0,U )ϕi ’s bijections for every i ∈ I, the indexing set of the covering. {ϕi ∶ Ui → U }i∈I is a covering for U in τ , then {ϕi ∶ ϕi → idU }i∈I is a covering of idU in τ ↓ U . Hence, πntop (g∣U , y0,U ) gives bijections on a covering of idU . Then, since it is a morphism of sheaves, then by lemma ??, we see that πntop (g∣U , y0,U )idU is a bijection. Therefore, g is local weak equivalence.  Example 2.25. Global weak equivalences are local weak equivalences. That, for global weak equivalences, the conditions of Def. 2.20 are satisfy on the level of pre-sheaves. Then, since the sheafification functor is exact, the conditions of the definition hold for the associated sheaves. In effect, let f ∶ X → Y be a global weak equivalence, then fU ∶ X(U ) → Y(U ) is a weak equivalence of simplicial sets for all U ∈ C, hence π0 (∣fU ∣) ∶ π0 (∣X(U )∣) → π0 (∣Y(U )∣) is a bijection ∀U ∈ C and πnpre (f ∣U , x0,U )ϕ∶V →U ∶ πn (∣X(V )∣, x0,V ) → πn (∣Y(V )∣, f0,V (x0,V ))

RA FT

are isomorphisms ∀n ≥ 1, U ∈ C, x0,U ∈ X0 (U ), and ϕ ∈ C ↓ U . Hence, f∗ ∶ π0pre (X) → π0pre (Y) , and f∗ ∶ πnpre (X∣U , x0,U ) → πnpre (Y∣U , f0,U (x0,U )) are isomorphism of pre-sheaves ∀n ≥ 1, U ∈ C, and x0,U ∈ X0 (U ). Then, since the sheafification functor is exact, the above isomorphisms induce isomorphisms of sheaves. Hence, f is a local weak equivalence. Remark 2.26. Notice that the converse is not correct. For instance, local isomorphisms are local weak equivalence. In particular, for a simplicial pre-sheaf X, the sheafification morphism ηb cX ∶ X → Xa , that is a local isomorphisms, is a local weak equivalence. However, it is not a global weak equivalence. Proof. [Jar07, p.6].



˘ Example 2.27. Cech weak equivalences are local weak equivalences.

D

Proof. Let X be a simplicial pre-sheaf, and let ηX ∶ X → Xa be the canonical sheafification morphism. The sheafification functor is given by filtered colimits of equalisers (finite limits). Then, since homotopy sheaves and the sheaf path connected components commutes with filtered colimits, and with limits in general we have πntop (Xa ∣U , x0,U ) ≅ (πntop (X∣U , x0,U ))a ≅ πntop (X∣U , x0,U ), n ≥ 0, U ∈ C, x0,U ∈ X0 (U ). π0top (Xa ) ≅ (π0top (X))a ≅ π0top (X). hence, ηX is a local weak equivalence.



Remark 2.28. A direct result of examples 2.25 and 2.27 above, is that local injective fibrations are global fibrations. However, the converse is not true. That, local injective fibrations have the RLP with respect to the class of local injective weak cofibrations which is strictly larger than the class of global weak cofibrations. Example 2.29. We saw in example 1.8 that sPSh(0) = sSet. Then, one can easily see that local weak equivalences in the case when C = 0 coincide with weak equivalences of simplicial sets. Lemma 2.30. Homotopy sheaves commute with colimits. Proof. This is a direct result of the facts that geometric realisation commutes with colimits for being a left adjoint, and topological homotopy groups commutes with colimits, by lemma 1.20.  2.1.2. Relative Homotopy Groups. Although the definition of above homotopy sheaves comes naturally from topology, it is not always convenient to work with. Verifying that a morphism is local weak equivalence seems to be one of the main troubles one might run to when proven the below lemmas. We see later that, in the proof of the main theorem of this chapter, theorem ??, we usually need to prove that a certain morphism is a local weak equivalence, providing it is an injective cofibration. Then, it is natural to look for a relative version of the above definition, and provide an alternative condition, which is recalled below in corollary 2.34 and lemma 2.35.

138

Definition 2.31 (Sheaf of Relative Homotopy Groups). Let n ≥ 0, U ∈ C. Then, the n-pre-sheaf of relative homotopy ’groups’ at U is defined to be the pre-sheaves πnpre (−∣U , −∣U , ∗) ∶ sPSh2 (C)U → PShSet∗ (C ↓ U )

(45) given by

πnpre (−∣U , −∣U , ∗) = πnTop (−, −, ∗)∗ ○ (−∣∗U , −∣∗U , ∗) where πnTop (−, −, ∗)∗ is the composition with the topological relative homotopy groups of pairs of simplicial sets, and (−∣∗U , −∣∗U , ∗) is the pointed restriction of pairs given in (33). Then, the functor of n-sheaves of relative homotopy groups at U is defined to be the functor ∗ πntop (−∣U , −∣U , ∗) ∶ sPSh2 (C)U → ShvSet τ ↓U (C ↓ U )

given by the sheafification of πnpre (−∣U , −∣U , ∗) πntop (−∣U , −∣U , ∗) = −# ○ πnpre (−∣U , −∣U , ∗). Hence, for every injective cofibration K ↪ L, and for every vertex k0,U ∈ K0 (U ), we have the sheaf of pointed sets πntop (L∣U , K∣U , k0,U ) = (πnTop (−, −, ∗) ○ (L∣∗U , K∣∗U , k0,U ))# ∶ (C ↓ U )op → Set∗ i.e. the sheafification with respect to Cτ ↓ U of the the n-pre-sheaf of relative homotopy groups given by (C ↓ U )op ϕ∶V →U

Ð→ Set∗ ↦ πn (∣L(V )∣, ∣K(V )∣, k0,V )

RA FT

πnpre (L∣U , K∣U , k0,U ) ∶

ψ

V

ϕ

  U

/ V′

πn (∣L(ψ)∣)

↦

.

ϕ′

Having a morphism f ∶ (L, K, k0,U ) → (L′ , K′ , f0,U (k0,U ), it is convenient to abuse notation and write f∗ for its image with respect to either of πnpre (−∣U , −∣U , ∗) or πntop (−∣U , −∣U , ∗) , if no confusion can arise. Based on [Swi75, §.3], then for n ≥ 2, one can see easily that πnpre (−∣U , −∣U , ∗) and πntop (−∣U , −∣U , ∗) lift to functors of pre-sheaves of groups and sheaves of groups, respectively. Recall that for a pointed pair spaces (L, K, k), the 0th -relative homotopy group is defined by:

D

π0 (L, K, k) = π0 (L, k)/π0 (K, k)

Remark 2.32. Sheaves of relative homotopy groups provide a convenient way of verifying whether a morphism of simplicial pre-sheaves is a weak equivalence or not, as in lemma 2.63. Those results are based mainly of the following lemmas. Lemma 2.33. Let U ∈ C, K ↪ L be a monomorphism of simplicial pre-sheaf. Then, for every U ∈ C, k0,U ∈ K0 (U ), there exist s long sequences pre πn+1 (L∣U, K∣U , k0,U ) → πnpre (K∣U , k0,U ) → πnpre (L∣U , k0,U ) → πnpre (L∣U, K∣U , k0,U ) → ⋯ pre π1 (L∣U, K∣U , k0,U ) → π0pre (K∣U , k0,U ) → π0pre (L∣U , k0,U ) → π0pre (L∣U, K∣U , k0,U ) → ∗

(46)

⋯→ ⋯→

(47)

top ⋯ → πn+1 (L∣U, K∣U , k0,U ) → πntop (K∣U , k0,U ) → πntop (L∣U , k0,U ) → πntop (L∣U, K∣U , k0,U ) → ⋯ top ⋯ → π1 (L∣U, K∣U , k0,U ) → π0top (K∣U , k0,U ) → π0top (L∣U , k0,U ) → π0top (L∣U, K∣U , k0,U ) → ∗

Moreover, both sequences are long exact. The exactness of both sequences is in the sense of exactness in definition ??. Proof. A straightforward application of [ibid., p.38], as in [PB13b, Lem.22].



Corollary 2.34. Let i ∶ K → L be an injective cofibration. Then, i is a local weak equivalence iff ∀n ≥ 0, U ∈ C, k0,U ∈ K(U ) the sheaf of relative homotopy ’groups’ is trivial, i.e. πntop (L∣U , K∣U , k0,U ) = 0 Lemma 2.35. Let i ∶ K → L be an injective cofibration. Then, f is a local weak equivalence iff for every ∀n ≥ 0, U ∈ C, and every commutative square (48)

∣∂∆ n ∣ _  ∣∆n ∣

α

/ ∣K(U )∣ _ ∣iU ∣

β

 / ∣L(U )∣

139

there is a covering {ϕi ∶ Ui → U }i∈I , such that for every ϕi ∶ Ui → U , ∣ϕ∗i ∣β is homotopic (rel ∣∂∆n ∣) to a map ∣∆n ∣ → ∣K(Ui )∣ , i.e. there exists commutative diagrams ∣∂∆ n ∣ _ s

(49)

∣ϕ∗i ∣α

d0

% ∣∂∆n ∣ × ∣∆1 ∣ _

c∣ϕ∗ ∣α i

∣iUi ∣

 ∣∆n ∣ × ∣∆1 ∣ 9  + ∣∆n ∣

hϕi

 3/ L(Ui )

∣ϕ∗i ∣β

d0

∣∂∆ n ∣ _

(50)

/+ K(Ui )

θϕi

/ ∣K(Ui )∣ : _ ∣iUi ∣

 / ∣L(Ui )∣ 1

RA FT

 ∣∆n ∣

∣ϕ∗i ∣α

hϕi d

where c∣ϕ∗ ∣α is the constant homotopy at i

∣ϕ∗i ∣α.

Proof. Let n ≥ 0, U ∈ C, E be a commutative square ∣∂∆ n ∣ _  ∣∆n ∣

n

α

/ ∣K(U )∣ _ ∣iU ∣

β

 / ∣L(U )∣.

n

D

Then, β is pointed map of pair spaces β ∶ (∣∆ ∣, ∣∂∆ ∣, ∗) → (∣L(U )∣, ∣K(U )∣, β(∗)). Hence, [β] ∈ πn (∣L(U )∣, ∣K(U )∣, β(∗)) or [β] ∈ π0 (∣L(U )∣, ∣K(U )∣), for n = 0. Now, having πntop (L∣U , K∣U , k0,U ) to be trivial is equivalent of having πnpre (L∣U , K∣U , k0,U ) to be locally trivial. That in turn is equivalent to the existence of a covering {ϕi ∶ Ui → U }i∈I such that for every ϕi in the covering ∣ϕ∗i ∣β = 0 ∈ πn (∣L(Ui )∣, ∣K(Ui )∣, ∣ϕ∗i ∣β(∗)). Then, using [Swi75, Prop. 3.14], this equivalent to having ∣iUi ∣

∣ϕ∗i ∣β homotopic (rel ∣∂∆n ∣) to a map θϕi ∶ ∣∆n ∣ → ∣K(Ui )∣ ↪ ∣L(Ui )∣. The existence of a homotopy (rel ∣∂∆n ∣) of ∣ϕ∗i ∣β is equivalent to the existence of the commutative diagram (49), whereas having ∣ϕ∗i ∣β homotopic (rel ∣∂∆n ∣) to θϕi , through hϕi is equivalent to the existence of the commutative diagram (50). Based on the above argument, this lemma is a direct result of the above corollary 2.34.



Definition 2.36. We say that a morphism of simplicial pre-sheaves f ∶ X → Y satisfies the local relative homotopy lifting property if it satisfy the condition of the above lemma. Lemma 2.37. Sheaves of relative homotopy ’groups’ commute with colimits. Proof. This is a direct result of the facts that geometric realisation commutes with colimits for being a left adjoint, and topological path connected components commutes with colimits, by lemma ??.  Base-free Homotopy Sheaves. On way to avoid the trouble of specifying base points and working with comma categories is obtain by considering the base-free homotopy sheaves, given in [Jar07, §.2]. Definition 2.38. Let n ≥ 0. The, the functor of the n-base-free homotopy pre-sheaves is defined to be the functor πnpre (−) ∶ sPSh(C) → PSh(C) given by πnpre (−) = πnTop (−)∗

140

where πnTop (−)∗ is the composition with πnTop (−) given in (102), i.e. πnpre (X)(U ) = πnTop (X(U )) . For X ∈ sPSh(C) and U ∈ C. Then, the n-base-free homotopy sheaves is defined to be the functor πnpre (−) ∶ sPSh(C) → Shvτ (C) given by the sheafification of πnpre (−)

πntop (−) = # ○ πnpre (−)

We abuse notation and denote πntop (f ) and πnpre (f ) by f∗ for a morphism of simplicial pre-sheaves f ∶ X → Y, if no confusion arise. Consider the functor

RA FT

()0 ∶ sPSh(C) → PSh(C) that assign to each simplicial pre-sheaf its pre-sheaf of vertices. Then, ∀n ≥ 0, we see readily that there is a natural transformation p ∶ πnpre (−) → ()0 given on object by (pX )(U ) ∶ πnpre (X)(U ) → X0 (U ) that sends each element of πnpre (X)(U ) = πn (∣X(U )∣, x0,U ) to the base-point with respect to which it ⋃ x0,U ∈X0 (U )

is obtained. Hence, we have a natural transformation

paτ ∶ πntop (−) → (()0 )aτ .

Then, having a morphism of simplicial pre-sheaves f ∶ X → Y, for n ≥ 0, we have a commutative square of pre-sheaves on C (51)

πnpre (X) pX

D

 X0

f∗

↺

f0

/ π pre (Y) n pY

 / Y0

and a commutative square of sheaves on Cτ (52)

πntop (X) (pX )a τ

 (X0 )aτ

f∗

↺ (f0 )a τ

/ π top (Y) n (pY )a τ

 / (Y0 )a τ

Then, one might would like to follow similar argument of the one used in simplicial sets to link Joyal’s homotopy sheaves and the base-free homotopy sheaves. We see below that this generalisation is possible, however, the core difference of dealing with pre-sheaves (rather than sets) requires some attention. That, for a morphism of pre-sheaves i ∶ U → X on Cτ , an arbitrary choice of xU ∈ X(U ) for every U ∈ C does not necessary define a natural transformation ∗ → X that induces xU ’s in its image. On the other hand, fixing U ∈ C and choosing xU ∈ X(U ) yield at least a natural transformation that induces x0 in its images. However, such natural transformation is not unique in general, precisely when ∃V ∈ C such that C(V, U ) = ∅ and X(V ) ≇ ∗. In either cases, a categorical analogue of i−1 (x), in , can not be defined. However, that can be redeemed by restricting the pre-sheave to the slice categories, as follows. For every U ∈ C, xU ∈ X(U ), then there exists precisely a unique natural transformation ιxU ∶ ∗ → X∣U given on object by im ((ιxU )ϕ ) = X(ϕ)(xU ) for ϕ ∈ C ↓ U . Then, the pre-image of xU ∈ X(U ) along i could be defined to be a pre-sheaf that fits in the following pull-back square of pre-sheaves on C ↓ U  / U∣U U xU ⌜ i∣U

  ∗

ιxU

 / X∣U .

141

Since pull-back of pre-sheaves is given object-wise, then for such definition and for ϕ ∶ V → U ∈ C ↓ U we have the cartesian square  / U(V ) UxU (ϕ) ⌜ iV

  ∗

ιxV

 / X(V ).

where xV ∶= X(ϕ)(xU ), hence UxU (ϕ) ≅ (U(V ))xV (the later is in the pre-image of maps of sets iV ). However, in order to avoid the unnecessary troubles or working up-to isomorphism, we simply define UxU to be the pre-sheaf on C ↓ U given for every ϕ ∶ V → U ∈ C ↓ U by UxU (ϕ) = (U(V ))xV = (iV )−1 (xV ). That such UxU fits in the above cartesian diagram. Now, having a commutative diagram of pre-sheaves on C U

g

p

i

 /Y

RA FT

 X

/V

f

then, ∀U ∈ C, xU ∈ X(U ), the universal property of pull back induced the unique natural transformation gxU ∶ UxU → VfU (xU ) that makes the following diagram commutative gxU

/ Vf

>

"

D

g∣U

U∣U

i∣U

 ∗ q

ιxU

r

U (xU )

>

U xU  q

#

 ∗ r

ιfU (xU )

 X∣U

f ∣U

#

/ V∣U p∣U

$  / Y∣U

Then, in particular ∀ϕ ∶ V → U ∈ C ↓ U, γ ∈ UxU (ϕ) = (U(V ))xV ⊂ U∣U (ϕ) = U(V ), we have (gxU )ϕ (γ) = (g∣U )ϕ (γ) = gV (γ) = (gV )xV (γ) hence (53)

(gxU )ϕ = (gV )xV

Based on the above notations, we recall the analogue of lemma 10.4, where the below two lemmas encode the link between Joyal’s homotopy sheaves and base-free homotopy sheaves. Actually, they also connect Joyal’s homotopy sheaves with Morel’s and Voevodsky’s homotopy sheaves to be seen later. Lemma 2.39. Let U i

 X

g

D

f

/V p

 /Y

be a commutative square of pre-sheaves on the category C. Then, the following statements are equivalents ● D is a cartesian square. ● ∀U ∈ C, xU ∈ X(U ), the induces natural transformation gxU is an isomorphism.

142

Proof. Assume that D is a cartesian square, then ∀V ∈ C, the object-wise induced square U(V )

(54)

gV

pV

iV

 X(V )

/ V(V )

fV

 / Y(V )

is a cartesian square. Then, by lemma 10.4, ∀xV ∈ X(V ), the induced morphism (gV )xV ∶ U(V )xV → V(V )fV (xV ) is a bijection. Then, by (53) ∀U ∈ C, xU ∈ X(U ), ∀ϕ ∶ V → U ∈ C, the map (gxU )ϕ = (gV )xV is a bijection, hence gxU is an isomorphism. on the other hand, assume that ∀U ∈ C, xU ∈ X(U ), gxU is an isomorphism. To show that D is cartesian it is enough to show that (54) is cartesian for every V ∈ C. Let V ∈ C, ∀xV ∈ X(V ), gxV is an isomorphism, so in particular (gxV )idV is a bijection, hence by (53) (gV )xV is a bijection. Then, by lemma 10.4, (54) is cartesian for every V ∈ C, hence D is cartesian. 

RA FT

Since homotopy sheaves are given by sheafification of homotopy pre-sheaves, we need to understand the above lemma on the level of sheafification, hence we need to understand the ’objects’ given in the above notation. First, sheafification commutes with restriction, hence for U ∈ C, we have (U∣U )aτ↓U = Uaτ ∣U

Then, since the sheafification functor is left exact, it commutes with finite limits, particularly with pull-backs, hence for xU ∈ X(U ) we have (UxU )aτ = (Uaτ )(xU )aτ where (xV )aτ denotes the image of xV through the canonical map

ηX,V ∶ X(V ) → Xaτ (V ).

Hence,

(UxU )aτ↓U (ϕ) = (Uaτ (V ))(xV )aτ = ((iV )aτ )

−1

((xV )aτ ).

D

Now, having a commutative diagram of pre-sheaves on C U

g

p

i

 X

/V

f

 /Y

then, ∀U ∈ C, xU ∈ X(U ), there exists natural transformations gxU ∶ UxU → VfU (xU ) induced from the above square and (gτa )(xU )aτ ∶ (Uaτ )(xU )aτ → (Uaτ )(fU (xU ))aτ induced from the sheafification of the above square. Since sheafification commutes with pull-backs, and gxU is the morphism induced between pull-backs (by the universal property of pull-backs) then (gxU )aτ↓U = (gτa )(xU )aτ Before passing to the analogue lemma on level of sheafification, observe that the second condition of the above lemma is redundant, that (gxu )ϕ = (gV )xV = (gxV )idV hence, it is equivalent to requiring (gxU )idU being bijection for every V ∈ C. A similar result holds on the level of sheafification, as per the below lemma, and that makes the proof of lemma 2.42 below much more concise. Lemma 2.40. Let Cτ be a site, and U i

 X

g

D

f

/V p

 /Y

be a commutative square of pre-sheaves on the category C, U ∈ C, ϕ ∶ V → U in C, x′U ∈ Xaτ (U ). Then, if ((gx′V )aτ↓V )id is an bijection, where x′V = Xaτ (ϕ)(xU ), then ((gx′U )aτ↓V )ϕ is a bijection. V

143

Proof. It is a straightforward application of lemma ?? for X = Xaτ , xU = x′U , FU,xU = Ux′U , GU,xU = VfU (x′U ) , and fU,xU = gx′U . That, fU,xU (ϕ) = (gx′U )ϕ = (gV )x′ = (gx′V )id = fV,xV (idV ). V

((fV,xV )aτ↓V )id V

Then, since is a bijection.

=

((gx′V )aτ↓V )id V

V

is a bijection, lemma ?? implies that ((gx′U )aτ↓V )ϕ = ((fU,xU )aτ↓U )ϕ 

Corollary 2.41. Let Cτ be a site, and

i

 X

/V

g

U

p

D

 /Y

f

be a commutative square of pre-sheaves on the category C. Then, (gx′U )aτ↓U ’s are isomorphisms of sheaves ∀U ∈ C, x′U ∈ Xaτ (U ) iff ((gx′U )aτ↓U )id ’s are bijection of sets ∀U ∈ C, x′U ∈ Xaτ (U ). U



RA FT

Proof. One direction is immediate, whereas the other is a direct result of the previous lemma 2.40. Lemma 2.42. Let Cτ be a site, and

i

 X

/V

g

U

p

D

 /Y

f

be a commutative square of pre-sheaves on the category C. Then, the following statements are equivalents ● Dτa , the sheafification of D with respect to τ , is a cartesian square of sheaves. ● ∀U ∈ C, xU ∈ X(U ), (gxU )aτ↓U , the sheafification of the induces natural transformation gxU with respect to τ ↓ U , is an isomorphism of sheaves.

D

Proof. Assume that Dτa is a cartesian square. Then, by the previous lemma 2.39, ∀U ∈ C, x′U ∈ Xaτ (U ) the induced morphism (gτa )x′U is an isomorphism. Hence, in particular, ∀xU ∈ X(U ), (xU )aτ ∈ Xaτ (U ), then for the choice x′U = (xU )aτ , we have an isomorphism (gxU )aτ↓U = (gτa )(xU )aτ . On the other hand, assume that ∀U ∈ C, xU ∈ X(U ), (gxU )aτ↓U is an isomorphism, i.e. (gτa )(xU )aτ = (gxU )aτ↓U . is an isomorphism. That would have been enough, if every x′U ∈ Xaτ (U ) is given by x′U = (xU )aτ for some xU ∈ X(U ), but that is not the case in general. Therefore, we still need to show that (gτa )x′U is an isomorphism for every x′U ∈ Xaτ (U ), then applying lemma 2.39, to see that Dτa is a cartesian square. Let x′U ∈ Xaτ (U ), and observe that, by lemma ??, there exists a covering {ϕi ∶ Ui → U }i∈I in τ , and xUi ∈ X(Ui ), ∀i ∈ I such that x′Ui = (xUi )aτ . {ϕi ∶ Ui → U }i∈I is a covering for U in τ , then {ϕi ∶ ϕi → idU }i∈I is a covering of idU in τ ↓ U . Then, we have ((gτa )x′U )ϕ = ((gτa )x′U ) i

i

idUi

= ((gτa )(xUi )aτ )

idUi

(gτa )x′U

is a bijection, ∀i ∈ I, i.e. gives bijections on a covering of idU . Then, since it is a morphism of sheaves, then by lemma ??, we see that ((gτa )x′U )id is a bijection. Then, by corollary ?? we see that (gτa )x′U is an U isomorphism for every x′U ∈ Xaτ (U ), then by lemma 2.39, Dτa is cartesian square.  Corollary 2.43. Let f ∶ X → Y be a morphism of simplicial pre-sheaves, n ≥ 0. Then, the following statements are equivalent (1) ∀U ∈ C the induced natural transformation f∗ ∶ πntop (X∣U, x0,U ) → πntop (Y∣U , f0,U (x0,U )) is an isomorphism for every x0,U ∈ X0 (U ).

144

(2) The induced square, constructed in (52), is a cartesian square. πntop (X) (pX )a τ



(X0 )aτ

/ π top (Y) n

f∗

(pY )a τ

↺



(f0 )a τ

/ (Y0 )a τ

Proof. It is a direct result of the previous lemma 2.42 for the choice U′ ∶= πnpre (X) X′ ∶= X0 ′ ′ g ∶= f∗

V′ ∶= πnpre (Y) Y′ ∶= Y0 ′ ′ f ∶= f0

′

′

′

′

and ′ xU ′ ∶= x0,U for U ∈ C, where ′ ′ are used to distinguish the corresponding symbols of lemma 2.42. That, (Ux0,U )aτ↓U = πntop (X∣U, x0,U ), (Vf0,U (x0,U ) )aτ↓U = πntop (Y∣U , f0,U (x0,U )), and gxU = f∗ on the level of pre-sheaves, and (gxU )aτ↓U = f∗ on the level of sheaves. 

RA FT

Corollary 2.44. Let f ∶ X → Y be a morphism of simplicial pre-sheaves. Then, the following statements are equivalent ● f is a local weak equivalence. ● ∀n ≥ 0, the induced square, constructed in (52), is a cartesian square. πntop (X)

(pX )a τ



(X0 )aτ

f∗

/ π top (Y) n

↺

(f0 )a τ



(pY )a τ

/ (Y0 )a τ

Proof. A direct result of the above corollary 2.43.



D

2.1.3. Morel and Voevodsky’s Homotopy Sheaves. homotopy Sheaves are used to define local weak equivalences, but the above definition by Joyal is not necessary the easiest to work with in all occasions. Therefore, we recall here differed definition of a closely related concept due to Morel and Voevodsky. Ambiguously enough it is also called homotopy sheaves, but fortunately denoted with a capital Π’s. The main difference between these two definitions is that the later consider all possible base point at once. That, in Joyal’s definition the homotopy sheaves on the site (Cτ ↓ U ) for each U ∈ C and x0,U ∈ X0 (U ), whereas Morel and Voevodsky consider a bigger site that encodes the data of C and all vertices of X, namely h− ↓ X0 , where h− ∶ C → PSh(C) is Yoneda embedding. Therefore, we start by recalling the relation between objects of this category and vertices of X, and the induced topology of this category. Then, we recall the homotopy sheaves on this site. We have to direct the readers attention that the below explanation represent the author’s understanding of Morel and Voevodsky approach. For simplicity, we start with with object-wise definitions (for a given simplicial pre-sheaf), then we extend the definitions for morphism of simplicial pre-sheaves. Lemma 2.45. There is a natural bijection between the objects of h− ↓ X0 and vertices of X, i.e. there is a bijection ϑX ∶ Ob(h− ↓ X0 ) → ⊔ X0 (U ). U ∈C

Proof. Define

ϑX ∶

Ob(h− ↓ X0 )

→

⊔ X0 (U )

U ∈C

f ∶ hU → X0 ↦ fU (idU ) ϑX is injective that for morphisms of pre-sheaves f ∶ hU → X0 , f ′ ∶ hU ′ → X0 such that ϑX (f ) = ϑX (f ′ ), i.e. fU (idU ) = fU ′ (idU ′ ), then U = U ′ . Then, ∀V ∈ C, ϕ ∈ hU (V ) we have fV (ϕ)

= fV (ϕ∗ idU ) = fV (hU (ϕ)(idU )) = X0 (ϕ)(fU (idU )) = X0 (ϕ)(fU′ (idU )) = fV′ (hU (ϕ)(idU ) = fV′ (ϕ∗ idU ) = fV′ (ϕ).

hence, f = f ′ , and ϑX is injective.

145

On the other hand, let x0,U ∈ ⊔ X0 (U ). Then, there exists U ∈ C such that x0,U ∈ X0 (U ), then we can U ∈C

define the morphism of pre-sheaves f ∶ hU → X0 given by f (ϕ) = X0 (ϕ)(x0,U ). Then ϑX (f ) = x0,U . Hence, ϑ is a bijection. We could have just used the fact that representable pre-sheaves by an object are determined by the image of the identity of that object. 

Hence, it is natural to abuse notation and denote f ∶ hU → X0 in h− ↓ X0 by x0,U , where x0,U = fU (idU ). Lemma 2.46. Let X be a simplicial pre-sheaf. Then, the topology τ on C induce a topology on the comma category (h− ↓ X0 ) denoted by τ ↓ X0 , giving a site denoted by (h− ↓ X0 )τ .

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Proof. The proof uses the language of covering families. However, one can equivalently use covering sieves. Let x0,U ∶ hU → X0 be an object in h− ↓ X0 . Notice that for any ϕ ∶ V → U ∈ C, x0,V ∶= X0 (ϕ)(x0,V ) ∶ hV → X0 ∈ (h− ↓ X0 ), and that hϕ ∶ x0,V → x0,U is a morphism in h− ↓ X0 . Then, define Covτ (x0,U ) to consist of families {hϕi ∶ x0,Ui → x0,U }i∈I in h− ↓ X0 , for every covering family {ϕi ∶ Ui → U }i∈I ∈ Covτ (U ). One sees readily that this collection of families defines a pre-topology on h− ↓ X0 , and hence a topology. We denote this induced topology by τ ↓ X0 , and denote the resulting site by (h− ↓ X0 )τ . 

Remark 2.47. The above definition of τ ↓ X0 makes the canonical functor

(55)

p−1 ∶ h− ↓ X0 x0,U hϕ

→ ↦ ↦

C U ϕ

D

a continuous functor from the site (h− ↓ X0 )τ to Cτ . Is it a morphism of sites? Remark 2.48. The above lemmas show that (h− ↓ X0 )τ ’almost’ contains ∣ ⊔ X0 (U )∣ copies of the site U ∈C

Cτ . By ’almost’ we mean that these are not a full identical copies of the site, but rather the different comma sites (Cτ ↓ U ) for each U ∈ C. Hence, (h− ↓ X0 ), in a way, glues ∣X0 (U )∣ copies of each (Cτ ↓ U ) taking care automatically of base point computability. To define homotopy (pre-)sheaves of pointed sets, one can define a pre-sheaf of pointed simplicial sets, then take the post-composition with homotopy groups of pointed simplicial sets, and sheafify, as explained below. Definition 2.49. Let X be a simplicial pre-sheaf. Then, we define the associated pre-sheaf of pointed simplicial sets to X to be the pre-sheaf SX ∶

(h− ↓ X0 )op x0,U ∶ hU → X0 hV

(56)

x0,V

hϕ

  X0

/ hU

Ð→ sSet∗ ↦ (X(U ), x0,U ) ↦

X(ϕ)

x0,U

Notice that SX is well-defined that: ● Since h− is fully faithful embedding, then for every morphism ψ ∶ hV → hU , there exists a unique ϕ ∶ V → U such that ψ = hϕ .

146

● Also, X(ϕ) preserves the base point. That, x0,U is a morphism of pre-sheaves, and hϕ being a morphism in (h− ↓ X0 ) for ϕ ∶ V → U in C, then we have the commutative diagram (57)

hU (U ) z X0 (U ) X0 (ϕ)

idV ∈

hV (V )

hϕ,V

x0,V (V )

$  z X0 (V )

∋ idU

x0,U (U ) hU (ϕ)

 / hU (V )

∋ ϕ.

x0,U (V )

In fact, we have a bigger commutative diagram, but the above is sufficient for the current argument. Using the above diagram, we see that = X0 (ϕ) ((x0,U (U ))(idU )) = (x0,U (V )) (hU (ϕ)(idU )) = (x0,U (V ))(ϕ) = (x0,U (V )) (hϕ,V (idV )) = (x0,V (V ))(idV ) = x0,V .

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X0 (ϕ)(x0,U )

● It is obvious that functoriality of SX comes from functoriality of X. Hence, SX is well-defined.

Definition 2.50 (Morel and Voevodsky Homotopy Sheaves). Let X be a simplicial pre-sheave, n ≥ 0. Then, the n-homotopy pre-sheaf of X is defined to be op Πpre → Set∗ n (X) ∶ (h− ↓ X0 )

(58) given by

Top Πpre n (X) = πn (−, ∗) ○ SX

D

and the n-homotopy sheaf of X to be the sheafification of Πpre n (X) with respect to the induced topology on h− ↓ X0 , i.e. pre a Πtop n (X) = (Πn (X))τ ↓X0 . Morel’s and Voevodsky’s homotopy sheaves can be used in a similar manner of base-free homotopy sheaves to define local weak equivalence. To make this argument concrete, we need to make sense of X0 and Πtop n (Y) as pre-sheaves on h− ↓ X0 , as down below. X0 is a pre-sheaf on C. However, we have the a continuous functor p−1 in (55), with respect to which its direct image p∗ (X0 ) is a pre-sheaf on h− ↓ X0 . Then, it is obvious that the resulting pre-sheaves can be pointed as follows p∗ (X0 )∗ ∶ (h− ↓ X0 )op → Set∗ x0,U ↦ (X0 (U ), x0,U ) hϕ ↦ X0 (ϕ) Then, we have the trivial natural transformation pX ∶ Πpre n (X) → p∗ (X0 )∗ i.e. for each object x0,U ∈ h− ↓ X0 , it send all elements of Πpre n (X)(x0,U ) = πn (∣X(U )∣, x0,U ) to the base point x0,U . Since p∗ is exact and commutes with colimits, then it commutes with sheafification, i.e. p∗ (X0 )aτ↓X0 ≅ p∗ ((X0 )aτ ), hence the sheafification with respect to τ ↓ X0 induces a natural transformation a (pX )aτ↓X0 ∶ Πtop n (X) → p∗ ((X0 )τ )∗

On the other hand, let f ∶ X → Y be a morphism of simplicial pre-sheaves, and consider the functor g −1 ∶= h− ↓ f ∶ (h− ↓ X0 ) → (h− ↓ Y0 ) given by post-composition with f0 on object, and constant on morphisms. It is a continuous functor that

147

● For every x0,U ∈ h− ↓ X0 , every covering family of x0,U in τ ↓ X0 is of the form {hϕi ∶ x0,Ui → x0,U }i∈I for a covering family {ϕi ∶ Ui → U }i∈I ∈ Covτ (U ), then applying g −1 yield a covering family {hϕi ∶ f0 ○ x0,Ui → f0 ○ x0,U }i∈I = {hϕi ∶ f0,U (x0,Ui ) → f0,U (x0,U )}i∈I for g −1 (x0,U ) = f0,U (x0,U ) in τ ↓ Y0 . ● g −1 preserve pull-backs. Hence, the direct image g∗ ∶ PSh(h− ↓ Y0 ) → PSh(h− ↓ X0 ) preserves sheaves. Notice, that ∀n ≥ 0, we have the canonical natural transformation pre pre Πpre n (f ) ∶ Πn (X) → g∗ (Πn (Y ))

given object-wise for x0,U ∈ h− ↓ X0 by the commutative diagram Πpre n (X)(x0,U )

Πpre n (f )x0,U

(fU )∗

/ πn (∣Y(U )∣, f0,U (x0,U )).

RA FT

πn (∣X(U )∣, x0,U )

/ g∗ (Πpre (Y ))(x0,U ) n

Since g∗ is exact and commutes with colimits, then it commutes with sheafification, i.e. top a (g∗ (Πpre n (Y )))τ ↓X0 ≅ g∗ (Πn (Y )),

hence the sheafification of Πpre n (f ) induces natural transformation of sheaves top top Πtop n (f ) ∶ Πn (X) → g∗ (Πn (Y ))

Similarity, we see that we have the evident natural transformation of sheaves p∗ (f0 )∗ ∶ p∗ (X0 )∗ → p∗ (Y0 )∗ .

(59)

D

Then, a straightforward calculations shoes that the below diagram of pre-sheaf commutes ∀n ≥ 0 Πpre n (X) pX

 p∗ (X0 )∗

Πpre n (f )

/ g∗ (Πpre (Y)) n

↺ p∗ (f0 )∗

pY

 / p∗ (Y0 )∗

That, for every x0,U ∈ h− ↓ X0 , γ ∈ Πpre n (X)(x0,U ) (Uf0 pX )x0,U (γ) = (pY Πpre n (f ))x0,U (γ) = f0,U (x0,U ), hence the associated diagram of sheaves (60)

Πtop n (X) (pX )a τ ↓X

0

 p∗ ((X0 )aτ )∗

Πtop n (f ) ↺ p∗ ((f0 )a τ )∗

/ g∗ (Πtop (Y)) n (pY )a τ ↓X

 / p∗ ((Y0 )a )∗ τ

0

commutes ∀n ≥ 0. Having the above understanding of (60), we abuse notation and denote it by (61)

Πn (X)  X0

f∗ ↺

f0

/ Πn (Y)  / Y0

Joyale’s homotopy sheaves and Morel’s and Voevodsky’s homotopy sheaves are related in a similar manner of the relation between Joyale’s and base-free homotopy sheaves as seen below.

148

Lemma 2.51. Let f ∶ X → Y be a morphism of simplicial pre-sheaves, n ≥ 0. Then, the commutative square of sheaves (61) / Πn (Y)

f∗

Πn (X)

↺

 X0

 / Y0

f0

is a cartesian square iff ∀U ∈ C, x0,U ∈ X0 (U ) (Joyale’s) morphism of sheaves πntop (f ∣U, x0,U ) is an isomorphism. Proof. It is a result of lemma 2.42 for the choice ′ C′ ∶= h− ↓ X0 ,′ τ ′ ∶= τ ↓ X0 U′ ∶= Πpre n (X) X′ ∶= p∗ (X0 )∗ ′ ′ g ∶= f∗

V′ ∶= g∗ (Πpre n (Y)) Y′ ∶= p∗ (Y0 )∗ ′ ′ f ∶= p∗ (f0 )∗

′

′

′

′

RA FT

and ′ xU ′ ∶= x0,U for ′ U ′ ∶= x0,U ∈ h− ↓ X0 for U ∈ C, where ′ ′ are used to distinguish the corresponding symbols of lemma 2.42. That, it implies the above commutative square is cartesian iff all (Πtop n (f )x0,U ) are isomorphism for all x0,U ∈ h− ↓ X0 . On the other hand, observe that the evident morphism of sites i ∶ Cτ ↓ U → (h− ↓ X0 )τ ↓X given by the functor i−1 ∶ (h− ↓ X0 ) ↓ x0,U → C ↓ U hϕ ∶ x0,V → x0,U ↦ ϕ hψ ∶ hϕ′ → hϕ ↦ ψ ′

is an isomorphism of sites. Moreover, by the construction of UxU above, we have ′ UxU (ϕ)′ =′ (U(V ))xV , i.e., pre pre (Πpre n (X))x0,U (hϕ ) = (Πn (X)(x0,V ))x0,V = πn (∣X(V )∣, x0,V ) = πn (X∣U , x0,U )(ϕ).

Hence,

i∗ (πnpre (X∣U , x0,U )) = (Πpre n (X))x0,U

and since i∗ commutes with sheafification, then we also have

i∗ (πntop (X∣U , x0,U )) = (Πtop n (X))x

Similarly, we have

D

i∗ (πntop (f ∣U , x0,U )) = (Πtop n (f ))x

Since, i is an isomorphism of sites, then

πntop (f ∣U , x0,U )

0,U

0,U

.

is an isomorphism iff i∗ (πntop (f ∣U , x0,U )) = (Πtop n (f ))x

Hence, the above commutative square is cartesian iff all X0 (U ).

πntop (f ∣U , x0,U )

0,U

.

are isomorphism for all U ∈ U, x0,U ∈ 

Corollary 2.52. Let f ∶ X → Y be a morphism of simplicial pre-sheaves. Then, f is a local weak equivalence iff for every n ≥ 0, the the commutative square of sheaves (61) f∗

Πn (X)

/ Πn (Y)

↺

 X0

 / Y0

f0

is a cartesian square. Proof. A direct result of the above lemma 2.51.



2.1.4. Stalks of Simplicial Pre-sheaves. The above definition of local weak equivalences is not the easiest to work with, but it is the most general. However, when the site have enough points, an alternative more accessible definition can be provided for local weak equivalences, as seen below. Definition 2.53. Let p = (p∗ , p∗ , φ) ∶ Set → Shvτ (C) be a point for the site Cτ , X be a simplicial pre-sheave on C, and −aτ ∶ PSh(C) → Shvτ (C) be the sheafification functor. The ppre = (p∗pre , p∗ , φp re) ∶ Set → PSh(C) defines a point of the pre-sheaf topos PSh(C), as seen in (??). Then, the post-composition with p∗pre and p∗ define a morphism of pre-sheaf topoi sPSh(C) = (PSh(C))

∆op

o

(p∗pre )∗ (p∗ )∗

/ Set∆op = sSet

149

where (p∗pre )∗ , (p∗ )∗ are the post-composition with p∗pre , p∗ , respectively. We abuse notation and denote this morphism of pre-sheaf topoi by ppre = (p∗pre , p∗ , φpre ) ∶ sSet → sPSh(C) In particular, for a simplicial pre-sheaf X, one has (p∗pre (X))n = p∗ ○ (Xn )aτ . Having a morphism of simplicial pre-sheaves f ∶ X → Y, then for every point p = (p∗ , p∗ , φ) of Cτ , then we have a morphism of simplicial sets p∗pre (f ) ∶ p∗pre (X) → p∗pre (Y). It is natural to call f a ’point-wise weak equivalence’ if the induced morphisms p∗pre (f ) are weak equivalences of simplicial sets for all points p of Cτ . In fact, the below lemma shows that if the site Cτ has enough points, then ’point-wise weak equivalences’ coincide with local weak equivalences. Moreover, it explains [MV99, Rem.1.3], where it allows us to refine the definition of ’point-wise weak equivalence’, where it is enough to restrict attention to a conservative set of points of Cτ . The point of the last statement is not tautological, that morphism of topoi ppre = (p∗pre , p∗ , φpre ) ∶ sSet → sPSh(C) is not a point itself, and if p∗pre (f ) happen to be weak equivalences of simplicial sets for a conservative set of points of Cτ that does not priory imply that it is so for all points of Cτ .

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Lemma 2.54. Let Cτ be a small site with enough points, f ∶ X → Y be a morphism of simplicial pre-sheaves on Cτ . Then, f is a local weak equivalence iff the induced morphisms of simplicial sets p∗pre (f ) ∶ p∗pre (X) → p∗pre (Y) are weak equivalences for all points p = (p∗ , p∗ , φ) in a conservative set of points of Cτ . Proof. Let p = (p∗ , p∗ , φ) be a point of C. Applying p∗pre on f yield a morphism of simplicial set p∗pre (f ) ∶ p∗pre (X) → p∗pre (Y).

Recall that p∗pre ≅ Lu , for the flat functor u = p∗pre ○ h− . Since u is flat, then El(u) is filtering, Lu (X), Lu (Y) are filtered colimits, as seen in ??, and Lu (f ) is obtained by the universal property of colimit, as in the commutative diagram (p∗pre (X))n

D

colim(Xn ○ πuop )

(p∗pre (f ))n

/ (p∗ (Y))n pre

op colim(fn ○πu )

.

/ colim(Yn ○ π op ) u

Also, the geometric realisation functor commute with colimits for being left adjoint, and π0 , πn (−, ∗) commutes with filtered colimits for all n ≥ 1, hence p∗ (π0top (f ))

= p∗pre (π0pre (f )) = Lu (π0pre (f )) = Lu (π0 (−) ○ ∣−∣ ○ f−,− ) = colim(π0 (−) ○ ∣−∣ ○ f−,- ○ πuop ) = π0 (∣colim(f−,- ○ πuop )∣) = π0 (∣p∗pre (f )∣)

where p∗pre in the last term is the above induced functor p∗pre ∶ sPSh(C) → sSet, and the first equation is due to the (??). Similar to the discussion preceding the lemma, the point p, gives rise to the morphism of topoi, defined in similar manner, and denoted with the same symbol ppre = (p∗pre , p∗ , φpre ) ∶ Set∗ → PSh(C)∗ Also, for every U ∈ C, we have morphism of topoi, defined in similar manner, and denoted with the same symbol ppre = (p∗pre , p∗ , φpre ) ∶ sSet∗ → sPSh(C)U The, ∀n ≥ 1, U ∈ C p∗∣U (πntop (f ∣U , x0,U ))

∗

= p∗∣U ,pre (πnpre (f ∣U , x0,U )) = (p∗pre ○ ∣U )(πn (−, ∗) ○ ∣−∣∗ ○ (∣∗U , ∗) ○ (f−,− , x0,− )) = p∗pre (colim(πn (−, ∗) ○ ∣−∣∗ ○ (∣∗U , ∗) ○ (f−,− , x0,− ) ○ π−op )) = colim (colim(πn (−, ∗) ○ ∣−∣∗ ○ (∣∗U , ∗) ○ (f−,− , x0,− ) ○ π−op )) ○ πuop = πn (−, ∗) ○ ∣−∣∗ ○ colim((∣∗U , ∗) ○ (f−,− , x0,− ) ○ π−op ○ πuop ) = πn (−, ∗) ○ ∣−∣∗ ○ p∗pre (colim((∣∗U , ∗) ○ (f−,− , x0,− ) ○ π−op ) ∗

for every x0,U ∈ X0 (U ). Where p∣U is the induced point on Cτ ↓ U , as in example ??. Since ∣U ∣U ∗ (f ) = f , then colim(∣U ○ f−,− ○ π−op ) = f hence

∗

p∗∣U (πntop (f ∣U , x0,U )) = πn (−, ∗) ○ ∣−∣∗ ○ p∗pre (f, ∣U (x0,U )) = πn (∣p∗pre (f )∣, p∗∣U ,pre (x0,U )).

150

One direction holds even if C does not have enough point (but at least has a point, so it makes sense). Assume that f is a local weak equivalence, and p = (p∗ , p∗ , φ) be a point of C. Then, π0top (f ) is an isomorphism of sheaves, and ∀n ≥ 1, U ∈ C, x0,U ∈ X0 (U ), then πntop (f ∣U , x0,U ) are isomorphisms of sheaves. Then, p∗ (π0top (f )) and p∗ (πntop (f ∣U , x0,U )) are isomorphisms of sets, hence π0 (∣p∗pre (f )∣) and πn (∣p∗pre (f )∣, p∗pre (ιx0,U )) are so. However, that is not enough to show that p∗pre (f ) is a weak equivalence of simplicial set, that we need to show πn (∣p∗pre (f )∣, x0 ) is a bijection for every vertex x0 ∈ (p∗pre (f ))0 . However, that is remedied in a same way used in the proof of lemma 2.24. Then, p∗pre (f ) is a weak equivalence of simplicial sets. On the other hand, when C has enough points, let p∗pre (f ) be weak equivalence of simplicial sets for every point p = (p∗ , p∗ , φ) is a a conservative set of points of Cτ . Then, for every such point p∗ (π0top (f )) is a bijection, and ∀n ≥ 1, U ∈ C, p∗ (πntop (f ∣U , x0,U )) are bijections ∀x0,U ∈ X0 (U ), hence π0top (f ) and πntop (f ∣U , x0,U ) are isomorphisms of sheaves, that conservative set of points of Cτ induces conservative set of points of Cτ ↓ U, ∀U ∈ C. Hence, f is a local weak equivalence.  2.2. The Local Injective Model Structure. The below construction of model structure on sPSh(C), for a given Grothendieck topology τ , depends on these of the simplicial sets given in chapter 3; and it gives a cofibranty generated model categories.

DR AF

T

Hereby we recall Jardine construction [Jar87] of local injective model structure, which was called at the time global model structure, there the term local structure refers to this of the category of fibrant objects. We recall the classes of morphisms forming the local injective model structure and the properties needed to prove that they equip sPSh(C) with a model structure. The core notation here is of the local weak equivalences which is defined through the sheaves of homotopy groups, as seen above. This model structure is given by the classes of injective cofibrations, local injective fibrations, and local weak equivalences, where local injective fibration is the class of morphism that has the RLP with respect to the class of local injective weak equivalences. Recall that injective cofibrations are object-wise cofibrations, i.e. a they are injections of simplicial sheaves. Notice that the injective cofibrations are independent of the topology of the site. Definition 2.55 (Local Injective Fibrations). Let p ∶ X → Y be a morphism of simplicial pre-sheaves. Then, p is called an local injective fibration if it has the right lifting property with respect to local injective weak fibrations. Hence, the local injective fibration depends on the topology τ .

Definition 2.56. Let f be a morphism of simplicial pre-sheaves. Then, f is called local injective weak cofibration if f is both local weak equivalence and an injective cofibration, whereas it is called local injective weak fibration if f is both local weak equivalence and an local injective fibration. Most of the axioms of theorem 2.57 are either formally true, or relatively easy. However, factorisation axioms proved to be more involved. Therefore, it is being split to different lemma that collectively prove it. The lemmas provided below in this section are also important and useful for other purposes rather than proving the existence of model structure. However, since our aim in this section is basically to prove theorem 2.57, we follow the unconventional order of recalling the proof of the theorem first, followed by the lemmas it is based on. In order to prove the factorisation axioms, one usually uses the standard tool of transfinite small object argument, lemme ??, which gives a factorisation in terms of I-cell and I-inj for a given class of morphisms I that satisfy the lemma hypotheses. Therefore, in order to use the transfinite small object argument, one need to identify two classes of morphisms of simplicial pre-sheaves I and J, such that: ● I-cell are injective cofibrations and I-inj are local injective weak fibrations. ● J-cell are local injective weak cofibrations and J-inj are local injective fibrations. We see below the following classes of injective cofibrations of simplicial pre-sheaves both admit the transfinite small object argument and satisfy the above desired condition. I ∶= {i ∶ ∂∆nU ↪ ∆nU ∣n ≥ 0, U ∈ C} . J ∶= {j ∶ U ↪ V∣j is a local injective weak cofibration, and V is κ-bounded} for κ > 2∣M or(C)∣

151

Theorem 2.57. Let Cτ be a small site. Then, the structure (C, F, W ) of the classes of injective cofibrations, invective local fibrations, and local weak equivalence, respectively, equip sPSh(C) with a model structure. Proof. CM1 Section 1.1 implies that the sPSh(C) is a complete and cocomplete category. CM2 (2-out-of-3) Lemma 2.24 shows that the 2-out-of-3 property is satisfied for local weak equivalences. CM3 (Stability under retract) Let f ∶ X → Y, and g ∶ X′ → Y′ be morphisms of simplicial pre-sheaves such that f is a retract of g. Then, there exists a commutative diagram: X

(62)

e0

/ X′

r0

E

g

R

f

 Y

e1

 / Y′

r1

/X f

 /Y

(63)

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with horizontal composite being the identities. Evaluating at U yield fU is a retract of gU in sSet, ∀U ∈ C. (a) Suppose that g is an injective cofibration. Then, by definition gU is a cofibration of simplicial sets ∀U ∈ C. Then, theorem 4.1. CM3a implies that fU is a cofibration of simplicial sets ∀U ∈ C. Hence, f is an injective cofibration. (b) Suppose that g is an local injective fibration. Since the class of local injective fibrations F is noting but (C ⋂ W )-inj, then, by corollary ??, F is stable under retract. f is a retract of g and g ∈ F , hence f ∈ F , i.e. f is a local injective fibration. (c) Suppose that g is a local weak equivalence. Hence, g∗ ∶ π0top (X′ ) → π0top (Y′ ), and g∗ ∶ πntop (X′ ∣U , x0,U ) → πntop (Y′ ∣U , g0,U (x0,U ) are isomorphisms of sheaves for all n ≥ 1, U ∈ C, and x0,U ∈ X0 (U ). Applying the functor of sheaves of path connect component, we have the commutative diagram with horizontal composite being the identities πntop (X∣U , x0,U ) f∗



D

πntop (Y∣U , f0,U (x0,U ))

e0∗

/ πntop (X′ ∣U , x′0,U )

r0∗

E∗

g∗

R∗

e1∗

/ πntop (Y′ ∣U , g0,U (x′0,U ))



r1∗

/ πntop (X∣U , x0,U ) f∗



/ πntop (Y∣U , f0,U (x0,U )).

Also, since r0 d0 = idX and r1 d1 = idY , then applying the functor of sheaves of homotopy groups for n ≥ 1, U ∈ C yield commutative diagrams with horizontal composite being the identities

(64)

πntop (X∣U , x0,U ) f∗



πntop (Y∣U , f0,U (x0,U ))

e0∗

/ πntop (X′ ∣U , x′0,U )

r0∗

E∗

g∗

R∗

e1∗

/ πntop (Y′ ∣U , g0,U (x′0,U ))



r1∗

/ πntop (X∣U , x0,U ) 

f∗

/ πntop (Y∣U , f0,U (x0,U ))

for every x0,U ∈ X0 (U ), where x′0,U = e0,0,U (x0,U ). Since r0∗ e0∗ = idπntop (X∣U ,x0,U ) and r1∗ e1∗ = idπntop (Y∣U ,f0,U (x0,U )) in all diagrams (63) and (64) for every n ≥ 1, U ∈ C, and x0,U ∈ X0 (U ), then all e0∗ ’s are monomorphisms and all r1∗ ’s are epimorphisms. All g∗ ’s are isomorphisms, so in particular they are monomorphism. Hence, all g∗ e0∗ ’s are so so. Then, the commutativity of the sub-diagrams E∗ ’s implies that all f∗ ’s are monomorphisms. On the other hand, viewing g∗ ’s as epimorphisms, we see that all r1∗ g∗ ’s are so. Then, the commutativity of the sub-diagrams R∗ ’s implies that all f∗ ’s are epimorphisms, hence isomorphism. Therefore, f is a local weak equivalence. CM4 (Lifting) Let E be a commutative square of morphisms of simplicial pre-sheaves A i

 B

e0

E e1

/X p

 /Y

where i is an injective cofibration, p is an local injective fibration, and either of i or p is a local weak equivalence. Then we need to show that E admits a lift h ∶ B → X.

152

(a) Let i be a local weak equivalence. Then, by the definition of local injective fibrations, E admits a lift h ∶ B → X. (b) Let p be a local weak equivalence. Then, the proof of CM1, CM2, CM4a, CM5 and lemma 2.60 implies that the conditions of Joyal’s trick, lemma ??, are satisfied. Hence, local injective weak fibrations has the RLP with respect to injective cofibrations, E admits a lift h ∶ B → X. CM5 (factorisation) The proof is based on the transfinite small object argument: (a) I admits the transfinite small object argument by corollary 2.59. Then there is a functorial factorisation (α, β) such that for every morphism of simplicial pre-sheaves f ∶ X → Y, f = β(f )α(f ) where β(f ) ∈ I-inj and α(f ) ∈ I-cell. Then, by lemma 2.67 and corollary 2.61 β(f ) is a local injective weak fibration and α(f ) is an injective cofibration. (b) J also admits the transfinite small object argument by corollary 2.59. Then there is a functorial factorisation (γ, δ) such that for every morphism of simplicial pre-sheaves f ∶ X → Y, f = δ(f )γ(f ) where δ(f ) ∈ J-inj and γ(f ) ∈ J-cell. Then, by lemmas 2.66 and corollary 2.64 δ(f ) is a local injective fibration and γ(f ) is an local injective weak cofibration. 

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2.2.1. Injective Cofibrations. As seen above the proof of the factorisation axiom CM5 is based on using transfinite small object argument on the classes of morphisms I and J. Therefore, we recall hereby that I and J admit transfinite small object argument in the below lemma. Lemma 2.58. Let κ be an infinite cardinal, such that κ > 2∣M or(C)∣ . Then, κ-bounded simplicial pre-sheaves are small relative to injective cofibrations. Proof. Let A be a κ-bounded simplicial pre-sheaf. We need to show that for any κ- filtered ordinal λ, and any λ-sequence F ∶ λ → sPSh(C) of injective cofibrations, the canonical morphism (22) ϑ ∶ colim sPSh(C)(A, F− ) → sPSh(C)(A, colim F ) λ

λ

is a bijection. Since colimits of sets are quotients of disjoint unions, then to show that ϑ is surjective, we need to show that for any morphism of simplicial pre-sheaves f ∶ A → colim F there exist β < λ and morphism of λ

simplicial pre-sheaves A → Fβ , for which we have the commutative diagram

D

A



f

Fβ

' / colim F

ldots

λ

where Fβ → colim F is the evident transfinite composition. λ

Since colimits in sPSh(C) are given object-wise, then ∀n ∈ Z≥0 , U ∈ C, f induces maps of sets fn,U ∶ An (U ) → colim(F− )n (U ) λ

where (F− )n (U ) ∶ λ → Set is the evident λ-sequences of injections of sets. The transfinite composition of invective cofibrations is so as we see later in lemma 2.60, also the same holds for transfinite composition of injections of sets as in lemma ??. Hence, all evident transfinite compositions Fβ ↪ colim F , and (Fβ )n (U ) ↪ colim (F− )n (U ) ′ ′ β

β

′

are invective cofibrations and injections, respectively, ∀β < β ≤ λ. Then, by the definition of colimits of sets ∀xn,U ∈ colim(F− )n (U ) there exist βxn,U < λ such that xn,U ∈ (Fβn,U )n (U ) ⊆ colim(F− )n (U ). In particular, λ

λ

∀an,U ∈ An (U ), there exists βan,U < λ such that fn,U (an,U ) ∈ (Fβan,U )n (U ). Then, setting βn,U = sup{βan,U ∣an,U ∈ An (U )} we notice that βn,U < λ because λ is a κ-filtered ordinal, βan,U < λ and ∣An (U )∣ < κ, recall definition ??. Then, fn,U (an,U ) ∈ (Fβan,U )n (U ) ⊂ (Fβn,U )n (U ). Hence, maps fn,U factorise through (Fβn,U )n (U ). Now, put β = sup{βn,U ∣n ∈ Z≥0 , U ∈ C}. Since ∣Z≥0 × Ob(C)∣ = max(∣Ob(C)∣, ℵ0 ) ≤ max(∣M or(C)∣, ℵ0 ) < κ, and βn,U < λ, ∀n ∈ Z≥0 , U ∈ C, then β < λ. Hence, all maps fn,U factorise through gn,U ∶ An (U ) → (Fβ )n (U ). Now, in order to show that f itself factorises through Fβ , it is enough to show that gn,U ’s give rise to a morphism of simplicial pre-sheaves, i.e. they should

153

be natural in both arguments, which is evident from the commutativity of the below diagram, and the fact that (Fβ )m (V ) ↪ colim(F− )m (V ) is an injection of sets (i.e. a monomorphism, cancels from the left), for any λ

µ ∶ [m] → [n] in ∆, and ϕ ∶ V → U in C An (U )

. fn,U gn,U

Aµ (ϕ)

%  (Fβ )n (U )

 Am (V )

* / colim(F− )n (U )

ldots

λ

(Fβ )µ (ϕ)

colim(F− )µ (ϕ) λ

gm,V

fm,V

%   (Fβ )m (V )

 * / colim(F− )m (V )

ldots

λ

λ

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Therefore, there exists g ∶ A → Fβ , given object-wise above, such that f = ϑ([g]), where [g] is the class of g in colim sPSh(C)(A, F− ). On the other hand, to show that ϑ is injective, let g ∶ A → Fβ , g ′ ∶ A → Fβ ′ such that ϑ([g]) = ϑ([g]), then we need to show that [g] = [g ′ ]. By the definition of ϑ cλβ ○ g = ϑ(g) = ϑ(g ′ ) = cλβ′ ○ g ′

where cλβ ∶ Fβ ↪ colim F , cλβ′ ∶ Fβ ′ ↪ colim F are the evident transfinite compositions. Without loose of λ

λ

generality, let β ≤ β ′ , since cλβ = cλβ′ ○ cββ , and cλβ′ cancels from the left for being an injective cofibration (a ′

monomorphism in sPSh(C)), then we have cββ ○ g = g ′ . But, [cββ ○ g] = [g], hence [g] = [g ′ ]. ′

′



Corollary 2.59. The classes I and J defined above admits the transfinite small object argument in

D

sPSh(C).

Proof. Direct result of lemma 2.58, above. That, I and J are classes of injective cofibrations, and for κ > 2∣M or(C)∣ , ∂∆nU are κ-bounded, and domains of J are κ-bounded, by construction.  Hereby, we recall some of the properties of injective cofibrations and local injective weak cofibrations. The below lemmas enable use to use Joyal’ trick, and hence prove CM4b axiom. It also gives an insight about the nature of the morphisms of I-cell, as the corollary shows. Lemma 2.60. The class of injective cofibrations is closed under transfinite composition, push-outs, and pull-backs. Proof. Let λ be an ordinal, and F ∶ λ → sPSh(C) be a λ-sequence of injective cofibrations. Since, sPSh(C) has all small colimits, then the transfinite composition of F exists given by F0 → colim F in sPSh(C). Then, λ

for every U ∈ C, we have the induced λ-sequence of simplicial sets FU ∶ λ → sSet given by FU,β = Fβ (U ), ∀β < λ. Since F is a λ-sequence of injective cofibrations, then FU ’s are λ-sequences of cofibrations of simplicial sets. Also, since colimits in sPSh(C) are taken object-wise, we have colim FU = (colim F ) (U ) λ

λ

Then, the components of the transfinite composition F0 → colim F are nothing but FU,0 → colim FU , for U ∈ C. λ

λ

Then, def 2.1 and lemma ??. ?? implies that the transfinite composition FU,0 → colim FU is a cofibration of λ

simplicial sets, for all U ∈ C. Hence, F0 → colim F is an injective cofibration. λ

154

Let i ∶ K → L be an injective cofibrations, and f ∶ K → X be a morphism of simplicial pre-sheaves. Then, by 1.1, there exists a push-out square of simplicial pre-sheaves K

f

i

/X k

 L

⌟  / Y.

where Y = L ∐ X. Since colimits in sPSh(C) are given object-wise, then the above push-out square induces K

the push-out square of simplicial sets K(U )

fU

iU

/ X(U ) kU

⌟  / Y(U )

 L(U )

RA FT

for every U ∈ C. By the definition of injective cofibrations, iU is an cofibration of simplicial sets for every U ∈ C. Then, def 2.1 and lemma ??. ?? imply that kU is a cofibration of simplicial sets for every U ∈ C. Hence, k is an injective cofibrations. On the other hand, that pull-back of monomorphisms is so in any category, as in lemma ??. So, in particular it holds for sPSh(C), and since injective cofibrations are monomorphisms, then their pull-backs are injective cofibrations.  Corollary 2.61. Morphisms of I-cell are injective cofibrations.

Proof. Every morphism of I is an injective cofibration. Then, by the previous lemma 2.60 morphisms of I-cell are so.  The below lemmas and corollary gives an insight about the nature of the morphisms of J-cell

D

Lemma 2.62. Let i ∶ U → V be a local injective weak cofibration. Then, the following statements are equivalents: ● Push-outs of i are local injective weak cofibrations. ● Push-outs of i along injective cofibrations are local injective weak cofibrations. Proof. One direction is immediate. For the other, assume that push-outs of i along injective cofibrations are local injective weak cofibrations, and let f ∶ U → X be a morphism of simplicial pre-sheaves, then we need to show that the push out of i along f is a local injective weak cofibration. Consider the push-out square of simplicial pre-sheaves U

f

i

/X k

 V

⌟  / Y.

where Y = V ∐ X, which exists due to 1.1. Observe that the first part of the proof of the factorisation axiom U

2.57. CM5a does not depend on this lemma or its consequences, hence it can be applied here for f . So, it factorises as f = qj where j ∶ U → K is an injective cofibration, and q ∶ K → X is a local injective weak fibration. Then, by lemma ?? the above push-out square can be realised as consecutive push-out squares U

j

i

 V

/K

q

l

j′

⌟  /L

/X k

q′

⌟  / Y.

where L = V ∐ K, and Y ≅ L ∐ X. Applying the assumption to the left push-out square implies that l is a U

K

local injective weak cofibration. Hence, on the level of sheaf of path connected component we have π0top (L) ≅ π0top (K) ≅ π0top (X).

155

Then, since the sheaf of path connected component commutes with colimits, we have π0top (Y) ≅ π0top (L)

π0top (X) ≅ π0top (X)

∐ π0top (K)

also, similar result hold on the level of sheaves of homotopy groups, ∀n ≥ 1, U ∈ C, k0,U ∈ K0 (U ), we have πntop (Y∣U , (kq)0,U (k0,U )) ≅ πntop (L∣U , (l)0,U (k0,U ))

πntop (X∣U , q0,U (k0,U )) ≅ πntop (X∣U , q0,U (k0,U ))

∐ top πn (K∣U ,k0,U )

That is not enough to show that k is weak equivalence, yet. That, we need to show the same for arbitrary base point x0,U , not necessary in the image of q, as q0,U (k0,U ) is. Then, a step-by-step imitation of the proof of lemma 2.24 implies that k is a weak equivalence.  Lemma 2.63. The class of local injective weak cofibrations is closed under transfinite composition, and push-outs. Proof. Let λ be an ordinal, and F ∶ λ → sPSh(C) be a λ-sequence of local injective weak cofibrations. Since, sPSh(C) has all small colimits, then the transfinite composition of F exists given by f ∶ F0 → colim F λ

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in sPSh(C). Then, by lemma 2.60, the transfinite composition is an injective cofibration. Composition with the functor π0top induces a λ-sequence of sheaves of sets π0top ○ F ∶ λ → Shvτ (C). Since, F is a λ-sequence of local weak equivalence, then π0top ○ F is a sequence of isomorphisms of sheaves, the by lemma ?? its transfinite composition is so. Since, π0top commutes with filtered colimits, then ∼

f∗ ∶ π0top (F0 ) → colim π0top (F− ) ≅ π0top (colim F ) λ

λ

is an isomorphism of sheaves. Similarly, ∀n ≥ 1, U ∈ C, composition with the functors πntop (−∣U , ∗)’s yield λ-sequences of sheaves of groups πntop (−∣U , ∗) ○ F ∶ λ → ShvGrp τ ↓U (C ↓ U ) for every x0,U ∈ (F0 )0 (U ). Since, F is top a λ-sequence of local weak equivalence, then πn (−∣U , ∗) ○ F are sequences of isomorphisms of sheaves, hence their transfinite compositions are so. Since, πntop (−∣U , ∗)’s commute with filtered colimits, then ∼

f∗ ∶ πntop (F0 ∣U , x0,U ) → colim πntop (F− ∣U , ∗) ≅ πntop (colim F, f0,U (x0,U )) λ

λ

are all isomorphisms of sheaves. Hence f is a local injective weak cofibration.

D

Using lemma 2.62, it is enough to show that local injective weak cofibrations are stable along injective cofibrations. Let i ∶ U ↪ V be a local injective weak cofibrations, and j ∶ U ↪ K be an injective cofibration. Consider the push-out square  j / K _ U _ i

  V

′

i′

j′

⌟  / L.

′

where L = V ∐ K. Then, by lemma 2.60 i , j are injective cofibrations, hence we just need to show that i′ is a U

local weak equivalence. Since colimits in sPSh(C) are given object-wise, and the geometric realisation functor commutes with finite colimits for being left adjoint, then the above push-out square induces the push-out square of topological spaces   ∣jU ∣ / ∣K(U )∣ ∣U(U )∣ _ _ ∣i′U ∣

∣iU ∣

  ∣V(U )∣

′ ∣jU ∣

⌟  / ∣L(U )∣

for every U ∈ C. We will use lemma 2.35 to show that i′ is a local weak equivalence. Hence, let D be a commutative diagram of topological spaces ∣∂∆ n ∣ _  ∣∆n ∣

α

/ ∣K(U )∣

D

∣i′U ∣

β

 / ∣L(U )∣

156

for n ≥ 0, U ∈ C. Then, there exist subdivision ∣L∣ ≅ ∣∆n ∣ (see [Spa66, §.3.3]), and a subdivision ∣K∣ ≅ ∣∂∆n ∣, such that β is homotopic (relative to ∣∂∆n ∣ to continuous map ∣L∣ → ∣L(U )∣ that maps the realisation of every simplex of L either to ∣K(U )∣ or ∣V(U )∣, i.e. there are commutative diagrams (65)

∣∂∆ n ∣ _ s

d0

α

% ∣∂∆n ∣ × ∣∆1 ∣ _

d0

 /3 L(U )

hL

β

/ ∣K(U )∣ _

α

n ∣K∣ ≅ ∣∂∆ ∣ _

(66)

K(U ) _ ∣i′U ∣

 ∣∆n ∣ × ∣∆1 ∣ 9  + ∣∆n ∣

+/

cα

∣i′U ∣

 / ∣L(U )∣

RA FT

 ∣L∣ ≅ ∣∆n ∣

1

hL d

such that hL d1 maps the realisations of simplexes of L either to ∣K(U )∣ or ∣V(U )∣. Notice that L is obtain from K be attaching n simplexes each of which is homeomorphic to ∣∆n ∣, hence, there is a sequence of sub-complexes of L K = K0 ⊂ K1 ⊂ K2 ⊂ ⋯ ⊂ Kn = L such that each Kk+1 is obtained from Kk by attaching a simplex. Then, diagram (66) induce the diagram α

/ ∣K(U )∣ _

↺

∣i′U ∣

∣K∣D _ k  ∣Kk ∣

D

(67)

 / ∣L(U )∣

βk

1

where βk is the composition of hL d with the inclusion Kk ⊂ L. we will prove the conditions of lemma 2.35 by induction on the indices of the subcomplexes Kk . For k = 0, then U has the covering {idU }, and there is the constant homotopy (relative ∣K∣ ≅ ∣∂∆n ∣) at β0 , such that D0 itself admits the obvious lift. Now, assume that there exist a local relative homotpy lifting of Dk compatible with attaching simplexes of subdivisions, i.e. a covering {ϕi ∶ Ui → U }i∈I of U in τ and commutative diagrams (68)

∣K∣  r _

∣ϕ∗i ∣α

d0

$ ∣K∣ ×  ∣∆1 ∣ _

c∣ϕ∗ ∣α i

i

(69)

 /3 L(Ui )

hk,ϕi

∣ϕ∗i ∣βk

d0

∣K∣ _  ∣Kk ∣

∣ϕ∗i ∣α θk,ϕi

/ ∣K(Ui )∣ ; _ ∣i′U ∣ i

hk,ϕ d

K(U i ) _ ∣i′U ∣

 ∣Kk ∣ × ∣∆1 ∣ :  , ∣Kk ∣

+/

 / ∣L(Ui )∣ 1

157

such that for σ ∈ Kk ● if βk (∣σ∣) ⊂ ∣V(U )∣, then hk,ϕi (∣σ∣ × ∣∆1 ∣) ⊂ ∣V(Ui )∣. ● if βk (∣σ∣) ⊂ ∣K(U )∣, then hk,ϕi is constant at ∣σ∣. Notice that if βk (∣σ∣) ⊂ ∣V(U )∣ ⋂ ∣K(U )∣ = ∣U(U )∣ then these conditions are compatible. Obviously, this is automatically satisfied for k = 0. Then, based on the above assumption, we extend the above local relative homotopy lifting of Dk to a local relative homotpy lifting of Dk+1 compatible with attaching simplexes of subdivisions as follows: Since Kk+1 is obtained from Kk by attaching a simplex σ given in a push-out square / Kk ∂∆ n_ _  ∆n

⌟  / Kk+1

σ

If βk+1 (∣σ∣) ⊂ ∣K(U )∣, then hk,ϕi is extended to homotopy hk+1,ϕi ∶ ∣Kk+1 ∣ × ∣∆1 ∣ → L(Ui ) that is constant on ∣σ∣, in which case the lift θk,ϕi is evident. However, if βk+1 (∣σ∣) ⊂ ∣V(U )∣, but not contained in ∣K(U )∣, then hk,ϕi can be extended to homotopy hk+1,ϕi ∶ ∣Kk+1 ∣ × ∣∆1 ∣ → L(Ui ) such that hk+1,ϕi (∣σ∣ × ∣∆1 ∣) ⊂ ∣V(Ui )∣, as follows: θk,ϕi

RA FT

Notice that ∣σ∣ ≅ ∣∆n ∣. Also, ∂∣σ∣ ⊂ Kk → , then since ∣V(Ui )∣ ⋂ ∣K(Ui )∣ = ∣U(Ui )∣, we have θϕi (∂∣σ∣) ⊂ ∣U(U )∣. Then, we have a commutative diagram ∣∂∆n ∣ ≅ ∂∣σ∣ _ ∂

θϕi

/ ∣U(Ui )∣

∣iUi ∣



∣∆n ∣ ≅ ∣σ∣

σ

 / ∣V(Ui )∣.

Since i is a local injective weak equivalence, then by lemma 2.35, there exist covering {ψij ∶ Uij → Ui }j∈Ji of Ui ’s in τ and commutative diagrams ∣∂∆n ∣ ≅ ∂∣σ∣ _ t

' ∣∂∆n ∣ × ∣∆1 ∣ _  ∣∆n ∣ × ∣∆1 ∣ 7

*

n ∣∆ ∣ ≅ ∣σ∣ 

(71)

∗ ∣θϕi ∣ψij

d0

D

(70)

∂∣σ∣ _  ∣σ∣

d0

∣ϕ∗i ∣α θσ,ψij

hσ,ψij d1

c

∣ψ ∗ ∣θϕ i ij

+/

K(U ij ) _ ∣iUij ∣

hσ,ψij

 3/ V(Ui )

∣ϕ∗ij ∣σ

/ ∣U(Uij )∣   _ 7 ∣iUij ∣

  / ∣V(Uij )∣ 

jUij

/ ∣K(Uij )∣ _ ∣i′U ∣ ij

 / ∣L(Uij )∣.

∗ Then, we have a covering {ϕi ψij ∶ Uij → Ui → U }j∈Ji ,i∈I of U in τ . Hence, hk+1,ϕi ψij is the extension of ψij hk,ϕi 1 ∗ by hσ,ψij , then hk+1,ϕi ψij (∣σ∣ × ∣∆ ∣) ⊂ ∣L(Uij )∣, also there exist lifts θk+1,ϕi ψij given as extension of ψij θk,ϕi by jUij θσ,ψij . Hence, there exists a local relative homotopy lifting of Dk+1 compatible with attaching simplexes of subdivisions, in the above sense. Therefore, such local relative homotopy lifting exist for Dn , i.e. for ∣∆n ∣ ≅ ∣L∣ = ∣Kn ∣. Then, by lemma 2.35 i′ is a local injective weak cofibration. 

However, the class of local injective weak cofibrations is not closed under pull-backs. Corollary 2.64. Morphisms of J-cell are local injective weak cofibrations. Proof. Every morphism of J is a local injective weak cofibration. Then, by the previous lemma 2.63 morphisms of J-cell are so. 

158

We have seen that injective cofibrations and local injective weak cofibrations are closed under transfinite composition, and push-outs, and that injective cofibrations are closed under pull-backs. Then, it is natural to ask if local injective weak cofibrations, are so. That is not correct in general as seen ion the below counter example. However, the following lemma provides a remedy, which is the core of proving the factorisation axiom. Local weak equivalences are not necessary closed under pull-back and push-out. However, the below lemma show that a weaker result is obtained if attention is restricted to a subclass of local injective weak cofibrations. Lemma 2.65 (Bounded Cofibration Condition). Let i ∶ U ↪ V be a local invective weak cofibration, κ > 2∣M or(C)∣ an infinite cardinal, and j ∶ K ↪ V be a κ-bounded simplicial sub-pre-sheaf of V. Then, there exists a κ-bounded simplicial sub-pre-sheaf L ↪ V such that j decomposes as K ↪ L ↪ V and the the projection L ⋂ U → L is a local invective weak cofibration. Proof. This lemma can be proven by induction, where we construct a sequence of κ-bounded sub-objects of V L0 ⊂ L1 ⊂ ⋯ ⊂ Lk ⊂ ⋯V. Then, we show that L ∶= ⋃ Lk is a also a κ-bounded sub-objects of V such that the projection L ⋂ U → L is k∈N

a local invective weak cofibration.

(72)

RA FT

Since K is κ-bounded sub-objects of V, we start for k = 0 by putting L0 ∶= K, and let j0 ∶ K ↪ be the injective cofibration j. Then, having a κ-bounded simplicial pre-sheaf Lk constructed with an injective cofibration jk ∶ Lk ↪ V, we construct a κ-bounded simplicial pre-sheaf Lk+1 with an injective cofibration jk+1 ∶ Lk+1 ↪ V that factorise jk , as follows: Consider the Cartesian square Lk  ⋂ _ U ⌜



pr2

pr1

  Lk

/U _

i

 /V

jk

D

Then, lemma 2.60 shows that the projection pr1 ∶ Lk ⋂ U → Lk is an injective cofibration. If pr1 in (72) is a local weak equivalence, we put Lk+1 = Lk and jk+1 = jk . If we reach this case the proof is technically done. Otherwise, let U ∈ C, n ≥ 0, u0,U ∈ (Lk ⋂ U)0 (U ) ⊂ U0 (U ). Then, the injections in (72) induce a morphism of pre-sheaves jk ∗

πnpre (Lk ∣U , (Lk ⋂ U)∣U , u0,U ) Ð→ πnpre (V∣U , U∣U , u0,U ). The main idea of the paragraph below is use the above induced morphism, properties of sheafification and commutativity of sheaves of homotopy groups with colimits to construct Lk+1 which is big enough to take elements of sections of πnpre (Lk ∣U , (Lk ⋂ U)∣U , u0,U ) to ∗, but still κ-bounded. Let γ ∈ πnpre (Lk ∣U , (Lk ⋂ U)∣U , u0,U )(idU ) = πn (∣Lk (U )∣, ∣(Lk ⋂ U)(U )∣, u0,U ). Then, jk ∗ (γ) ∈ πnpre (V∣U , U∣U , u0,U )(idU ) = πn (∣V(U )∣, ∣U(U )∣, u0,U ). since i is a local injective weak cofibration, then corollary 2.34 implies that πntop (V∣U , U∣U , u0,U ) is a trivial sheaf, hence πnpre (V∣U , U∣U , u0,U ) is locally trivial pre-sheaf. Hence, there exist a covering {ϕi ∶ ϕi → idU }i∈I of idU in τ ↓ U , i.e. a covering {ϕi ∶ Ui → U }i∈I of U in C such that ϕ∗i jk ∗ (γ) = ∗ ∈ πn (∣V(Ui )∣, ∣U(Ui )∣, u0,Ui ). Since every simplicial pre-sheaf is a filtered colimit of its κ-bounded sub-objects, as in corollary ??, geometrical realisation commutes with colimits for being left adjoint, πn commute with filtered colimits by lemma 1.20, and based on the definition of colimits of sets, then there exist Vγ,i a κ-bounded sub-object of V such that ϕ∗i jk ∗ (γ) = ∗ ∈ πn (∣Vγ,i (Ui )∣, ∣(Vγ,i ⋂ U)(Ui )∣, u0,Ui ). Since ∣I∣ < κ that κ > 2∣M or(C)∣ , then by lemma 1.25 the simplicial pre-sheaf Vγ ∶= Lk ⋃(⋃ Vγ,i ) i∈I

159

is a κ-bounded with an injective cofibration to V such that jk factorise through Vγ ⊂ V. Then, ∀i ∈ I, ϕ∗i jk ∗ (γ) maps to the trivial element of πn (∣Vγ (Ui )∣, ∣(Vγ ⋂ U)(Ui )∣, u0,Ui ). Notice that ∣N∣ < κ, since κ is infinite; ∣(Lk ⋂ U)0 (U )∣ < κ; ∣Ob(C)∣ ≤ ∣M or(C)∣ < κ; ∣(Lk ⋂ U)0 (U )∣ ≤ ∣(Lk )0 (U ) < κ∣, ∀U ∈ C, hence ∣πn (∣Lk (U )∣, ∣(Lk ⋂ U)(U )∣, u0,U )∣ < κ, ∀n ≥ 0, U ∈ C, u0,U ∈ (Lk ⋂ U)0 (U ). Hence, be lemma 1.25 the simplicial pre-sheaf Lk+1 ∶=

⎞⎞ ⎛ ⎛ Vγ ⋃ ⋃ ⎠⎠ ⎝ ⎝ n≥0,U ∈C u0,U ∈(Lk ⋂ U)0 (U ) γ∈πn (∣Lk (U )∣,∣(Lk ⋂ U)(U )∣,u0,U ) ⋃

is κ-bounded with an injective cofibration jk+1 ∶ Lk+1 toV. By construction Lk+1 contains Lk as a sub-object, hence jk factorise through jk+1 . Hence, Lk is constructed as desired for every k ≥ 0, where we also have L0 ⊂ L1 ⊂ ⋯ ⊂ Lk ⊂ ⋯V. Then, applying lemma 1.25 again implies that the simplicial pre-sheaf L ∶= ⋃ Lk

(73)

k≥0

T

is a κ-bounded with an injective cofibration j ′ ∶ L → V. Now, let n ≥ 0, U ∈ Cu0,U ∈ (L ⋂ U)0 (U ), and γ ∈ πnpre (L∣U , (L ⋂ U)∣U , u0,U )(idU ) = πn (∣L(U )∣, ∣(L ⋂ U)(U )∣, u0,U ). Then, using (73), and the facts that geometric realisation commute with colimits for being a left adjoint and topological homotopy groups commutes with colimits as in lemma 1.20, then ∃k ≥ 0 such that u0,U ∈ (L ⋂ U)0 (U ) and

DR AF

γ ∈ πn (∣Lk (U )∣, ∣(Lk ⋂ U)(U )∣, u0,U ).

Then, by the construction of Lk+1 , there exist a covering {ϕi ∶ ϕi → idU }i∈I of idU in τ ↓ U , i.e. a covering {ϕi ∶ Ui → U }i∈I of U in C such that ϕ∗i jk ∗ (γ) vanish in πn (∣Lk+1 (Ui )∣, ∣(Lk+1 ⋂ U)(Ui )∣, u0,Ui ), ∀i ∈ I, hence they vanish in πnpre (L∣U , (L ⋂ U)∣U , u0,U )($i ) = πn (∣L(Ui )∣, ∣(L ⋂ U)(Ui )∣, u0,Ui ), ∀i ∈ I i.e. πnpre (L∣U , (L ⋂ U)∣U , u0,U )’s are locally trivial pre-sheaf, hence πntop (L∣U , (L ⋂ U)∣U , u0,U )’s are trivial sheaves, and the projection L ⋂ U → L is a local injective weak equivalence.  2.2.2. Local Injective Fibrations. The main results that we aim to show here are lemmas 2.66 and 2.67, below. Lemma 2.66. Morphisms of J-inj are local injective fibrations. Lemma 2.67. Morphisms of I-inj have the RLP with respect to all injective cofibrations. Moreover, they are local injective weak fibrations. The proofs of the above lemmas are relatively involved, and similar to some extent. Therefore, it is thought to be better split using the below set of auxiliary lemmas. The main idea of both proofs is to consider weak partial lifting - to be recalled later - of particular commutative squares. Then, the non-trivial part of the proofs is to show that there exists such weak partial lifting in either case. This will differ between the two proofs, and it is given by lemma 2.70 and 2.71, respectively. Afterwards, applying Zorn’s lemma induces the needed lift for both proofs. Therefore, we start by recalling partial lightings. Definition 2.68 (Partial lifting). Let D be a commutative diagram U _

e0

i

D

 V

e1

/X p

 /Y

where i is an injective cofibration, which is not an isomorphism. Then, if there exists a commutative diagram pl ∶

i

uU  _ U

e0

j

h

 W _

p

k

 V

/X >

e1

 /Y

160

such that j and k are injective cofibrations and j is not an isomorphism, then such diagram is called partial lifting of D, and we say that D admits partial lifting pl. Moreover, pl is called a weak partial lifting of D if j and k are also local weak equivalence. Hence, in such case i has to be a weak equivalence. Definition 2.69 (Order of partial lifting). Let pl, pl′ be (weak) partial liftings of D, pl ∶

i

uU  _ U

e0

j

h

 W _

/ X pl′ ∶ >

j′

p

k

 V

uU  _ U  ′ W _

i

e1

h′ p

k′

 /Y

 V

/X >

e0

e1

 / Y.

Then, it is said that pl ≤ pl iff there exist an injective cofibration W ↪ W′ such that the following diagrams commute h /  k /  j / W _ W _ U p > X W _ >V. ′

  W′ W′

 / W′

h′

k′

RA FT

j′

Notice that these condition arise naturally when pl′ is obtained as a (weak) partial lifting of the bottom quadrangle of pl, as in the proofs of lemmas 2.66 and 2.67, respectively. The below two lemmas shows that J-inj and I-inj has right weak partial lifting with respect to local injective weak cofibrations. Lemma 2.70. Let p ∶ X → Y be a morphism of simplicial pre-sheaves such that p ∈ J-inj, and let D be a commutative square e0 /X U _ i

D

 V

p

D

 /Y

e1

with i being a local injective weak cofibration which is not an isomorphism. Then, D admits a weak partial lifting. i

Proof. Since U ≇ V, then im i ≠ V, hence ∃n ≥ 0, U ∈ C such that ∃vn,U ∈ Vn (U ) ∖ im in,U . Let Wvn,U = im ιvn,U ⊂ V, since ∆nU is κ-bounded as in example 1.23, and ∣(im ιvn,U )m (V )∣ ≤ ∣(∆nU )m (V )∣, then Wvn,U is also κ-bounded. Hence, the conditions of lemma 2.65 apply to the below diagram U i

Wvn,U

 /V



i.e., there exist a κ-bounded simplicial pre-sheaf W′ such that the bottom morphism decomposes as Wvn,U ↪ W′ ↪ V and i′ in the below Cartesian square  / U W′ ⋂ _ U _ ⌜ i

i′

   W′

 /V

is a local injective weak cofibration, i.e. i′ ∈ J, then the below square admits the dotted lift  / U e 0 /6 X (74) W′ ⋂ _ U i′

   W′

p

h0

/V

e1

 / Y.

161

Then, considering the below solid commutative diagrams, and the universal property of push-outs, then there exist dotted morphisms that makes the whole diagrams commute (75)

W′ ⋂ _ U  

W′ ⋂ _ U  

/U _ o j

i′

   W′  s

j

i′

⌟  / W′ ⋃ U q

   W′

i e1 i k

/U _ ⌟  / W′ ⋃ U

",  V

pe0 h

h0

e1

e0

 -# X

ph0

 +Y

p

 - Y.

Since i′ is an injective cofibration, then its push-out j is so. Consider the diagram (76)

U  _u U

e0

j

h

RA FT

 W _

/X >

i

p

k

 V

e1

 /Y

where W ∶= W′ ⋃ U. Then, the commutativity of the diagrams (75) implies the commutativity of the two triangles sub-diagrams of (76), whereas the commutativity of the bottom quadrangle sub-diagram of (76) is due to the commutativity of D, the commutativity of (75), and the universal property of push-outs. Hence, (76) is j

commutative. Also, U ≇ W ⊂ V that vn,U ∈ (Wvn,U )n (U ) ⊂ W′n (U ) ⊂ Wn (U ) whereas vn,U ∉ (im i)n (U ). Hence, (76) is a partial lifting of D.

D

Moreover, since local injective weak cofibration are stable under push-outs, by lemmas 2.63, j is a local injective weak cofibration, and the two-out-of-three property, lemma 2.24, implies that k is so. Hence, (76) is a weak partial lifting of D.  Lemma 2.71. Let p ∶ X → Y be a morphism of simplicial pre-sheaves such that p ∈ I-inj, and let D be a commutative square K _

e0

i

D

 L

e1

/X p

 /Y

with i being an injective cofibration which is not an isomorphism. Then, D admits a partial lifting. i

Proof. Since K ≇ L, then im i ≠ L, hence ∃n ≥ 0, U ∈ C such that ∃ln,U ∈ Ln (U ) ∖ im in,U . Let n be smallest such n. Then, considering the canonical morphisms ιln,U

∂∆nU ↪ ∆nU ↪ L notice that the image of (∂ni , idU )’s by the above morphism lies in im i. That is due to the choice of n, and that the images of (∂ni , idU )’s are (n − 1) simplex of L. Hence, one obtains the evident commutative diagram ∂∆ nU _

/ K . _

 ∆nU

 /L

i

ιvn,U

162

Then, the below square admits the dotted lift /K

∂∆ nU _

(77)

/6 X

e0

p

h0

 ∆nU

ιln,U

/L

 / Y.

e1

Then, considering the below solid commutative diagrams, and the universal property of push-outs, then there exist dotted morphisms that makes the whole diagrams commute (78)

n ∂∆ _ U

/K  _ n

 ∆nU

⌟  / M o

/K _

n ∂∆ _ U

j

ιln,U

j

i e1 i k

⌟  /M

 ∆nU

 ,L

e0 pe0 h

h0

 ,X

ph0

e1

RA FT

 *Y

p

 *Y

where M = ∆nU ∐ K. Since ∂∆nU ↪ ∆nU is an injective cofibration, then its push-out j is so. ∂∆n U

diagram (79)

uU  _ K

e0

j

h

 M _

i

D

/X =

p

k

 L

Consider the

e1

 / Y.

Then, the commutativity of the diagrams (78) implies the commutativity of the two triangles sub-diagrams of (79), whereas the commutativity of the bottom quadrangle sub-diagram of (79) is due to the commutativity of D, the commutativity of (78), and the universal property of push-outs. Hence, (79) is commutative. Also, j

K ≇ M ⊂ L that ln,U ∈ Mn (U ) whereas ln,U ∉ (im i)n (U ). Hence, (79) is a partial lifting of D.  The above two lemmas show the existence of (weak) partial liftings of certain commutative squares that appears naturally in the proofs of the lemmas 2.66 and 2.67. Now, we show that the class of such lifting admits Zorn’s lemma in both cases, followed by the proof of those lemmas. Lemma 2.72. Let D be a commutative diagram U _

e0

i

D

 V

e1

/X p

 /Y

where i is a local injective weak cofibration, and let W P L be the class of weak partial liftings of D, then ≤ defined in def 2.69 is a partial order on W P L. Moreover, it is inductive ordering. Proof. Lemma 2.70 implies that W P L ≠ ∅. It is easy to see that ≤ defines a partial order on W P L. Moreover, it is an inductive ordering on W P L that for a chain of weak partial lifting C ⊂ W P L, indexed by the ordinal λ, we put Wλ = ⋃ Wβ ⊂ V. For any ordinal β < λ, let wβ ∶ Wβ ↪ Wβ+1 be the canonical injection β<λ

163

for which wplβ ≤ wplβ+1 as in def 2.69. Then, the 2-out-of-3 property and the commutative diagrams  U p

jβ

/ Wβ _

jβ+1

!  Wβ+1

shows that wβ is a local weak equivalence as well. Then, by the universal property of colimts there exist jλ , lλ and hλ that makes the following diagram commutative U  _u U

wplλ ∶

/X >

e0

jλ h λ

 W _ λ

i

p

kλ

 V

 /Y

e1

RA FT

where kλ is the evident injection i.e. an injective cofibration, and jλ is a transfinite composition of local injective weak cofibrations, hence by lemma 2.63 it is so. Then, using the 2-out-of-3 property on the side triangle, we see that kλ is also a local weak equivalence. Also, since im jβ ≠ Wβ ∀β < λ and Wλ = ⋃ Wβ , then im jλ ≠ Wλ . β<λ

Hence, wplλ ∈ W P L, and it is also easy to see that Wβ ≤ Wλ , ∀β < λ, i.e. wplλ is an upper bound for C in W P L, and ≤ is an inductive ordering on W P L.  Lemma 2.73. Let D be a commutative diagram K _

e0

i

D

D

 L

e1

/X

p

 /Y

where i is an injective cofibration, and let P L be the class of partial liftings of D, then ≤ defined in def 2.69 is a partial order on P L. Moreover, it is inductive ordering. Proof. Lemma 2.71 implies that P L ≠ ∅. It is easy to see that ≤ defines a partial order on P L. Moreover, it is an inductive ordering on P L that for a chain of partial lifting C ⊂ P L, indexed by the ordinal λ, we put Mλ = ⋃ Mβ ⊂ L. For any ordinal β < λ, let mβ ∶ Mβ ↪ Mβ+1 be the canonical injection for which plβ ≤ plβ+1 β<λ

as in def 2.69. Then, by the universal property of colimts there exist jλ , lλ and hλ that makes the following diagram commutative K  _u U

plλ ∶

e0

/X =

jλ hλ

i

 M _λ

p

kλ

 L

e1

 /Y

where kλ is the evident injection i.e. an injective cofibration, and jλ is a transfinite composition of injective cofibrations, hence by lemma 2.60 it is so. Also, since im jβ ≠ Mβ ∀β < λ and Mλ = ⋃ Mβ , then im jλ ≠ Mλ . β<λ

Hence, plλ ∈ P L, and it is also easy to see that Mβ ≤ Mλ , ∀β < λ, i.e. plλ is an upper bound for C in P L, and ≤ is an inductive ordering on P L.  Now we have the enough data to prove lemmas 2.66 and 2.67, we recall the proof below:

164

Proof of lemma ( 2.66). Suppose that p ∶ X → Y ∈ J-inj, we need to show that p has the RLP with respect to all local injective weak cofibrations. Consider a commutative square U _

e0

i

D

 V

/X p

 /Y

e1

where i ∶ U ↪ V is a local injective weak cofibration. If i is an isomorphism, then D admits the left e0 i−1 ∶ V → X. Otherwise, when i is not an isomorphism, let, W P L denotes the class of weak partial lifting of D, then lemma 2.72 W P L is none-empty and satisfies the conditions of Zorn’s lemma. Hence, W P L has a maximal element. Let e0 /X wpl ∶ U  _u U > j

h

 W _

i

p

k

 V

 /Y

RA FT

e1

be a maximal element of W P L. Assume for the sake of contradiction that k is not an isomorphism, then applying 2.70 to the bottom quadrangle sub-diagram we get the weak partial lifting h

W  _u U

j′

h′

 W ′_

k

/X >

p

k′

 V

e1

 /Y

D

Which induces the weak partial lifting of D

wpl′ ∶

i

U  _u U

e0

j′j

h′

 W ′_

/X > p

k′

 V

 /Y

e1

where wpl ≤ wpl′ and wpl ≠ wpl′ which contradict with having wpl a maximal element of W P L in wpl. Therefore, k is an isomorphism and D admits the lift hk −1 ∶ V → X. Hence, p is a local injective fibration. The proof of the converse statement of lemma 2.66 is immediate result of the definition of local injective fibrations. Proof of lemma ( 2.67 ). Suppose that p ∶ X → Y ∈ I-inj, we need to show that p has the RLP with respect to all injective cofibrations, as a result it has the RLP with respect to all local injective weak cofibrations, i.e. it is a local injective fibration. Then, we show that p is a global fibration. Then, example ?? shows it is a local weak equivalence which proves the lemma. First, consider a commutative square e0 /X K _ i

 L

D e1

p

 /Y

where i ∶ K ↪ L is an injective cofibration. If i is an isomorphism, then D admits the left e0 i−1 ∶ L → X. Otherwise, when i is not an isomorphism, let, P L denotes the class of partial lifting of D, then lemma 2.73 P L

165

is none-empty and satisfies the conditions of Zorn’s lemma. Hence, P L has a maximal element. Let pl ∶

uU  _ K

e0

j

h

 M _

i

/X > p

k

 L

 /Y

e1

be a maximal element of P L. Assume for the sake of contradiction that k is not an isomorphism, then applying 2.71 to the bottom quadrangle sub-diagram we get the partial lifting

j′

h′

 M ′_

k

/X =

h

uU  _ M

p

k′

Which induces the partial lifting of D

e1

uU  _ K

e0

j′j

h′

DR AF

pl′ ∶

 /Y

T

 L

i

 M ′_

k′

 L

e1

/X =

p

 /Y

where pl ≤ pl′ and pl ≠ pl′ which contradict with having pl a maximal element of P L in pl. Therefore, k is an isomorphism and D admits the lift hk −1 ∶ L → X. Hence, p has the RLP with respect to all injective cofibration. In particular, it has the RLP with respect to local injective weak cofibrations, hence it is a local injective fibration. On the other hand, p is a global weak equivalence, that ∀U ∈ C, and for every commutative diagram of simplicial sets (80)

∂∆ n_

e0

/ X(U ) pU

i

 ∆n

e1

 / Y(U )

it induces a commutative square of simplicial pre-sheaves (81)

∂∆ nU _

e0

p

i

 ∆n

/X

e1

 /Y

where i is the canonical inclusion, and e0 , e1 given by (e0 )m,V (µ, ϕ) = X(ϕ)(e0,m (µ)), and (e1 )m,V (µ, ϕ) = X(ϕ)(e1,m (µ)). One sees easily that e0 , e1 are well-defined and that the square is commutative. Since p ∈ I-inj, then (81) admits a lift h ∶ ∆n → X. Then, (80) admits a lift h′ ∶ ∆n → X(U ), given by h′m (µ) = hm,U (µ, idU ) that e0,m (µ) = (e0 )m,V (µ, idU ), e1,m (µ) = (e1 )m,V (µ, idU ), and the same applies for i and pU . Hence, pU has the RLP for all injections ∂∆n ↪ ∆n for all U ∈ C. Then, by corollary 3.41 pU is a weak Kan fibration for all

166

U ∈ C, in particular they are weak equivalences of simplicial sets. Hence, p is a global weak equivalence, then by lemma 2.25 it is local weak equivalence. Therefore, p is a local injective weak fibration. Notice that we can not use the fact that p in the above proves is a global injective fibration to deduce being local injective fibration, see remark 2.28. The proof of the converse statement of lemma 2.67 is not automatic, and I am not sure if it holds. Proof. The above arguments give all the components of the proof of theorem 2.57.



3. The Model Structures on Simplicial Sheaves The below construction of model structure on sShvτ (C), is due to Joyal. It can be proven based on local injective model structure on sPSh(C), for the given Grothendieck topology τ , although, historically, it was constructed before the later. It also gives a cofibranty generated model categories.

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Definition 3.1. Let f ∶ X → Y be a morphism of simplicial sheaves. Then, f is called an injective cofibrations of sheaves, or local weak equivalence of sheaves if it is so when considered a morphism of simplicial pre-sheaves. Moreover, it is called an local injective fibration of sheaves if it has the RLP with respect to all local injective weak cofibrations of sheaves. Lemma 3.2. Let p ∶ X → Y be a morphism of simplicial sheaves. Then, p is a local injective fibration of sheaves iff it is a local injective fibration, when considered as a morphism of simplicial pre-sheaves. Proof. One direction is immediate, for the other, let E be a commutative square in sPSh(C) e0

A

i

E

 B

e1

/X

p

 /Y

D

where i is a local injective weak cofibrtion, and A and B are not necessary simplicial sheaves. Then, applying the sheafiifcation functor, we have the commutative diagram A e0 ηA

!

Aa

ea 0

 a B =

X p

ia

i

(/

ea 1

 /6 Y

ηB

 B

e1

Since the sheafification functor is left exact, then ia is an injective cofibration. By example 2.27, both of ηA and ηB are local weak equivalences, then by the 2-out-of-3 2.57.CM2 ia is a local weak equivalence as well. Hence, there is a left h ∶ Ba → X, which defines a lift of E given by hηB ∶ B → X. Hence, p is a local injective fibration, when considered a morphism of simplicial pre-sehaves.  Based on the above lemma, the use of the terms ’of sheaves’ can be suppressed from the name of local injective fibrations of sheaves. We see below that the classes injective cofibrations, local injective fibrations, and local weak equivalences define the local injective model structure on sShvτ (C). Most of the axioms of theorem 3.4 are based on those of 2.57 or their supporting lemmas, In addition to example 2.27 and the below lemma. Lemma 3.3. Let p ∶ X → Y be a local injective fibration of simplicial pre-shaves, where Y is a simplicial pre-sheave. Then, the sheafification pa ∶ Xa → Y is a local injective fibration of simplicial shaves.

167

Proof. let E be a commutative square in sShvτ (C) A i

 B

e0

E e1

/ Xa pa

 /Y

where i is a local injective weak cofibration. Then, there is a simplicial pre-sheave A′ = X ×Xa A. Since the sheafification functor is left exact then it commutes with finite limits, hence (A′ )a = Xa ×Xa A ≅ A and the below diagram commutes in sPSh(C) / Xa >

e0

>A ηA′

ηX i

A′

/X

e′0

p

 B

i′

pa

 /Y

e1

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Then, by 2-out-of-three 2.57.CM2 i′ is a local weak equivalence. Therefore, there exist a lift h ∶ B → X of the below face sub-diagram, hence ηX h is a lift of E and pa is local injective fibration.  Theorem 3.4. Let Cτ be a small site. Then, the structure (C, F, W ) of the classes of injective cofibrations, invective local fibrations, and local weak equivalence, respectively, equip sShvτ (C) with a model structure.

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D

Proof. CM1 Reflective subcategories of complete/cocomplete catgeories are so. And, sShvτ (C) is a seflective subcategory of sPSh(C), with the reflector being the composition with sheafification functor. Then, by theorem 2.57.CM1, we see that sShvτ (C) is complete and cocomplete. CM2 (2-out-of-3) Let f ∶ X → Y, g ∶ Y → Z be morphisms of simplicial sheaves. Then, if two out of f , g, and gf are local weal equivalence, then they are so considered as morphisms of simplicial pre-sheaves. Then, 2.57.CM2 implies that the third is local weal equivalence, considered as a morphism of simplicial pre-sheaves, hence is so as a morphism of simplicial sheaves. CM3 (Stability under retract) Let f ∶ X → Y, and g ∶ X′ → Y′ be morphisms of simplicial sheaves such that f is a retract of g. Then, there exists a commutative diagram of simplicial sheaves: X

e0

/ X′

r0

E

g

R

f

 Y

e1

 / Y′

r1

/X f

 /Y

with horizontal composite being the identities. Suppose that g is an injective cofibration, local injective fibration, local weak equivalence. Then, by theorem 2.57.CM3a,CM3b,CM3c, f is an injective cofibration, local injective fibration, local weak equivalence, respectively. CM4 (Lifting) Let E be a commutative square of morphisms of simplicial sheaves A i

 B

e0

E e1

/X p

 /Y

where i is an injective cofibration, p is an local injective fibration of sheaves, and either of i or p is a local weak equivalence. Then, by 2.57.CM4 there is a lift of simplicial pre-sheaves h ∶ B → X. Since, sShvτ (C) is full sub-category of sPSh(C), then h is a morphism of simplicial sheaves. Hence, E admits a lift. CM5 (factorisation) Let f ∶ X → bcY be a morphism of simplicial sheaves. In particular it is a morphism of simplicial pre-sehaves. Hence, 2.57.CM5 it admits two functorial factorisations (as a morphism of simplicial pre-sehaves) f = β(f )α(f ), and f = δ(f )γ(f )

168

where β(f ) is a local injective weak fibration, α(f ) is an injective cofibration, δ(f ) is a local injective fibration and γ(f ) is an local injective weak cofibration. Let either of these factorisations be represented in the below diagram /Y >F

f

X

p

i

ia

Z

pa

ηZ

  Za where i is an injective cofibration and p is a local injective fibration, and either of them a local weak equivalence. Notice that Z is not necessary a simplicial sheaf. Since the sheafification functor is left exact, then ia is an injective cofibration. Also, by lemma 3.3 pa is a local injective fibration. Now, if i is a local weak equivalence, then the composition of local weak equivalences is so, hence ia = ηZ i is a local weak equivalence. Otherwise, providing that p a local weak equivalence, then the 2-out-of-3 CM2 implies that pa is so. Hence, f admits two functorial factorisations in sShvτ (C), namely

D

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f = (β(f ))a (α(f ))a , and f = (δ(f ))a (γ(f ))a 

APPENDIX A

Simplicial and Cosimplicial Objects The equivalence between simplicial sets and CW complexes...the link between combinatorics and topology...the introduction of invariants under homotopy equivalence) 1. Triangulated Spaces th

For every subset I ⊆ [n], the I -face of ∆ntop is defined to be {(t0 , t1 , ..., tn ) ∈ ∆ntop ∣ti = 0 for i ∉ I}. For every increasing map ν ∶ [m] → [n], we define: ∆m top

→

∆ntop

u

↦

t,

ti = {

uν −1 (i) 0

T

∆νtop ∶

i ∈ im ν otherwise

DR AF

We distinguish points vi = (0, 0, .., 1, .., 0) ∈ ∆ntop , zeros apart from the ith -coordinate,for i = 0..n, and we call them the vertices of the simplex. Notice that ∆νtop maps injectively vj to vν(j) . For every such ν, we understand ∆νtop as an identification of a m-dimensional simplex with a face of n-dimensional simplex, namely im ∆νtop , i.e. n an embedding of ∆m top on the im ν-face of ∆top . We define gluing data of simplices to be a contra-variant functor X ∶ ∆op inc → Set,

where ∆inc is the category of finite sets and increasing maps between them. The of elements of X(n) ∶= X([n]) is understood to be the set of n-dimensional simplices to be glued, for every increasing ν ∶ [m] → [n], and X(ν) specify which ∆m top to be identify with the im ν-face of each of the ∆ntop . The requirement of the functoriality of X means that X(id[n] ) = idX(n) , i.e. different elements of X(n) are understood to represent different n-dimensional simplices, so that the cardinality of X(n) is minimal. and X(νµ) = X(µ)X(ν), so that the im µ-face of im ν-face of a simplex is a face, namely the im νµ-face. For every gluing data X, we define the topological space ∞

∣ X ∣= ( ⊔ (∆ntop × X(n),Dis )) /R n=0

where X(n),Dis s the discrete topogicpace with the underlying set X(n) R is the smallest equivalence relation that n identify (u, y) ∈ ∆m top × X(m),Dis and (t, x) ∈ ∆top × X(n),Dis , such that there is a increasing map ν ∶ [m] → [n] and y = X(ν)(x), and t = ∆νtop (u). ∣ X ∣ is topologised with the weakest topology that makes the canonical map ∞

ΠX ∶ ⊔ (∆ntop × X(n),Dis ) →∣ X ∣

(83)

n=0

continuous, we call this topology the canonical topology on ∣ X ∣. We call the gluing data a triangulation of ∣ X ∣, and we call the induced topological space with the triangulation a triangulated space. Definition 1.1 (Triangulable Spaces). Let (X, τ ) be a topological space, we say that X is triangulable if it is homeomorphic to a triangulated space. . Remark 1.2. In the previous construction, we do not identify two proper faces of two simplices directly, but we rather identify each of them with one simplex of the same dimension of the common face. 169

170

Example 1.3 (S 2 ). Let X be a gluing data given by X(0) = {P1 , P2 , P3 }, X(1) = {L1 , L2 , L3 }, X(2) = {F1 , F2 }, and X(n) = ∅ otherwise, and the maps: ν ∶ [0] → [1], ν(0) = 0 ∶ ν ∶ [0] → [1], ν(0) = 1 ∶

X(ν)(L1 ) = P1 , X(ν)(L2 ) = P2 , X(ν)(L3 ) = P2 X(ν)(L1 ) = P2 , X(ν)(L2 ) = P2 , X(ν)(L3 ) = P3

ν ∶ [1] → [2], ν(0) = 0, ν(1) = 1 ∶ ν ∶ [1] → [2], ν(0) = 0, ν(1) = 2 ∶ ν ∶ [1] → [2], ν(0) = 1, ν(1) = 2 ∶

X(ν)(F1 ) = L1 , X(ν)(F2 ) = L3 X(ν)(F1 ) = L2 , X(ν)(F2 ) = L2 X(ν)(F1 ) = L1 , X(ν)(F2 ) = L3

and identities and compositions otherwise. Then, we can see readily that the triangulated space ∣ X ∣ is given by identifying the lines of the same colors in the figure, identifying the end points of the red line. Then, we can see that ∣ X ∣ is homeomorphic to the sphere S 2 .

ν ∶ [0] → [1], ν(0) = 0 ∶ ν ∶ [0] → [1], ν(0) = 1 ∶

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Remark 1.4. Note that the same space is homeomorphic to the geometric realisation given by the triangulation X, such that X(0) = {P1 , P2 , P3 }, X(1) = {L1 , L2 , L3 }, X(2) = {F1 , F2 }, and X(n) = ∅ otherwise, and the maps:

ν ∶ [1] → [2], ν(0) = 0, ν(1) = 1 ∶ ν ∶ [1] → [2], ν(0) = 0, ν(1) = 2 ∶ ν ∶ [1] → [2], ν(0) = 1, ν(1) = 2 ∶

X(ν)(L1 ) = P1 , X(ν)(L2 ) = P2 , X(ν)(L3 ) = P3 X(ν)(L1 ) = P2 , X(ν)(L2 ) = P3 , X(ν)(L3 ) = P1 X(ν)(F1 ) = L1 , X(ν)(F2 ) = L1 X(ν)(F1 ) = L3 , X(ν)(F2 ) = L3 X(ν)(F1 ) = L2 , X(ν)(F2 ) = L2

D

and identities and compositions otherwise. Then, we can see readily that the triangulated space ∣ X ∣ is given by identifying the lines of the same colors in the figure, which is also homeomorphic to the sphere S 2 .

2. Simplicial Category

[Mor03, §.2.2] and [May92] Consider the simplicial category ∆, i.e. the category of finite ordered sets and order-preserving maps between them (non-decreasing maps). Then, we distinguish the below sets of morphisms: For each n ≥ 1, 0 ≤ i ≤ n, the ith -coface, ∂ni ∶ [n − 1] → [n], the unique injective map skipping the value i, given by ∂ni (j) = {

j j
For each n ≥ 0, 0 ≤ i ≤ n, the ith -codegenerations, σni ∶ [n + 1] → [n], the unique surjective map repeating the the value i, given by σni (j) = {

j j≤i j−1 j >i

Then, it is readily seen that these morphisms satisfies the relations:

(84)

j i ∂n+1 ∂ni = ∂n+1 ∂nj−1 j+1 i σnj σn+1 = σni σn+1 ⎧ ∂ i σ j−1 ⎪ ⎪ ⎪ n−1 n−2 j i σn−1 ∂n = ⎨ id[n−1] ⎪ ⎪ i−1 j ⎪ ∂n−1 σn−2 ⎩

for i < j for i ≤ j for i < j for i = j or i = j + 1 for i > j + 1

171

We have the flowing diagram of faces and degenerations: ∂20 ∂10

[0] o

(85)

σ00 ∂11

o / /

[1] o

/

σ11 ∂21 σ10

∂30 σ22

o

/

/

∂31 σ21 ∂32 σ20

/ [2] o

∂22

/

o

/

[3]...

/

∂33

The importance of these morphisms that they generate all the morphisms of the category ∆, as shown in the below lemma: Lemma 2.1. Any morphism µ ∶ [m] → [n] of ∆ is a composition of faces and degenerations, and can be written uniquely: jt jt−1 j1 i2 is µ = ∂ni1 ∂n−1 ...∂n−s+1 σm−t σm−t+1 ...σm−1 where n − m = s − t, n ≥ i1 > i2 > ... > is ≥ 0 and m − 1 ≥ j1 > j2 > ... > jt ≥ 0.

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Idea of the proof. [GM03, P15]. It can be proven by induction on the both the number of skipped values, and the number of the repeated valued in the codomain.  Below is an illustrated example:

5⋅ ⋅ /5 ⋅ ⋅

⋅ ⋅ ⋅

[2]

µ

/ [3]

=

⋅ ⋅ ⋅

=

[2]

)⋅ /) ⋅

σ10

/ [1]

∂20

5⋅ ⋅ /⋅ /⋅

5⋅ 5⋅ ⋅ / [2]

∂32

/ [3]

D

Remark 2.2. Avoid the confusion might arise form the name of the superscript of faces, ∂10 does not map 0 to 0, it rather skips 0 and maps 0 to 1, and so on. 3. Extended Simplicial Category

Notice that [0] is a terminal object in ∆. However, ∆ does not admit an initial objects. In practice, it is useful to consider an initialisation ∆∅ of ∆, see 5.1.1 below. Let [−1] = {k ∈ Z∣ − 1 ≥ k ≥ 0} = ∅, and consider the canonical initialisation ∆∅ of ∆ with objects Ob(∆∅ ) = Ob(∆) ⊔{[−1]} and morphisms ⎧ ∆([m], [n]) ⎪ ⎪ ⎪ ⎪ ∆∅ ([m], [n]) = ⎨{εn } ⎪ ⎪ ⎪ ⎪ ⎩∅

m, n ≥ 0, m = −1, m ≥ 0, n = −1.

4. (Co)Simplicial Objects Hereby, we recall the notion of simplicial and cosimplicial objects in given categories. With comparison to triangulated spaces, cosimplicial objects provide the ”blocks” of realisation of simplicial objects, whereas the simplicial objects play the role of providing the gluing data of these realisation blocks. This will be explained later after providing the definitions and concrete examples Definition 4.1. Let C and D be categories, then we define the simplicial objects of C to be contra-variant functors: X ∶ ∆op → C and we define cosimplicial objects of D to be covariant functors: R∶∆→D

172

We write Xn , Rr instead of X([n]) and R([n]), respectively. Also, when it is not ambiguous, we will use the short notation: dni ∶= X(∂ni ), sni ∶= X(σni ) and din ∶= R(∂ni ), sin ∶= R(σni ) Then, (84) induce the same relations on din , sin , and the below corresponding relations on dni , sni : dni dn+1 = dnj−1 dn+1 j i n+1 n n si sj = sn+1 j+1 si n−1 ⎧ sn−2 ⎪ j−1 di ⎪ ⎪ dni sn−1 = ⎨ idXn−1 j ⎪ n−2 n−1 ⎪ ⎪ ⎩ sj di−1

(86)

for i < j for i ≤ j for i < j for i = j or i = j + 1 for i > j + 1

Then, for the cosimplicial objects R, we have a similar diagram of (85), substituting [n], ∂ni , and σni with Rn , din , and sin , respectively. Whereas, for the simplicial object X , (85) induces the flowing diagram:

X0 o

s00

/ X1 o

d10

o

/

s10 d21 s11

s20 d32 s21 d31

o X2 /

o

RA FT

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o

d11

o

d33

o

d22

o

d20

s22 d30

/ / X3 ... /

One can think of the simplicial object X of the category C as a set of objects of C, namely Xn with the morphisms dni , sni , satisfying (86), and all their possible composition. Whereas cosimplicial object R of the category D can be thought of a set of objects of D, Rn with the morphisms din , sin , satisfying equations similar to (84), and all their possible composition. op We define the category of simplicial objects in C to be the functor category C∆ , and the category of cosimplicial objects in D to be the functor category D∆ , i.e. the morphisms of (co)simplicial objects are natural transformations. Hence, the morphism of simplicial objects in C, f ∶ X → Y is a set of morphisms fn in C that makes the below diagram commutes: Xn

D

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Xµ

/ Xm

fn

 Yn



Yµ

fm

/ Ym

for any order preserving map µ ∶ [m] → [n], particularly, the following diagrams commute: Xn

dn i

fn

 Yn

/ Xn−1 

dn i

fn−1

/ Yn−1

Xn

sn i

fn

 Yn

/ Xn+1 

sn i

fn+1

/ Yn+1

We denote the category of simplicial objects of C by ∆op C, and the category of cosimplicial objects of D by ∆D. Consider have the functor (−)n ∶ ∆op C → C that sends X to Xn , and f to fn . Then, for every diagram functor F ∶ J → ∆op C, we denote Fn ∶= (−)n F . We use the same notation for cosimplicial objects. In the rest of this section we will direct our attention to two important examples of simplicial and cosimplicial objects. Where we study simplicial sets (a simplicial object in the category Set), and their geometric realisation, induced by ∆ a cosimplicial object of the category Top. we will recall the definition, basic properties and example in the following section. 5. The Simplicial Homotopy Theory of Simplicial Objects 5.1. Truncation and (co)skeleta adjunctions. Fix an integer n ≥ −1, and let ιn ∶ ∆≤n ↪ ∆ be the inclusion of the full subcategory ∆≤n in ∆, whose objects are [i], for i ≤ n, and let ι∞ ∶= id∆ . In particular, th ∆≤−1 ≅ ∅. Let C be a category, a functor ∆op ≤n → C is called a n -truncated simplicial object in C, denote

173 op op op th the category Funct(∆op ≤n , C) of n -truncated simplicial object by ∆≤n C. In particular, ∆≤−1 C ≅ ∗ and ∆≤∞ C ≅ op op ∆ C. Then, pre-composition with ιn induces a functor

tn ∶ ∆op C → ∆op ≤n C, called the nth -truncation functor, as it forgets the k-simplices for k > n. In particular, the ∞th -truncation is the identity functor. Assume that the category C admits finite colimits and finite limits, then nth -truncation functor admits both left and adjoint functors called the nth -skeleton and nth -coskeleton functors, respectively, see [GJ09, §.VII.1] and [Del74, §.5.I.1]. The nth -skeleton, denoted op skn ∶ ∆op ≤n C → ∆ C, th is given by the left Kan extension along the inclusion ιop n . For an n -truncated simplicial object X● in C,

(X● )([k]) = colim X● ○ Lkn skn (X● )k = Lanιop n

, for

k ∈ N,

∆op ≤n

op where Lkn ∶ ιop → is the canonical forgetful functor. The above colimits exist because the categories n ↓ [k] op op op ιn ↓ [k] are finite, for every k ∈ N. Notice that ιop is equivalent to the category of elements of the n ↓ [k] op op op th functor ∆ (ιn (−), [k] ) ∶ ∆≤n → Set. Dually, the n -coskeleton, denoted

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op coskn ∶ ∆op ≤n C → ∆ C,

th is given by the right Kan extension along the inclusion ιop n . For an n -truncated simplicial object X● in C,

(X● )([k]) = lim X● ○ Rnk coskn (X● )k = Ranιop n where Rnk ∶ [k]op ↓ ιop n →

∆op ≤n

, for

k ∈ N,

is the canonical forgetful functor.

The above triple of adjunctions

op skn ⊣ tn ⊣ coskn ∶ ∆op ≤n C → ∆ C

induces an adjunction

sqn ⊣ cosqn ∶ ∆op C → ∆op C

where

D

sqn ∶= skn ○ tn

, and

cosqn ∶= coskn ○ tn .

For n ≥ −1, let n ∶ sqn → id∆op C be the counit of the adjunction skn ⊣ tn . Since tn admits a right adjoint,  is a natural monomorphism, and hence, for a simplicial object X● ∈ ∆op C, one may think of sqn X● as a subobject of X● . X● is said to be generated by its nth -simplices if nX● is an isomorphism of simplicial objects. X● is said to be of finite combinatorical dimension if there exist an integer n for which X● is generated by its nth -simplices. In fact, for −1 ≤ m ≤ n ≤ ∞, the inclusion of categories ∆≤m ↪ ∆≤n induces inclusions of categories op op ιop ⊂ ιop for every integer k ≥ 0, which in turn induce a natural monomorphism sqm ↪ sqn . m ↓ [k] n ↓ [k] Hence, one obtains an infinite filtration n

̃ ≥−1 → End(∆op C), sq− ∶ Z visualised as

sq−1 ↪ sq0 ↪ sq1 ↪ sq2 ↪ ⋯ ↪ sqk ↪ ⋯ sq∞ ≅ id∆op C .

n op Let k be an integer. When 0 ≤ k ≤ n, id[k]op is a terminal object in ιop n ↓ [k] , hence sk (X● )k = k op op colim X● ○ Ln ≅ Xk . Whereas, when k > n, there does not exist a terminal object in ιn ↓ [k] , and the above colimit only depends on the codegeneracies, [GJ09, §.VII.1]. When C admits countable colimits, then one obtains a natural isomorphism colim sq−∣N ⋯ sq∞ ≅ id∆op C .

Dually, for n ≥ −1, let ηn ∶ id∆op C → cosqn be the unit of the adjunction tn ⊣ coskn . Since tn admits a left adjoint, ηn is a natural epimorphism, and hence, for a simplicial object X● ∈ ∆op C, one may think of sqn X● as a quotient object of X● . X● is generated by its nth -simplices if and only if ηn,X● is an isomorphism of simplicial objects.

174

For −1 ≤ m ≤ n ≤ ∞, the inclusion of categories ∆≤m ↪ ∆≤n induces inclusions of categories [k]op ↓ ιop m ⊂ [k] ↓ ιop for every integer k ≥ 0, which in turn induce a natural epimorphism cosq ↠ cosq . Hence, one n m n obtains an infinite cofiltration ̃ op → End(∆op C), cosq− ∶ Z ≥−1 op

visualised as id∆op C ≅ cosq∞ ↠ ⋯ ↠ cosqk ↠ ⋯ ↠ cosq2 ↠ cosq1 ↠ cosq0 ↠ cosq−1 . k Let k be an integer. When 0 ≤ k ≤ n, id[k]op is a initial object in [k]op ↓ ιop n , hence coskn (X● )k = lim X● ○Ln ≅ op op Xk . Whereas, when k > n, there does not exist an initial object in ιn ↓ [k] , and the above limit only depends on the cofaces, [GJ09, §.VII.1]. When C admits countable limits, then one obtains a natural isomorphism

id∆op C ≅ cosq∞ ≅ lim cosq−∣N . Two case are of special interest, namely when n = −1, 0. For n = −1, and for every integer k ≥ 0, the op categories ιop and [k]op ↓ ιop −1 ↓ [k] −1 are empty categories. Hence, sk−1 (X)k = colim XLk−1 = ∅C

, and

∅

k cosk−1 (X)k = lim XR−1 = ∗C ∀k ≥ 0, ∅

hence sk−1 (X) = ∅∆op C

cosk−1 (X) = ∗∆op C ,

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, and

identifying functors from the terminal category

∆op ≥−1

with their images.

For n = 0 t0 is the 0th -simplices functor. ∆op ≤0 is a terminal category, and hence there is a canonical isomorphism ∆op C ≅ C. For an object X ∈ C, let X also denote its associated 0th -truncated simplicial object ≤0 op op op in C. Since [0] is the initial object in ∆ , then for every integer k ≥ 0, ιop is a terminal category, 0 ↓ [k] k then the diagram XL0 consists of a unique vertex X and no non-trivial morphisms. Then, sk0 (X)k = colim XLk0 ≅ X = (Const X)k

, for

∗

k ∈ N.

D

Then, the constant functor Const ∶ C ↪ ∆op C is a 0th -skeleton functor, and hence sq0 − ≅ Const −0 . On the other hand, for every integer k ≤ 0, [k]op ↓ ιop 0 is a discrete category with k + 1 objects and no non-trivial morphisms. Then, for an object X ∈ C, the diagram XR0k consist of k + 1 vertices, each of which is X with no non-trivial morphisms. Then, cosk0 (X)● = ∏ X, [●]

with faces and degeneracies induces from the cofaces and codegeneracies of the Simplicial Category ∆, cf. [MV99, The Sing● functor, p.88]. Composing with the 0th -truncation, one finds cosq0 X● ≅ ∏ X0 , [●]

for every simplicial object X● in C, [Sta16, Tag 0182]. The constant functor Const admits a lift adjoint, as long as the category C admits coequalisers. It is called the connected components functor, and denoted by π0 ∶ ∆op C → C. In general, for a diagram category I, the very definition of colimits asserts that the existence of I-shaped colimits colimI in C is equivalent the existence of an adjunction colim ⊣ ConstI ∶ Funct(I, C) → C. I

However, for simplicial objects, the relations in (86) result in having colim X● ≅ π0 (X● ) ∶= coeq(d10 , d11 ∶ X1 ⇉ X0 ). Then, under the assumption that C admits finite colimits, on has a quadruple of adjunctions π0 ⊣ Const ≅ sk0 ⊣ t0 ≅ −0 ⊣ cosk0 ≅ ∏ − ∶ ∆op C → C, [●]

and as a result a triple of adjunction Const π0 ⊣ sq0 ⊣ cosq0 ∶ ∆op C → ∆op C. The last triple of adjunction saturates, i.e. the two adjunctions it induces are already present.

175

5.1.1. Relative truncation and (co)skeleta adjunctions. For a simplicial object X● , and for −1 ≤ n ≤ ∞, there the nth -truncation functor evidently induces a relative truncation functor op op ● tX n ∶ ∆ C ↓ X● Ð→ ∆≤n C ↓ tn X●

which admits a right adjoint, when C has finite limits. Its right adjoint, the relative nth -coskeleton op op ● coskX n ∶ ∆≤n C ↓ tn X● Ð→ ∆ C ↓ X●

is given by coskn , pulled-back along ηnX● ∶ X● → cosqn (X● ), see [Del74, §.5.I.1] and overcategory/32#Adjunction.

https://ncatlab.org/nlab/revision/

Example 5.1. Notice that there is a canonical isomorphism of categories ∆op (C ↓ S) ≅ ∆op C ↓ Const S S for S ∈ C, for which tConst coincide with the 0th -truncation in C ↓ S, and hence 0 S coskConst (X → S) = ∏ X → Const S. 0 S [●] 0 (Co)Augmentation. For −1 ≤ n ≤ ∞, objects of the comma category ∆op ≤n C ↓ tn sk are called augmented n -truncated simplicial objects in C. Since [−1] is an initial object in ∆∅ , then there is an isomorphism between op 0 th the category ∆op ≤n C ↓ tn sk of augmented n -truncated simplicial objects in C and the category ∆∅,≤n C of extended nth -truncated simplicial objects in C. The canonical isomorphism

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th

op 0 ∆op ≤n C ↓ tn sk Ð→ ∆∅,≤n C

D

+ sends α● ∶ X● → tn Const S, for X● ∈ ∆op ≤n C and S ∈ C, to the X● given on objects by ⎧ ⎪ ⎪X([m]) m ≥ 0, Xk+ ([m]) = ⎨ ⎪ S m = −1, ⎪ ⎩ and on morphisms by ⎧ ⎪ ⎪X(µ) µ ∈ ∆, + Xk (µ) = ⎨ ⎪ α µ ∶ [−1] → [m]. ⎪ ⎩ m An augmented nth -truncated simplicial object α● ∶ X● → tn Const S in C is also referred to as a nth -truncated simplicial resolution of S.

5.2. Mapping (co)cylinders, Mapping (co)cones, (co)suspension. Let C be a category that admits small coproducts, then ∆op C is a simplicial category. The function complexes of ∆op C Map(−, −) ∶ ∆op C × ∆op C → sSet

is given by Map(X● , Y● )n = ∆op C(X● ⊗ ∆n , Y● ), for X● , Y● ∈ ∆op C, where the tensor with simplicial sets

, for

n∈N

− ⊗ − ∶ ∆op C × sSet → ∆op C is given by (X● ⊗ K● )n = ∐ Xn ,

, for

n∈N

Kn

for X● ∈ ∆op C and K● ∈ sSet, with cofaces and codegenerations given by the cofaces and codegenerations of K● and the universal property of coproduct. Notice that X● ⊗ ∆0 ≅ X● for every X● ∈ ∆op C. TBC...

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APPENDIX B

Simplicial Sets Definition 0.1 (Simplicial Sets). A simplicial set X is defined to be simplicial object of Set: X ∶ ∆op → Set I.e. simplicial sets are simplicial objects in Set. Then, it is natural to define a morphism of simplicial sets to be a morphism in ∆op Set. We denote category of simplicial sets by sSet ∶= ∆op Set.

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One can think of simplicial sets as a graded set with maps between them, generated by dni and sni that satisfy (86). We denote a generic element of Xn by xn . We denote the c 1. Geometric Realisation and Singular Functors

We saw in 1 that topological simplices plays the role of the blocks of the topological intuition ”realisation” of the gluing data. We recall that the geometric realisation functor ∆ plays the same role and gives rise to the geometric intuition when dealing with simplicial sets. Therefore, we start by recalling the definition of ∆. Consider the functor: given by ∆([n]) =

∆ntop ,

∆ ∶ ∆ → Top the n -simplex, and for every order preserving map µ ∶ [m] → [n], ∆(µ) is defined by: th

∆(µ) ∶ ∆m top u

→ ↦

∆ntop t,

ti =

∑

uj .

D

j∈µ−1 (i)

n m n ∆(µ) is the unique affine map ∆m top → ∆top that sends the vertex vi ∈ ∆top to vµ(i) ∈ ∆top , [Mor03, p.11]. We id

denote ∆µtop ∶= ∆(µ). ∆ is a functor that ∆top[n] = id∆ntop , and for any composable pair of order preserving maps ν

µ

[l] → [m] → [n], and every v ∈ ∆ltop , let u = ∆νtop (v), t = δµ (u), and t′ = ∆µν top (v), then: ti =

∑

uj , and uj =

j∈µ−1 (i) ∆µν top .

∑

vk , hence ti =

k∈ν −1 (j)

∑

∑

vk =

j∈µ−1 (i) k∈ν −1 (j)

∑ k∈(µν)−1 (i)

vk = t′i

Hence,∆µtop ∆νtop = For every such morphism µ, we understand ∆µtop as an identification of the mdimensional simplex with a face of the n-dimensional simplex, namely im ∆µtop . Notice that ∆µtop also maps the vertices vj to vµ(j) , on the contrary to the case of triangulated spaces, this mapping is not necessary injective. That, when µ is not strictly increasing the dimension of the face is less than the dimension of ∆m top . Definition 1.1 (Geometric Realisation). Let X be a simplicial set. Then, define the topological space: ∞

∣ X ∣= ( ⊔ (∆ntop × Xn,Dis )) /R

(89)

n=0 n where R is the smallest equivalence relation that identify each (u, xm ) ∈ ∆m top × Xm,Disc and (t, xn ) ∈ ∆top × Xn,Disc , whenever there is a morphism µ ∶ [m] → [n] in ∆ such that

xm = Xµ (xn ), and t = ∆µtop (u).

(90)

∣ X ∣ is topologised the same way as in (83). Throughout these notes equivalence classes referred to by square brackets, so elements of ∣ X ∣ are denoted [(t, xn )] where t ∈ ∆ntop and xn ∈ Xn .Notice that ∣ X ∣ is the coequaliser in Top of the the obvious diagram, [Mor03, p.12] ∞

∞

m=0,l=0

n=0

m n ⊔ (∆top × Xl,Dis ) ⇉ ⊔ (∆top × Xn,Dis ) 177

178

Also, let f ∶ X → Y be a morphism of simplicial sets. Then, define (91)

∣ f ∣ ([(t, xn )]) = [(t, fn (xn ))]

Lemma 1.2. Let ∣ − ∣∶ sSet → Top be defined on object by (89) and on morphism by (91). Then, ∣ − ∣ is a well-defined functor. Proof. ∣ f ∣ is well-defined that ∀[(t, xn )], [(u, xm )] ∈∣ X ∣ such that [(t, xn )] = [(u, xm )], then there exist order-preserving maps µ ∶ [m] → [n], or µ′ ∶ [n] → [m] that satisfies (90). Without loose of generality, up to renaming, let µ ∶ [m] → [n] exists such that xm = Xµ (xn ), and t = ∆µtop (u). Using (88) we see that fm (xm ) = fm (Xµ (xn )) = Yµ (fn (xn )), hence [(t, fn (xn ))] = [(u, fm (xm ))], i.e. ∣ f ∣ ([(t, xn )]) =∣ f ∣ ([(u, xm )]). Consider the commutative diagram: ∞

n

⊔ (∆top × Xn )

πX

/∣X ∣

n=0 ∣f ∣



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f′

∞

n

⊔ (∆top × Yn )

n=0

∞

πY

 /∣Y ∣

where f ′ = ⊔ (id∆ntop × fn ), let V ⊂∣ Y ∣ open, (∣ f ∣ πX )−1 (V ) = (πY f ′ )−1 (V ). πY is continuous by definition, n=0

and one can readily see that f ′ is continuous, then (∣ f ∣ πX )−1 (V ) is open, hence ∣ f ∣−1 (V ) is open and ∣ f ∣ is continuous. f

g

One can readily see that ∣ idX ∣= id∣X∣ , and for any composable morphisms X → Y → Z, we have ∣ gf ∣=∣ g ∣∣ f ∣. 

D

Definition 1.3. The above defined functor

∣ − ∣∶ sSet → Top

is called the geometric realisation functor. The geometric realisation allows us to think of elements of Xn as n-simplices, glued using the above equivalence relation. Therefore, we call the elements of Xn the n-simplices of X, in particular we call elements of X0 the vertices of X. We drop the notation of the subscript Dis, and we use Xn to refer to both the discrete topological space and its underlying set, when the it is not ambiguous. This functor have a right adjoint, namely the singular homology functor. Both functor plays an essential role in the study of simplicial sets, and this adjuction induces an equivalence between the category of Kan complexes and the category of CW-complexes. Definition 1.4 (Singular Functor). Let X be a topological space. Then, for n ≥ 0, we define: (92)

S(X)n ∶= Top(∆ntop , X) = {fn ∶ ∆ntop → X∣fn is continuous},

and for every morphism µ ∶ [m] → [n] in ∆, we define S(X)µ ∶ S(X)n → S(X)m by: (93)

S(X)µ (fn ) = fn µ, ∀fn ∈ S(X)n .

Also, let f ∶ X → Y be a continuous map of topological spaces. Then, for n ≥ 0, we define (94)

S(f )n ∶

S(X)n fn

Ð→ S(Y )n z→ f ○ fn .

Lemma 1.5. Let X be a topological spaces S(X) ∶ Top → sSet be defined on object by (??) and (??), and given on morphism component-wise by (??). Then, ∣ − ∣ is a well-defined functor.

179

Proof. We can see readily that S(X)id[n] = idS(X)n , and S(X)µν = S(X)ν S(X)µ , for composable morµ

ν

phisms [l] → [m] → [n]. Hence, S(X) ∶ ∆op → Set is a simplicial set. Also, the collection of S(f )n defines a morphisms of simplicial sets S(f ) ∶ S(X) → S(Y ), with componenets S(f )n , that the following diagram commutes for every morphism µ ∶ [m] → [n] in ∆: S(X)n

S(f )n

.

S(Y )µ

S(X)µ

 S(X)m

/ S(Y )n

S(f )m

 / S(Y )m

On the other hand, one can easily see that S(idX ) = idS(X) , and that S(gf ) = S(g)S(f ) for composable f

g

contiuous maps X → Y → Z. Hence, S ∶ Top → sSet is a well-defined functor.



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Definition 1.6. The above defined functor S ∶ Top → sSet

is called the singular functor. Also, we call S(X) the singular simplicial set of X, and the n-simplices of S(X) are called singular n-simplices of X. The below lemma is the main result in this section. Where, it provides a useful tool that links combinatorial and topological concepts, and it is of particular usefulness in the study of model structure of the category of simplicial sets. Lemma 1.7. The geometric realisation functor is a left adjoint to the singular functor, i.e. there is a natural isomorphism: φ ∶ Top(∣ − ∣, −) → sSet(−, S(−))

..

D

Proof. ....



2. Generators of Simplicial Sets

2.1. Degeneracy. Let X be a simplicial set, xn+1 ∈ Xn+1 is called degenerate simplex if there is n ≥ i ≥ 0, and xn ∈ Xn such that xn+1 = sni (xn ). That, in the realisation ∣ X ∣, (t, xn+1 ) is identified with (sin (t), xn ) for every t ∈ ∆n+1 top , so xn+1 does not encode additional data more than these already given by xn and the degeneration. Notice that 0-simplices of X are all non-degenerate. The below lemma gives an alternative definition of degenerate simplices. Lemma 2.1. Let X be a simplicial set, xm ∈ Xm , then the following statements are equivalent: (1) xm is a degenerate simplex of X. (2) There exits n < m, xn ∈ Xn and a surjection µ ∶ [m] → [n], such that xm = Xµ (xn ). (3) There exits n < m, xn ∈ Xn and a map µ ∶ [m] → [n], such that xm = Xµ (xn ). Proof. 1 ⇒ 2, and 2 ⇒ 3 are straightforward. 3 ⇒ 1 Assume there exits n < m, xn ∈ Xn and a map µ ∶ [m] → [n], such that xm = Xµ (xn ). Then, using lemma 2.1, µ factorise as: jt jt−1 j1 i2 is µ = ∂ni1 ∂n−1 ...∂n−s+1 σm−t σm−t+1 ...σm−1 i If t ≤ 0, then n ≥ m, hence t > 0, i.e. µ can be writen µ = νσm−1 , for some 0 ≤ i ≤ m − 1, and ν ∶ [m − 1] → [n]. m−1 Then, xm = Xµ (xn ) = si (Xν (xn )). Since, Xν (xn ) ∈ Xm−1 , then xm is degenerate. 

Lemma 2.2 (Eilenberg-Zilber Lemma). Let X be a simplicial set, for every xm ∈ Xm a m-simplex of X, there exists a unique surjection µ ∶ [m] → [n], and a unique non-degenerate n-simplex of X, xn ∈ Xn , such that xm = Xµ (xn ). cite

180

Proof. type idea assume there are two non-degenerate...: Since xn ∈ Xn is a pure degeneracy of xp ∈ Xp , then xn = Xµ (xp ) for µ ∶ [n] → [p] given by: jt jt−1 j1 µ = σn−t σn−t+1 ...σn−1

where t = n − p, n − 1 ≥ j1 > j2 > ... > jt ≥ 0. Also, since xn ∈ Xn is a pure degeneracy of xq ∈ Xq , then xn = Xν (xq ) for ν ∶ [n] → [q] given by: kt−1 k1 ku ν = σn−u σn−u+1 ...σn−1

where u = n − q, n − 1 ≥ k1 > k2 > ... > ku ≥ 0. 

T

The above lemma says that a simplex of X is either non-degenerate, or the ”surjective”-image of unique nondegenerate given in the above canonical way. Therefore, in some occasions, it will be cnvennto direct our attention to the non-degenerate simplices of X, we denote the set of non-degenerate n-simplices by X(n) . Pure Degeneracy. Let X be a simplicial set, xm = Xµ (xn ) ∈ Xm for xn ∈ Xn and µ ∶ [m] → [n] in ∆, then we say that xm is a pure degeneration of xn if µ is writen in the form 2.1 with s = 0. Then every simplex is a pure degeneracy of itself. Apart from self-pure degeneracy, one can see easily that pure degenerated simplices are not faces of non-degenerate simplices.

DR AF

Remark 2.3. Let X be a non-empty simplicial set, x0 ∈ X0 , n ≥ 0, then there is a unique pure degeneracy of x0 of grade n, i.e. belongs to Xn , namely Xσn (x0 ), where σn is the unique morphism [n] → [0].We call Xσn (x0 ) the nth -pure degeneracy of x0 . Moreover, for any µ ∶ [m] → [n] in ∆, σn µ = σm , then: Xµ (Xσn (x0 )) = Xσm (x0 ). In particular, for µ being ∂ni or σni , we see that faces and degenerations of pure degeneracy of x0 is a pure degeneracy of x0 . In Eilenberg-Zilber Lemma lemma 2.2, xm is a pure degeneracy of xn Notice that the property of having only one pure degenerated simplex for each grade is not satisfied for other simplices rather than vertices. 2.2. Generator of Simplicial Sets. Let X be a simplicial set, S a graded subset set of X, i.e. Sn ⊂ Xn , ∀n ≥ 0, not necessary a simplicial subset. For [p] ∈ ∆, and θ ∶ [q] → [p] in ∆, we define < S >X,p = {xp ∈ Xp ∣∃m ≥ 0, ∃β ∶ [p] → [m] order-preserving , ∃xm ∈ Sm , such that xp = Xβ (xm )} ⊂ Xp . < S >X,θ (xp ) = Xθ (xp ). < S >X,θ is well defined that for xp ∈< S >X,p , ∃m ≥ 0, ∃β ∶ [p] → [m], ∃xm ∈ Sm , such that xp = Xβ (xm ), hence < S >X,θ (xp ) = Xθ (xp ) = Xθ (Xβ (xm )) = Xβθ (xm ) i.e. ∃m ≥ 0, ∃βθ ∶ [q] → [m] order-preserving, ∃xm ∈ Sm , such that < S >X,θ (xp ) = Xβθ (xm ), hence < S >X,θ (xp ) ∈< S >X,q . Since X is simplicial set, then < S >X , defined on objects and morphisms above, is so. Moreover, one can see readily that the above inclusions induce an inclusion of simplicial sets < S >X ⊂ X. < S >X is the smallest simplicial subset of X that contains S as a graded subset. Definition 2.4. We call < S >X defined above the simplicial subset of X generated by S, and we say that X is generated by S iff X =< S >X , then we also say S generates X, S is a ”generator” of X, and graded elements of S are the ”generators” of X. We call a generator S of X a minimal generator if any graded subset S ′ of S, that generates X, coincide with S. Lemma 2.5. Let X be a simplicial set, then X has a unique minimal generator. Proof. Let S be the set of ”generator”s of X, S ≠ ∅, that < X >X = X, i.e. X ∈ S. S is partially order by graded inclusion. Let T be a chain of totally ordered subset of S, and let S be the graded set defined by Sn = ⋂ Tn T ∈T

S is a graded subset of X, we need to show that S generates X. If we do, then by Zorn’s Lemma, S has a minimal element, i.e. X has a minimal generator...TBC 

181

In all consider examples, Sn = ∅ for all but finite many n ≥ 0, In most cases Sn ≠ ∅ only for one value of n. Remark 2.6. Avoid the confusion between a generator of simplicial sets with the the generators of morphisms in the simplicial category. Definition 2.7. Let X be a simplicial set, xn ∈ Xn , and xm ∈ Xm , we define a simplicial relation of xn , and xm to be the triple (p, α, β) where p ≥ 0, α ∶ [p] → [n], and β ∶ [p] → [m] in ∆, such that Xα (xn ) = Xβ (xm ) ∈ Xp . We say that there is a simplicial relation between xn , and xm in X, if such triple exists. We say that a simplicial relation (p, α, β) factorises through a simplicial relation (p′ , α′ , β ′ ) between xn , and xm , if there exists i ∶ [p] → [p′ ] in ∆ such that the following diagram commutes: [p] i

 [p′ ]

α

 } [n]

!  [m]

β′

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α′

β

We say that a simplicial relation between xn , and xm is minimal if it does not factorise through different simplicial relation. Notice that when (p, α, β) factorise through (p′ , α′ , β ′ ), then: Xα (xn ) = Xβ (xm ) is a redundant simplicial relation that it can be deduced from Xα′ (xn ) = Xβ ′ (xm ). Moreover, let S be a graded subset of X, we define the set minimal simplicial relations on S in X to be the sets of all minimal simplicial relation between pairs of elements of S. Lemma 2.8. Let X be a simplicial set, xn ∈ Xn , and xm ∈ Xm , then there is a unique minimal simplicial relation between xn , and xm . Proof. TA..



D

Lemma 2.9. Let X be a simplicial set, S is the generator of X, then there is a unique set of minimal simplicial relations on S in X. Proof. TBC.



Lemma 2.10. Let X, and Y be simplicial sets, X generated by S. Then, giving a morphism of simplicial sets f ∶ X → Y is equivalent to giving a graded map f ′ ∶ S → Y , that respects the set of minimal simplicial relations on S in X. Moreover, f ′ is the restriction of f to S. Proof. It is clear that if f ∶ X → Y is a morphism of simplicial sets, i.e. a natural transformation, then it is a graded map respects the set of minimal relations on S, and its restriction is so. Then, f ′ = f ∣S . On the other hand, assume that there is a graded map f ′ ∶ S → Y , that respects the set of minimal simplicial relations on S in X. Since X is generated by S, then ∀n ≥ 0, ∀xp ∈ Xp , ∃n ≥ 0, ∃α ∶ [p] → [n], ∃xn ∈ Sn , such that xp = Xα (xn ). Then we define: fp (xp ) = Yα (fn′ (xn )) fp is well-defined. That, if there exist m ≥ 0, ∃β ∶ [p] → [m], xm ∈ Sm , such that xp = Xβ (xm ). Then we have the simplicial relation between xn and xm given by Xα (xn ) = Xβ (xm ). Since f ′ respects minimal simplicial relations on S in X, then it respects all simplicial relations on S in X. Hence, Yα (fn′ (xn )) = f ′ (Xα (xn )) = ′ f ′ (Xβ (xm )) = Yβ (fm (xm )). Then we have the graded map f ∶ X → Y , defined object-wise above. One sees readily that f ′ = f ∣S . We need to show that f is a morphism of simplicial sets, i.e. the following diagram commutes for every θ ∶ [p] → [q] Xp

Xθ

fp

 Yp

/ Xq fq

Yθ

 / Yq

182

that

= fq (Xθ (Xα (xn ))) = fq (Xαθ (xn )) = Yαθ (f ′ (xn )) = Yθ (Yα (f ′ (xn ))) = Yθ (fp (Xα (xn ))) = Yθ (fp (xp )). Hence, f is the desired morphism of simplicial sets. fq (Xθ (xp ))



Corollary 2.11. Equality of morphisms of simplicial sets is determined on generators. Remark 2.12. The same concept and results can be generalised for any functor F ∶ C → sSet, not necessary simplicial sets. 2.3. Generators and Geometric Realisations. The geometric realisation is the generator simplices glued together by the minimal relations, the rest is just technical redundant information. Lemma 2.13. Let X be a simplicial set, generated by S, then ∞

∣ X ∣= ( ⊔ (∆ntop × Sn,Dis )) /R n=0

Where, R is the smallest equivalence relation generated by the set of minimal simplicial relations of S in X. 3. The Category of Simplicial Sets

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Since the category Set is a complete and cocomplete category, then by lemma ?? the category of simplicial sets is a complete and cocomplete category with limits and colimits are given object-wise. I.e. for a digaram functor F ∶ J → sSet, we have: (lim F )n = lim Fn , and (colim F )n = colim Fn J

J

J

J

D

Where Fn = (−)n F for (−)n ∶ sSet → Sets with (X)n = Xn and (f )n = fn . Whereas, faces and degeneracies are induced by universal property of limit and colimit. 3.0.1. Initial and Terminal Simplicial Sets. Let X be a simplicial set, if there exists an integer n0 such that Xn0 = ∅, then Xn = ∅∀n ≥ 0, we call this simplicial set the empty simplicial set and we denote it by ∅. Also, we have the simplicial set X given by X = ∗, ∀n ≥ 0, where ∗ is a singleton, with the evident maps. We denote this simplicial set ∗. One can readily see that ∅, and ∗ are the initial and terminal objects of sSet, respectively. Later on we will see that ∗ = ∆0 , and ∅ = ∂∆0 . 3.0.2. Morphisms of Simplicial Sets. Natural transformations are monomorphisms, epimorphisms, or isomorphisms if they are monomorphisms, epimorphisms, or isomorphisms object-wise, respectively. Since, monomorphisms in sSet are just injections, we call a monomorphism of simplicial sets X ↪ Y a simplicial injection, and we denote it X ⊂ Y . In particular, simplicial inclusion X ⊂ Y is given by inclusions Xn ⊂ Yn , ∀n ≥ 0 that makes the following diagram commute for every µ ∶ [m] → [n] in ∆ Xn



in

Xµ

  Xm

/ Yn Yµ

im

 / Yn

where in ’s are the inclusion maps. We say that X is a simplicial subset of Y . The above commutative diagram implies that X is given on morphism as a restriction of Y , that Xµ = Yµ ∣Xn . Notice that not every such family of inclusions makes the above diagram commute. For example, let X = ∗, ∗ ≠ Y ≠ ∅, then there exist m≠ 0 such that #Ym > 1. Let µ ∶ [m] → [n] be in ∆ for some n ≥ 0, yn ∈ Yn , ′ ym ∶= Yµ (yn ), then choose ym ∈ Ym different from ym , and consider inclusions in ∶ ∗ → Yn , such that {yn } = im in , ′ and {ym } = im im , then the above diagram does not commute and the collection of maps in does not define a morphism of simplicial sets. 3.0.3. Operations on Simplicial Sets. Operations on simplicial sets are defined object-wise, as follows: Let X and Y be simplicial sub-sets of Z, then we define: (X ⋂ Y )n ∶= Xn ⋂ Yn , (X ⋃ Y )n = Xn ⋃ Yn , (X ∖ Y )n = Xn ∖ Yn for every n ≥ 0. One can easily see that they are well-defined simplicial sets, and that there exist injections X ⋂ Y ⊂ X, X ⊂ X ⋃ Y , and X ∖ Y ⊂ X. Quotient Simplicial Sets:

183

Let X be a simplicial sets, and ∼ be a graded equivalence relation on X that respects faces and degenerations, i.e. ∀xn ∼n x′n ∈ Xn , then dni xn ∼n−1 dni x′n , and sni xn ∼n+11 sni x′n Then, one can define the simplicial set X/ ∼ by (X/ ∼)n = Xn /∼n , dni ([xn ]) = [dni xn ] , and sni ([xn ]) = [sni xn ] Then, the projection X → X/ ∼ is the initial universal morphism of simplicial sets f ∶ X → Y such that fn (xn ) = fn (x′n ) for xn ∼n x′n . 3.1. Finite Limits and Colimits of Simplicial Sets. We recall here the description of the building blocks of finite (co)limits, namely (co)product, (co)equaliser: Products. Let X, Y be simplicial sets, then the product of X and Y is given by (X × Y )n = Xn × Yn Hence, n-simplices of X × Y are ordered pairs (xn , yn ) where xn ∈ Xn and yn ∈ Yn . Faces and degeneracy are given component-wise by: dni (xn , yn ) = (dni xn , dni yn ) , and sni (xn , yn ) = (sni xn , sni yn )

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Products of simplicial sets plays an important rule in the study of the model category of simplicial sets, particularly in simplicial homotopy. Therefore, we study it in more details in the next paragraph. Equaliser : Let f, g ∶ X → Y be morphisms of simplicial sets, then eq(f, g), the equaliser of f and g is the simplicial subset of X given by: eq(f, g)n = {xn ∈ Xn ∣fn (xn ) = gn (xn )} ⊂ Xn

Fibre Product: Let f ∶ X → Y , f ′ ∶ X ′ → Y be morphisms of simplicial sets. Then, the fibre product of X ×Y X ′ is given the equaliser of f pr1 ∶ X × X ′ → Y and f ′ pr2 ∶ X × X ′ → Y . 3.1.1. Product of Simplicial Sets and Shuffles. In order to understand morphisms of simplicial sets from product of simplicial sets into a simplicial set, it is useful to obtain a description of the generators of the product in terms of the generators of the generators of its components, or at least a description of its non-degenerate simplices. A classical way to obtain this description is shuffles, which is recalled below:

D

Definition 3.1 ((p, q)-shuffles). A (p, q)-shuffle is a permutation π ∈ Σp+q of the set {0, ..., p + q − 1} that preserves the relative order of elements of each of the subsets {0, 1, ..., p − 1} and {p, p + 1, ..., p + q − 1}, i.e. π(i) < π(j) for 0 ≤ i < j ≤ p − 1 or p ≤ i < j ≤ p + q − 1

We denote (µ, ν) = π, µi = π(i − 1) for 0 ≤ i ≤ p − 1 and νj = π(p + j) for 0 ≤ j ≤ q − 1. One can think of shuffle as shuffling two sets of p and q elements together without changing the order f the elements of each set. For a (p, q)-shuffle (µ, ν), One can then define q+p−2 q p+q−1 p+q−2 p sµ = sq+p−1 µp−1 sµp−2 ...sµ0 , and similarly sν = sνq−1 sνq−2 ...sν0 .

Then, for any two simplicial sets X and Y , xp ∈ Xp , and yn ∈ Yq , we find that sν (xp ) ∈ Xp+q , and sµ (xq ) ∈ Yp+q . Hence, (sν (xp ), sµ (yq )) ∈ (X×)p+q Lemma 3.2. Let X, Y be simplicial sets, and xp ∈ Xp , yq ∈ Yq be non-degenerate simplicies of X, Y , respectively. Then (95)

(sν (xp ), sµ (yq )) ∈ (X × Y )p+q

is a non-degenerate simplex of X × Y , for (µ, ν) a (p, q)-shuffle. Proof. Let (µ, ν) be a (p, q)-shuffle, and xp ∈ Xp and yq ∈ Yq be non-degenerate simplicies. Assume for the sake of contradiction that (sν (xp ), sµ (yq )) is degenerate, then there exists (xp+q−1 , yp+q−1 ) ∈ (X × Y )p+q−1 such that (sν (xp ), sµ (yq )) = sp+q−1 (xp+q−1 , yp+q−1 ) = (sp+q−1 (xp+q−1 ), sp+q−1 (yp+q−1 )) i i i q+p−2 q p+q−1 p+q−2 p for some 0 ≤ i ≤ p + q − 1. We have sµ = sq+p−1 µp−1 sµp−2 ...sµ0 , and sν = sνq−1 sνq−2 ...sν0 for 0 ≤ µi < µj ≤ p + q − 1 and 0 ≤ νi < νj ≤ p + q − 1. Then, by Eilenberg-Zilber lemma 2.2, and using the relations (84), we can rearrange

184

sµj ’s and sνk ’s such that si = sµq+p−1 = sνp+q−1 for some 0 ≤ j ≤ p − 1 and 0 ≤ k ≤ p − 1, which contradict with j k having (µ, ν) a (p, q)-shuffle.  Lemma 3.3. Let X, Y be simplicial sets, (xn , yn ) = (sν (xp ), sµ (yq )) ∈ (X × Y )n , where xp , yq are nondegenerate simplices of X, Y , respectively, such that p + q ≤ n. Then (xn , yn ) is non-degenerate iff p + q = n and (µ, ν) is a (p, q)-shuffle. Proof. Let the condition be satisfied, then by lem 3.2 (xn , yn ) is non-degenerate. On the other hand, let (xn , yn ) ∈ (X × Y )n be a non-degenerate simplex. By lemma 2.1, we have: lu−1 kt−1 l1 k1 kt lu ...σn−1 , ν = σn−u σn−u+1 ...σn−1 µ = σn−t σn−t+1

where, n − 1 ≥ k1 > k2 > ... > kt ≥ 0, and n − 1 ≥ l1 > l2 > ... > lu ≥ 0. lu−j kt−i We have p + u = q + t = n, that xn = Xν (xp ), and yn = Yµ (yq ). We put µi = σn−t+i for 0 ≤ i ≤ p − 1 and νj = σn−u+j for 0 ≤ j ≤ q − 1. Assume for the sake of contradiction that p + q ≠ n, then p + q < n and t + u > n, hence the indices of µi ’s and νj ’s are not mutually disjoint, i.e. there exists 0 ≤ i ≤ t, 0 ≤ j ≤ u such that r = ki = lj . Then, using (84) we can rearrange factors of µ and ν such that

Hence,

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r r µ = µ′ σn−1 , and ν = ν ′ σn−1

(xn , yn ) = sn−1 r (Xν ′ (xp ), Yµ′ (yq ))

which contradict with having (xn , yn ) non-degenerate. Thus n = p + q, i.e. t = p and u = q. Also, assume for the sake of contradiction that (µ, ν) is not a (p, q)-shuffle then µi ’s and νj ’s are not mutually disjoint. Then, the above argument leads to a contradiction. Hence, (µ, ν) is a (p, q)-shuffle.  Notice that the inverse of the above lemma does not hold, i.e. non-degenerate simplices of product does not have to be written in the above form (95).

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Counter Example 3.4. Consider the simplicial set ∆n , to be defined later, subsection 4. Straightforward verification shows that (id[1] , id[1] ) ∈ (∆1 × ∆1 )1 , (σ20 , σ21 )) ∈ (∆2 × ∆2 )3 are non-degenerate. However, that cannot be written in the above form (95). Remark 3.5. Understanding the structure of the simplicial set ∆p × ∆q , in terms of finding its generators and relation, is important to understand the notation of simplicial homotopy, when studying the model structure on the category of simplicial sets. Therefore, it would be studied in more details after recalling the standard simplex 4. 4. Classical Examples of Simplicial Sets Hereby we review several examples of simplicial sets. Some of these examples plays an important role in the study of the canoincal model structure on the category of simplicial sets, in 3, namely n-simplex ∆n , skeleton, the boundary ∂∆n , and the r-horn Λnr . Also, we recall some other classical examples, interesting for their own. We might not mention in this section that certain morphism between objects of ∆ are order-preserving, to avoid long description, but it is always assumed to be the case, and it is essential in the following examples: n-Skeleton and the Dimension of a Simplicial set. Definition 4.1. Let X be a simplicial set, and n ≥ 0. For [p] ∈ ∆, and θ ∶ [q] → [p] in ∆, we define (skn X)p = {xp ∈ Xp ∣∃m ≤ n, ∃β ∶ [p] → [m] order-preserving , ∃xm ∈ Xm , such that xp = Xβ (xm )} ⊂ Xp . (skn X)θ (xp ) = Xθ (xp ). (skn X)θ is well defined that for xp ∈ (skn X)p , ∃m ≤ n, ∃β ∶ [p] → [m], ∃xm ∈ Xm , such that xp = Xβ (xm ), hence (skn X)θ (xp ) = Xθ (xp ) = Xθ (Xβ (xm )) = Xβθ (xm ) i.e. ∃m ≤ n, ∃βθ ∶ [q] → [m] order-preserving, ∃xm ∈ Xm , such that (skn X)θ (xp ) = Xβθ (xm ), hence (skn X)θ (xp ) ∈ (skn X)q .

185

Since X is simplicial set, then skn X, define on objects and morphisms above, is so. It is called the n-skeleton of X. Moreover, one can see readily that the above inclusions induce an inclusion of simplicial sets skn X ⊂ X. Notice that ∀p ≤ n, ∀xp ∈ Xp , choosing m = p, and β = id[p] , we see that xp ∈ (skn X)p . Hence, (skn X)p = Xp . Whereas, for p > n, ∀xp ∈ (skn X)p , by lemma 2.1, we see that xp is a degenerate simplex of X. Hence, the simplices of skn X are the p-simplices of X, with p ≤ n, and their degenerations. Definition 4.2 (Dimension of Simplicial Sets). Let X be a simplicial set, n ≥ 0 an integer such that X = skn X ≠ skn−1 X, then we say that X has the dimension n. When a simplicial set has dimension n, then all p-simplices are degenerate for p > n, and there exist a nondegenerate n-simplex. Notice that for m < n, we have the inclusion of simplicial sets skm X ⊂ skn X, and that n

X = colim sk− X = ⋃ skn X ∆

n=0

.

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We can define sk−1 X by extending the definition formally for n = −1, then sk−1 X = ∅. Lemma 4.3. Let X be a simplicial set of finite dimension, then the dimension of X coincide with the topological dimension of its geometric realisation ∣ X ∣. The Standard n-Simplex ∆n .

Definition 4.4. Let n ≥ 0, we define the simplicial set ∆n to be the represented functor by [n], i.e. ∆n = ∆(−, [n])

We will adopt the short notation ∆np ∶= ∆n ([p]), and ∆nθ ∶= ∆n (θ) for [p] ∈ ∆, and θ ∶ [q] → [p] in ∆. Then, we find that:

D

∆np = {α ∶ [p] → [n]∣α is order-presrving} ≅ {(t0 , t1 , ..., tp ))∣n ≥ tp ≥ ... ≥ t1 ≥ t0 ≥ 0}. ∆nθ (α) = αθ ∈ ∆nq for α ∈ ∆np .

The above mention bijection shows that #∆np = (

n+p+1 ). p+1

one can easily see that id[n] ∈ ∆nn is the generator of ∆n , with an empty set of minimal simplicial relations. Then, by lemma 2.10 we see that morphisms of simplicial sets f ∶ ∆n → X, are determined by a choice of xn ∈ Xn to be xn = f (id[n] ) Lemma 4.5. Let X be a simplicial set, then there is a canonical natural isomorphism ≅

ι ∶ X− Ð→ sSet(∆− , X) Proof. 1. This is a direct result of the Yoneda’s lemma ??.



Proof. 2. ∀n ≥ 0 ∆n is generated by id[n] , and the set of empty simplicial relations, then by lemma 2.10, morphisms of simplicial sets ∆n → X are in one-to-one correspondence with n-simplices of X, i.e.there is a bijection ≅

ιn ∶ Xn Ð→ sSet(∆n , X) such that (ιn (xn ))n (id[n] ) = xn , ∀xn ∈ Xn . Notice that ∀µ ∶ [m] → [n] in ∆ the following diagram commutes Xn

ιn

(∆µ )∗

Xµ

 Xm

/ sSet(∆n , X) .

ιm

 / sSet(∆m , X)

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That lemma 2.10, implies that morphisms of simplicial sets ∆m → X coincide iff they coincide on the generator id[m] of ∆m , and ∀xn ∈ Xn , we have: ((∆µ )∗ (ιn (xn )))m (id[m] )

= (ιn (xn )∆µ )m (id[m] ) = (ιn (xn ))m (∆µm (id[m] )) = (ιn (xn ))m (µ) = (ιn (xn ))m (∆nµ (id[n] )) = Xµ ((ιn (xn ))n (id[n] )) = Xµ (xn ) = (ιm (Xµ (xn )))m (id[m] ).

The last equality is due to the fact that Xµ (xn ) ∈ Xm . Then, (∆µ )∗ (ιn (xn )) = ιm (Xµ (xn )), ∀xn ∈ Xn . Hence, (∆µ )∗ ιn = ιm Xµ , and the above diagram commutes. Then, the family ιn ’s gives rise to the desired natural transformation.  Based on the above lemma, we can abuse notation and call morphisms of simplicial sets ∆n → X the n-simplices of X, referring to the above bijection ιn . ∀n ≥ 0, we adopted the convention of using the grade of a simplex of simplicial set in the name of that simplex, like xn ∈ Xn , hence the notation ιn (xn ) becomes redundant. Therefore, we shorten notation, and adopt the convention of writing ιx ∶= ιn (x), ∀x ∈ Xn , n≥ 0, whenever the grade of x is known, either appearing in the name like in xn , or can be calculated like Xµ (xn ), for µ ∶ [m] → [n] in ∆. We refer to the pth -component of ιxn by ιxn ,p . So, in particular, we have ιxn ,n (id[n] ) = xn , ∀xn ∈ Xn .

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id[n] ∈ ∆nn is non-degenerate, that if we assume for the sake of contradiction that it is degenerate, then there exists n′ < n, and order-preserving maps µ ∶ [n] → [n′ ], and α ∈ ∆nn′ such that id[n] = ∆nµ (α) = αµ. µ is not an injection because n′ < n, and that contradicts with the previous equality. Hence, id[n] is non-degenerate. For every integer p > n, every p-simplex is degenerate. ∀α ∈ ∆np , α = id[n] α = ∆nα (id[n] ). Since p > n, then α is degenerate, by lemma 2.1. Therefore, ∆n (n) ≠ ∅, and ∆n (p) = ∅ for p > n. Hence dim ∆n = n.

The n-Simplex is very important simplicial set,we will see later that ∆1 plays in simplicial homotopy theory the role of the unite interval in the topological homotopy theory. We recall some of the facts about it in the following lemmas:

D

Lemma 4.6. The geometric realisation of the standard n-simplex ∆n is the topological standard n-simplex

∆ntop .

Proof. Consider the maps: f∶

∆ntop t

Ð→ ∣ ∆n ∣ ↦ [(t, id[n] )]

g∶

∣ ∆n ∣ [(s, α)]

Ð→ ∆ntop ↦ ∆α top (s)

One can see readily that f is a continuous map. g is well-defined that ∀[(s, α)], [(s′ , α′ )] ∈∣ ∆n ∣, such that [(s, α)] = [(s′ , α′ )], then, without loose of generality and up-to renaming, ∃[p], [p′ ] ∈ ∆, and θ ∶ [p′ ] → [p], such that α ∈ ∆np , α′ ∈ ∆np′ , and s = ∆θtop (s′ ), and α′ = ∆nθ (α) = αθ Hence, ′ αθ ′ α θ ′ α g([(s′ , α′ )]) = ∆α top (s ) = ∆top (s ) = ∆top (∆top (s )) = ∆top (s) = g([(s, α)]) ′

Also, g is continuous, that projections and ∆α top ’s are continuous. In order to prove the lemma we need to see that g is the inverse of f : ∀t ∈ ∆, id

(gf )(t) = g([(t, id[n] )]) = ∆top[n] (t) = id∆ntop (t), hence gcircf = id∆ntop On the other hand ∀[(s, α)] ∈∣ ∆n ∣, ∃[p] ∈ ∆ such that α ∈ ∆np and s ∈ ∆ntop . Then, (f g)([(s, α)]) = f (δα(s) ) = [(δα(s) , id[n] )]. Since α = ∆nα (id[n] ) , then (f g)([(s, α)]) = [(δα(s) , id[n] )] = [(s, α)] = id∣∆n ∣ ([(s, α)]), hence f g = id∣∆n ∣ , and ∆ntop is homeomorphic to ∣ ∆n ∣.



187

The proof of above lemma shows that the id[n] ∈ ∆nn correspond to the the topological standard n-simplex ∆ntop , whereas all other simplices of ∆n are either faces of id[n] , or degenerations of id[n] and its faces. n The category of Standard Simplexes: For any µ ∶ [m] → [n] in ∆, [p] ∈ ∆ consider the map ∆µp ∶ ∆m p → ∆p µ m given by ∆p (β) = µβ for β ∈ ∆p . Then, once can see easily that the following diagram commutes for any θ ∶ [q] → [p]: ∆µ p

∆m p

/ ∆n p

∆m θ

∆n θ

 ∆m q

 / ∆n q

∆µ p

Hence, the set of maps ∆µp defines a morphism of simplicial sets ∆µ ∶ ∆m → ∆n . One can readily see that the ′ this defines a functor ∆− ∶ ∆ → sSet. ∆− is faithful full that: For µ, µ′ ∶ [m] → [n] such that ∆µ = ∆µ , then in ′ particular ∆µm = ∆µm , and ′ µ = ∆µm (id[m] = ∆µm (id[m] ) = µ′ .

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On other hand, morphisms of simplicial sets ∆m → ∆n are in one to one correspondence to m-simplices of ∆n , to be the image of id[m] , i.e. they are in one to one correspondence to morphisms µ ∶ [m] → [n] in ∆. Since, each such µ induces ∆µ , then each morphism of simplicial sets ∆m → ∆n is of the form ∆µ , for a µ in ∆. I.e. ∆− is a full embedding of categories. We tend to abuse notation and denote by ∆ both the simplicial category and the full subcategory the image of ∆− . Definition 4.7 (The Simplex category of X, (∆− ↓ X)). Let X be a simplicial set, we define the simplex category of X to be the comma category (∆− ↓ X), with objects being the morphisms of simplicial sets of the form ∆n → X, i.e. ιxn for all xn ∈ Xn , n ≥ 0, and morphisms being commutative diagrams:

ιxm

/ ∆n

f

∆m

!

D

X

ιxn

}

D

,induced by a unique morphism µ ∶ [m] → [n] in ∆, as we have see above. Then, we define the evident forgetful functor UX ∶ (∆− ↓ X) → sSet, given by UX (ιxn ) UX ( D )

= ∆n = f.

. The below lemma shows the significance of ∆n , and indicates that all simplicial sets are just a colimit of diagrams of ∆n ’s. Lemma 4.8. Let X be a simplicial set, then X = colim UX . (∆− ↓X)

ˆ ∶ (∆− ↓ X) → sSet given by Proof. Consider the constant functor X ˆ x ) = X , and X(D) ˆ X(ι = idX . n Define the family of morphisms of simplicial sets ηX,ιxn = ιxn ∶ ∆n → X. Then, by the definition of (∆− ↓ X) the below diagram commutes for any D ∶ ιxm → ιxn in (∆− ↓ X) ∆m = UX (ιxm )

ηX,ιxm =ιxm

/ X(ι ˆ x )=X m

UX (D)

 ∆n = UX (ιxn )

ηX,ιxn =ιxn

/ X(ι ˆ x )=X n

⋅ ˆ given component-wise above. Hence, this family of morphisms gives rise to a natural transformation ηX ∶ UX → X

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In order to show that the desired colimit is X, we need to how that the pair (X, ηX ) is initial universal among ⋅ such pairs. Let Y be a simplicial set and ηY ∶ UX → Yˆ a natural transformation, where Yˆ ∶ (∆− ↓ X) → sSet is the constant functor given by Yˆ (ιxn ) = Y , and Yˆ (D) = idY . We need to show that (Y, ηY ) factorise through (X, ηX ) uniquely. We can define the family of maps ξn ∶ Xn → Yn given by ξn (xn ) = ηY,ιxn ,n (id[n] ). For any µ ∶ [m] → [n] in ∆, the following diagram commutes: / Yn

ξn

Xn Xµ



 Xm

Yµ

/ Ym

ξm

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That ∀xn ∈ Xn : Yµ (ξn (xn )) = Yµ (ηY,ιxn ,n (id[n] )) = ηY,ιxn ,m (∆nµ (id[n] )) = ηY,ιxn ,m (id[n] µ) = ηY,ιxn ,m (µ) The second equality is due to the fact that ηY,ιxn ’s, the component of the natural transformation ηY , are morphisms of simplicial sets, i.e. natural transformations. On the other hand: (96)

ξm (Xµ (xn )) = ηY,ιXµ (xn ) ,m (id[m] ) = ηY,ιxn ∆µ ,m (id[m] )

Te second equality is due to the fact that ιn is a natural transformation (actually a natural isomorphism), i.e. ιm Xµ = (∆µ )∗ ιn , hence, ιXµ (xn ) = ιxn ∆µ . We have the following commutative diagram in (∆− ↓ X): ∆µ

D

∆m

ιxn ∆µ

!

D

X

/ ∆n

}

ιxn

Then, since ηY is a natural transformation, we have the commutative diagram: ∆m = UX (ιxn ∆µ )

ηY,ιxn ∆µ

/ Yˆ (ιx ∆µ ) = Y n

∆µ =UX (D)

 ∆n = UX (ιxn )

ηY,ιxn

/ Yˆ (ιx ) = Y n

Substituting in (96), we find that: ξm (Xµ (xn )) = ηY,ιxn ,m (∆µ (id[m] )) = ηY,ιxn ,m (µ) i.e. the family ξn ’s gives rise to a morphism of simplicial sets ξ ∶ X → Y , and hence to the constant natural ⋅ ˆ→ transformation ξˆ ∶ X Yˆ , with ξˆιxn = ξ. ˆ X , i.e. we need to show that ηY,ι ,m = ξm ηX,ι ,m , ∀m ≥ 0, ∀xn ∈ Xn , ∀n ≥ 0. We need to show that ηY = ξη xn xn n Then, ∀µ ∈ ∆m , we have: ξm (ηX,ιxn ,m (µ))

= ξm (ιxn ,m (µ)) = ξm (ιxn ,m (∆nµ (id[n] ))) = ξm (Xµ (ιxn ,n (id[n] ))) = ξm (Xµ (xn )) = Yµ (ξn (xn )) = Yµ (ηY,ιxn ,n (id[n] )) = ηY,ιxn ,m (∆nµ (id[n] )) = ηY,ιxn ,m (µ)

The above equalities follows from the commutativity of the diagram:

189

∆nn ∆n µ

 ∆nm

ηX,ιxn ,n

Xn



ξn

ηX,ιxn ,m

ηY,ιxn ,n

ηY,ιxn ,m

Xµ

 / Yn Yµ

  Xm

  / Ym

ξm

ˆ X. Hence, ηY = ξη ⋅ ˆ let ξˆ′ ∶ X ˆ → ˆ and To show the uniqueness of ξ, Yˆ be a natural transformation such that ηY = ξˆ′ ηX . Since X, ′ ′ ˆ ˆ ˆ Y are constant functors, then ξ is a constant natural transformation. Let ξ ∶= ξιxn , ∀xn ∈ Xn , ∀n ≥ 0. Then ′ ξm ηX,ιxn ,m = ηY,ιxn ,m , ∀m ≥ 0, ∀xn ∈ Xn , ∀n ≥ 0 Then, ∀n ≥ 0, ∀xn ∈ Xn , we have

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ξn′ (xn ) = ξn′ (ιxn ,n (id[n] ))ξn′ (ηX,ιxn ,n (id[n] )) = ηY,ιxn ,n (id[n] ) = ξn (xn ) Hence ξ = ξ ′ , and ξˆ = ξˆ′ . Therefore, X = colim UX . (∆− ↓X)



In the rest of this subsection, we recall some of the concepts related to the standard simplexes that we use in the study of the model structure on the category of simplicial sets, later on. Vertices:

Definition 4.9. Let X, and Y be simplicial sets, y0 ∈ Y0 a vertex of Y . We define the constant morphism on X to the vertex y0 to be the composition: X

/ ∗ = ∆0

ιy0

/Y .

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Notice that for a none-empty simplicial set X, and x0 ∈ X0 , ιx0 ,n (σn ) = Xσn (x0 ). Hence, the nth -pure degeneracy of x0 correspond to the unique constant morphism on ∆n to x0 : ∆n

∆σ n

/ ∆0

ιx0

/X

Definition 4.10 (Fibres of morphism of simplicial sets). Let f ∶ X → Y be a morphism of simplicial sets, y0 ∈ Y0 a vertex of Y , we define the fibre of f at y0 to be the unique pullback simplicial set Xy0 , up-to isomorphism, in the below Cartesian square:  /X Xy0    ∆0

ιy0

 /Y

f

Remark 4.11. The fibre of f ∶ X → Y at y0 ∈ Y0 is the universal (terminal) constant morphism to y0 that factors through f . Lemma 4.12. Let f ∶ X → Y be a morphism of simplicial sets, y0 ∈ Y0 a vertex of Y , then the canonical projection Xy0 → X is an injection. Moreover, Xy0 ,n = {xn ∈ Xn ∣fn (xn ) = ιyn ,n (σn )}. Proof. Since ∗ = ∆0 has one element at each grade, hence ιy0 ∶ ∗ → Y is an injection, hence a monomorphism. In arbitrary category monomorphisms are closed under pullback, hence the projection Xy0 → X is a monomorphism, hence an injection.  The Simplicial Set ∆p × ∆q : Lemma 3.3, show the condition for certain simplices of the product simplicial set to be, that satisfy certain criteria, to be non-degenerate. However, the simplicial set ∆p × ∆q is important in studying the model structure, and one would like to understand it in details, in the sense of understanding its generators and relations. That will be provided in the following lemma:

190

Lemma 4.13. The simplicial set ∆p ×∆q is generated by (sν (id[p] ), sµ (id[q] )), for all (µ, ν) being (p, q)-shuffle. Proof. Lemma ?? shows that (sν (id[p] ), sµ (id[q] )) is non-degenerate for(µ, ν) a (p, q)-shuffle. Let (αn , βn ) ∈ (∆p ×∆q )n a non-degenerate simplex, then by lemma ??, n ≤ p+q. Hence, ∆p ×∆q is of dimension p + q. By lemma 2.1, we have jt jt−1 j1 lu−1 i2 is k2 kr l1 lu αn = ∂pi1 ∂p−1 ...∂p−s+1 σn−t σn−t+1 ...σn−1 βn = ∂qk1 ∂q−1 ...∂q−r+1 σn−u σn−u+1 ...σn−1 where p − n = s − t, q − n = r − u, p ≥ i1 > i2 > ... > is ≥ 0, n − 1 ≥ j1 > j2 > ... > jt ≥ 0, q ≥ k1 > k2 > ... > kr ≥ 0, and n − 1 ≥ l1 > l2 > ... > lt ≥ 0.

TBC.  n

n

n

Boundary ∂∆ . Let n ≥ 0, we define the boundary of the n-simplex ∆ to be the (n − 1)-skeleton of ∆ , and we denote it ∂∆n , i.e. ∂∆n = skn−1 ∆n Then, for every p ≥ 0 we have:

∂∆nθ (α)

= {α ∶ [p] → [n]∣∃m ≤ n − 1, ∃β ∶ [p] → [m], ∃µ ∶ [m] → [n], such that α = µβ} = {α ∶ [p] → [n]∣ im α ≠ [n]} = {α ∶ [p] → [n]∣α is not surjective}. = αθ.

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∂∆np

one can easily see that ∂∆n is the simplicial subset of ∆n , generated by ∂ni ∈ ∆nn−1 for 0 ≤ i ≤ n with is following set of minimal relations: j−1 i ∂nj ∂n−1 = ∂ni ∂n−1 , i.e. n

In fact, ∂∆ is the union

⋃i∈[n] din (∆n−1 )

i dn−1 (∂nj ) = dn−1 i j−1 (∂n )

for i < j

of all the images, [Mor03, p.12]

D

Remark 4.14. Avoid the confusion of thinking about the following relations as a simplicial relations between the generators ∂ni ’s. ⎧ ∂ i σ j−1 for i < j ⎪ ⎪ ⎪ n−1 n−2 j i for i = j or i = j + 1 σn−1 ∂n = ⎨ id[n−1] ⎪ ⎪ i−1 j ⎪ ∂ σ ⎩ n−1 n−2 for i > j + 1 i i They are rather relations between ∂n−1 ’s, σn−1 ’s, and id[n−1] that their induced simplicial relations are: j dni (σn−1 )

i ⎧ sn−2 (∂n−1 ) ⎪ ⎪ ⎪ j−1 n = ⎨ ∆id[n−1] (id[n−1] ) ⎪ ⎪ n−2 i−1 ⎪ ⎩ sj (∂n−1 )

for i < j for i = j or i = j + 1 for i > j + 1

The generators ∂∆n are all (n − 1)-simplices, hence dim ∂∆n = n − 1. Recall that the boundary of the topological standard n-simplex ∂∆ntop is given by: ∂∆ntop = {t ∈ ∆ntop ∣∃0 ≤ i ≤ n, such that ti = 0}. Lemma 4.15. The geometric realisation of ∂∆n is the boundary of the topological standard n-simplex ∂∆ntop . Proof. Similar to the proof of lemma 4.6 using the well defined maps: f∶

∂∆ntop t

Ð→ ∣ ∂∆n ∣ ↦ [(t, ∂ni )]

g∶ i

where t = o, for some 0 ≤ i ≤ n

∣ ∂∆n ∣ [(s, α)]

Ð→ ∂∆ntop ↦ ∆α top (s) 

1

n

∂10

∆ plays the role of the unit interval, we distinguish from its boundary ∂∆ , two 0-simplices, corresponding to deleting the vertex v0 , so we call it (1), and ∂11 corresponding to deleting the vertex v1 , so we call it (0). When we write ∂∆n ↪ ∆n , we refer to the canonical inclusion, unless mentioned otherwise. One can see readily that ∀n ≥ 0, ∀µ ∶ [n] → [n], we have a morphism of simplicial sets iµ ∶ ∂∆n → ∆n that sends ∂ni to µ∂ni for 0 ≤ i ≤ n. Lemma 4.16. ∀n ≥ 0, let f ∶ ∂∆n → ∆n be a morphism of simplial sets. Then, there exists µ ∶ [n] → [n] such that f = iµ .

191

Proof. type.



Let X be a simplicial set, xn ∈ Xn , we define the boundary of xn to be the set ∂(xn ) ∶= (ιxn ,n (∂n0 ), ιxn ,n (∂n1 ), ..., ιxn ,n (∂nn )). r-horn Λnr . Let n ≥ 0, 0 ≤ r ≤ n, we define Λnr to be the simplicial subset of ∆n generated by ∂ni ∈ ∆nn−1 for 0 ≤ i ≠ r ≤ n with is following set of minimal relations: i dn−1 (∂nj ) = dn−1 i j−1 (∂n )

(97)

Lemma 4.17. Let n ≥ 0, 0 ≤ r ≤ n, then:

Λnr,p

for i < j, and i, j ≠ r

= {α ∶ [p] → [n]∣α is order-preserving, [n] ≠ im α ≠ [n] ∖ {r}} 

Proof. Lemma 4.18. Let n ≥ 0, 0 ≤ r ≤ n, the geometric realisation of the face r, opposite to the vertex r, deleted.

Λnr

is

Λnr,Top ,

i.e. the boundary of

∆ntop ,

with

When we write Λnr ↪ ∆n , we refer to the canonical inclusion, unless mentioned otherwise. Let X be a simplicial set, xn ∈ Xn , we define the rth horn of xn , or the horn concentrated at the vertex r to be n r the set Λr (xn ) ∶= (ιxn ,n (∂n0 ), ιxn ,n (∂n1 ), ..., ιxn̂ ,n (∂n ), ..., ιxn ,n (∂n )). 5. Function Complexes and Cartesian Product

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In this section we see that the category of simplicial sets is a Cartesian closed category, with an internal Hom, the so called ’function complex’. Definition 5.1. Let X and Y be simplicial sets. The function complex between X and Y , denoted by Hom(X, Y ), is defined to be the simplicial set: Hom(X, Y ) ∶= sSet(X × ∆− , Y ) i.e. it is given on object [n] ∈ ∆, and morphism µ ∶ [m] → [n] in ∆, by: Hom(X, Y )n Hom(X, Y )µ

= sSet(X × ∆n , Y ) = (idX × ∆µ )∗ ∶ sSet(X × ∆n , Y ) → sSet(X × ∆m , Y )

where (idX × µ)∗ is the precomposition with idX × µ.

D

One can see easily that we have the bifunctor:

Hom(−, −) ∶ sSetop × sSet Ð→ sSet

Consider the collection of maps of sets: evn ∶ Xn × Hom(X, Y )n → Yn , ∀n ≥ 0, given by: evn (xn , fn ) = fn,n (xn , id[n] ) Lemma 5.2. The above collection of maps of sets evn ’s gives rise to a morphism of simplicial sets: ev ∶ X × Hom(X, Y ) → Y given object-wise above. Proof. Notice that ∀µ ∶ [m] → [n] in ∆, the below diagram commutes,∀fn ∈ sSet(X × ∆n , Y ): Xn × ∆nn

fn,n

/ Yn

Xµ ×∆n µ

 Xm × ∆nm



fn,m

Yµ

/ Ym

Hence, the below diagram commutes: Xn × sSet(X × ∆n , Y )

evn

Xµ ×(idX ×∆µ )∗

 Xm × sSet(X × ∆m , Y )

/ Yn 

evm

Yµ

/ Ym

that, on the one hand: Yµ (evn (xn , fn ))

= Yµ (fn,n (xn , id[n] )) = fn,m ((Xµ × ∆nµ )(xn , id[n] )) = fn,m (Xµ (xn ), µ).

192

On the other hand: evm ((Xµ × (idX × ∆µ )∗ ) (xn , fn ))

= evm (Xµ (xn ), fn ○ (idX × ∆µ )) = fn,m ((idXm × ∆µm )(Xµ (xn ), id[m] )) = fn,m (Xµ (xn ), µ).

Thus, we have the desired morphism of simplicial sets ev ∶ X × Hom(X, Y ) → Y .



Proposition 5.3 (Exponential Law). There is a natural isomorphism, given component-wise ∼

ev∗,X,Y,Z ∶ sSet(Z, Hom(X, Y )) Ð→ sSet(X × Z, Y ) defined ,for g ∶ Z → Hom(X, Y ) by ev∗,X,Y,Z (g) = ev ○ (idX × g), natural in X, Y, and Z. I.e. the category of simplicial sets is a Cartesian closed category with interial hom being hom. Hence, for every simplicial set X, Hom(X, −) ∶ sSet → sSet is a right adjoint to the functor X × − ∶ sSet → sSet. Proof. ∀X, Y, and Z simplicial sets, let consider the collection of maps: veX,Y,Z ∶ sSet(X × Z, Y ) Ð→ sSet(Z, Hom(X, Y ))

T

defined for f ∶ X × Z → Y by veX,Y,Z (f )n (zn ) = f ○ (idX × ιzn ) for zn ∈ Zn .

DR AF

On the one hand, we notice that ev∗,X,Y,Z veX,Y,Z = idsSet(X×Z,Y ) . That, ∀f ∶ X × Z, Y, xn ∈ Xn , zn ∈ Zn , we have: ev∗,X,Y,Z (veX,Y,Z (f ))(xn , zn ) = ev ((idX × (veX,Y,Z (f )))(xn , zn )) = ev(xn , f ○ (idX × ιzn )) = f ((idX × ιzn )(xn , id[n] )) = f (xn , zn ) hence, ev∗,X,Y,Z (veX,Y,Z (f )) = f , and ev∗,X,Y,Z veX,Y,Z = idsSet(X×Z,Y ) . On the other hand, veX,Y,Z ev∗,X,Y,Z = idsSet(Z,Hom(X,Y )) . That, for g ∶ Z → Hom(X, Y ), zn ∈ Zn , xn ∈ Xn ,, we have: (veX,Y,Z (ev∗,X,Y,Z (g))(zn ))(xn , id[n] ) = (veX,Y,Z (ev(idX × g))(zn ))(xn , id[n] ) = ev(idX × g)(idX × ιzn )(xn , id[n] ) = ev(xn , g(zn )) = g(zn )(xn , id[n] ). hence, (veX,Y,Z (ev∗,X,Y,Z (g))(zn )) = g(zn ), and veX,Y,Z ev∗,X,Y,Z = idsSet(Z,Hom(X,Y )) .



Definition 5.4. Let p ∶ X → Y , and i ∶ K → L be morphisms of simplicial sets. Then, the below diagram commutes: p∗ / Hom(L, Y ) Hom(L, X) i∗

 Hom(K, X)

i∗

p∗

 / Hom(K, Y )

where i∗ is the precomposition with i × id∆− - we use this short notation when dealing with function complexes if no confusion is caused, whereas it should have been written as (i × id∆− )∗ , and p∗ is the composition with p. Thus, there is a canonical morphism, induced by the universal property of pullback: (i∗ , p∗ ) ∶ Hom(L, X) → Hom(K, X) ×Hom(K,Y ) Hom(L, Y ). Since Hom(K, X) ×Hom(K,Y ) Hom(L, Y ) is product of i∗ and p∗ in the category (sSet ↓ Hom(K, Y )), we use the short notation : i∗ ∏ p∗ ∶= Hom(K, X) ×Hom(K,Y ) Hom(L, Y ). Also, for i ∶ K → L, and j ∶ U → V morphisms of simplicial sets, the below diagram commutes: U ×K

idU ×i

j×idK

 V ×K

/ U ×L j×idL

idV ×i

 / V ×L

193

Thus, there is a canonical morphism, induced by the universal property of pushforward: j ◻ i ∶ (V × K) ∐ (U × L) → V × L. U ×K

Since (V × K) ∐U ×K (U × L) is the coproduct of j × idK and idU × i in the category (U × K ↓ sSet), we use the short notation : j ∐ i ∶= (V × K) ∐U ×K (U × L). Lemma 5.5. Let j ∶ U → V, i ∶ K → L, and p ∶ X → Y be morphisms of simplicial sets, then there is a one-to-one correspondence between the below commutative squares of the forms: U

/ Hom(L, X)

and

(i∗ ,p∗ )

j

j◻i

 V ×L

 / i∗ ∏ p∗

 V

j ∐i

/X  /Y

p

Moreover, if any of these commutative squares admit a lift, the the corresponding one of the other form does so. Proof. Joe 2.4.16 Alternatively apply lemma ?? on an appropriate cube.

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 6. Abstract Realisations

When we examine geometric realisation ??, we notice that the realisation makes sense in a more general settings. To see that, we notice first that the category of topological spaces Top are complete and cocomplete category, so are Topop and Top × Topop . We have the forgetful functor U ∶ Top → Set

and its left adjoint

F ∶ Set → Top

D

that sends each set S to the discrete topological space SDis . Then, notice that in ∣ X ∣, (u, y) ∈ ∆m top × X(m),Dis and (t, x) ∈ ∆ntop × X(n),Dis are identified when there exists a morphism ν ∶ [m] → [n] in ∆, such that y = X(ν)(x), and t = ∆νtop (u)

, when we consider X op ∶ ∆ → Topop , the above relations can be simplified to: (t, x) = ((∆, F op (X op ))(µ)) (u, y). Hence, we can see that the geometric realisation of the simplicial set X is given by ∣ X ∣= colim ∆ × (F op X op ) ∆

Actually, we call it the discrete geometric realisation, that it depends on F . Let D be a complete and cocomplete category, F ∶ C → D a functor, X a simplicial object in C, and R a cosimplicial object of D, then we can define the F -realisation of X with realisation ”blocks” R to be: ∣ X ∣R = colim R × (F op X op ) ∆

Notice that ∣ X ∣ is isomorphic to the coequaliser of the canonical morphisms l n ∐ R × F (Xm ) ⇉ ∐ R × F (Xn ) l,m∈N

n∈N

A case of particular interest and application, is when C = Set. Then, since D is complete and cocomplete, one can always define a functor F ∶ Set → D given on object by F (S) = ∐ ∗D S

194

6.1. Functional Simplicial Sets. Let D be a complete and cocomplete category, and R a cosimplicial object of D. Define the bifunctor SR ∶ Dop × D (X, Y )

→ sSet ↦ D(X × R● , Y )

For D = sSet, and R● = ∆● , SR (X, Y ) = MapsSet (X, Y ), in particular sSet(X, Y ) = SR (X, Y )0 . Notice that for C = Set, and F ∶ Set → D given by F (S) = ∐S ∗D , there is a adjunction X× ∣ − ∣R ∶ sSet ⇄ D ∶ SR (X, −) for every X ∈ D. In particular, for X = ∗, one has an adjunction ∣ − ∣R ∶ sSet ⇄ D ∶ SR (∗, −) = D(R● , −). The geometric realisation and singular functors are particular case of the above adjunction, for D = Top and R ● = ∆● . 7. Examples of Simplicial Objects

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Let X be a simplicial set, C ∶ Set → Ab a functor. Then, CX is a simplicial abelian group. The following examples is of a particular interest that it would be used to define homology theory of simplicial sets. Let X be a simplicial set, A an abelian group, consider the functor CA ∶ Set → Ab, that sends each set to its free generated group, with coefficients in A. Then CA X is a simplicial abelian group. The elements of Cn (X, A) ∶= CA X([n]) are finite formal sums of elements of X(n) , with coefficients in A, let c ∈ Cn (X, A), then c = ∑ ax x, ax ∈ A. x∈X(n)

8. Complexes of a Simplicial Sets

Chain Complexes. Let C ∶ ∆op → Ab be a simplicial abelian group. We define dn ∶ C(n) → C(n−1) : n

dn = ∑(−1)i C(∂ni ),

D

i=0

Which turns

dn+1

dn

... → C(n+1) → C(n) → C(n−1) → ...

into a chain complex that, for n ≥ 1, dn−1 dn

n n−1 ⎞ n ⎛n−1 j j ) = ∑ (−1)j C(∂n−1 ) (∑(−1)i C(∂ni )) = ∑ ∑ (−1)i+j C(∂ni ∂n−1 ⎠ i=0 ⎝ j=0 i=0 j=0 n i−1

n−1 n−1

j j = ∑ ∑ (−1)i+j C(∂ni ∂n−1 ) + ∑ ∑ (−1)i+j C(∂ni ∂n−1 ) i=1 j=0 n i−1

i+j

= ∑ ∑ (−1) i=1 j=0 n−1 k

= ∑ ∑ (−1)

k+j+1

k=0 j=0 n−1 n−1

= ∑ ∑ (−1) j=0 k=j

i−1 C(∂nj ∂n−1 )+

i=0 j=i n−1 n−1

j C(∂ni ∂n−1 )

i=0 j=i n−1 n−1

k C(∂nj ∂n−1 )+

k+j+1

i+j

∑ ∑ (−1)

k C(∂nj ∂n−1 )+

∑ ∑ (−1)

i+j

i=0 j=i n−1 n−1

∑ ∑ (−1)

j C(∂ni ∂n−1 )

i+j

j C(∂ni ∂n−1 )=0

i=0 j=i

This chain complex is denoted by C● , and elements of Cn ∶= C(n) are called n-chains, those n-chains c ∈ Cn that maps to zero by the boundary map dn are called n-cycles. Homology groups of the chain complex C● , are defined by Hn (C● ) = ker dn / im dn+1 . Elements of Hn (C● ) are called homology classes, each of which is represented by an n-cycle. n-chain that are in im dn+1 are called a boundaries, two n-chains with difference a boundary are called homological, even though they do not belong to ker dn .

195

9. Pointed Simplicial Sets Working in Top, one has the notion of homotopy groups. Since, the geometric realisation and singular functors allow to move between sSet, and Top. A natural question arises if we can carry the notion of homotopy groups to simplicial sets. Notice, that homotopy groups are defined on pointed topological spaces Top∗ . Hence, in order to curry the notion to simplicial sets, we need to have a corresponding ’pointed simplicial set’ sSet∗ , not to be confused with simplicial pointed sets ∆op (Set∗ ). In this section we recall the relation between (sSet)∗ and Top∗ , whereas homotopy groups will be provided later in ??. Recall that a pointed topological space is a pair (X, x), where X is a topological space, and x ∈ X is a fixed point, and a map of pointed topological spaces is a continuous map that preserves the base point. One, can think of (X, x) also to be a continuous map ιx ∶ ∗ → X such that ιx (∗) = x, and their morphisms to be evident commutative diagrams. It is easy to verify that this correspondence between (X, X) and ιx gives a isomorphism of categories Top∗ ≅ (∗ ↑ Top). Therefore, the desired category of pointed simplicial sets is natural to be given as in the below definition. Definition 9.1 (Pointed Simplicial Sets). The category of simplicial sets to be sSet∗ to be the comma category (∗ ↑ sSet).

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Hence, we can see that a pointed simplicial sets correspond to a paid (X, x0 ), where X is a simplicial set, and x0 is a vertex of X. Also, one sees that morphisms in (∗ ↑ sSet) corresponds to vertices-preserving morphisms of simplicial sets. 9.1. Pointed Geometric Realisation and Singular Functors. Notice that the restriction of ∣ − ∣ to (∗ ↑ sSet) defines a functor (98)

∣ − ∣∗ ∶ sSet∗ → Top∗

induced evidently by ∣ − ∣. ∣ − ∣∗ is called the pointed geometric realisation functor. Similarly, the restriction of the singular functor S to (∗ ↑ Top) defines a functor (99)

S∗ ∶ Top∗ → sSet∗

D

induced evidently by S. S∗ is called the pointed singular functor. Similar to the geometric realisation and singular functors we have the below adjunction. Lemma 9.2. The pointed geometric realisation functor is a left adjoint to the pointed singular functor, i.e. there is a natural isomorphism: φ∗ ∶ Top∗ (∣ − ∣∗ , −) → sSet∗ (−, S∗ (−)) Proof. TBC

 10. Topological Homotopy of Simplicial Sets

The adjunction between the geometrical realisation and the singular functors, provides a useful tool to transform the concepts understood in topology to the settings of simplicial sets, and to other settings as we will see later in chapter 4. 10.1. Topological Path Connected Components. Definition 10.1 (Topological Path Connected Components). The topological path connected components is defined to be the functor (100)

π0Top (−) ∶ sSet → Set

given by π0Top (−) = π0 (−) ○ Ho ○ ∣−∣ where Ho∗ is given in (4).

196

10.2. Topological Homotopy groups. Definition 10.2 (Topological Homotopy groups). Let n ≥ 1. Then, the n-topological homotopy group is defined to be the functor πnTop (−, ∗) ∶ sSet∗ → Grp given by πnTop (−, ∗) = πn (−, ∗) ○ Ho∗ ○ ∣ − ∣∗

(101) where Ho∗ is given in (4).

Also, for n = 0, the 0-topological homotopy of pointed sets is defined similarly. The 0-topological homotopy of pointed sets does not provide additional information to what topological path connected component provide. However, it is useful in formulating statement, where it becomes more consistent. In particular, observe that for a morphism f ∶ X → Y of simplicial sets, the induced map f∗ ∶ π0Top (X) → π0Top (Y ) is a bijection iff for every x0 ∈ X0 the induced map of pointed sets f∗ ∶ π0Top (X, x0 ) → π0Top (Y, f0 (x0 ))

RA FT

is an isomorphism. This is due to the fact that the underlying sets of π0Top (X, x0 ), π0Top (Y, f0 (x0 )) coincide with the set π0Top (X), π0Top (Y ), respectively, for every x0 ∈ X0 . Then, it becomes natural for a morphism of pointed simplicial sets to be called weak equivalences iff it induces isomorphisms on the level of its topological homotopy groups, for all n ≥ 1 and bijections on the level of its topological path connected components, i.e. if it realisation is a topological weak equivalence, as seen later on. 10.3. Base-free Topological Homotopy Sets.

(102) given on object by (103)

D

Definition 10.3 (Base-free Topological Homotopy Sets). Let n ≥ 0. Then, the n-base-free topological homotopy sets is defined to be the functor πnTop (−) ∶ sSet → Set

πnTop (X) = ⊔ πnTop (X, x0 ) x0 ∈X0

Top

as a coproduct in Set where πn (X, x0 ) are understood to be the underlying sets, i.e. after applying the forgetful functor, hence it is just a disjoint union; whereas on morphisms is given by the universal property of coproduct. The induced map πnTop (X) → πnTop (Y ) from the morphism f ∶ X → Y is denoted by f∗ . Notice in particular that πn (∅) is defined and it is ∅. There are a canonical surjective maps of sets pX ∶ πnTop (X) → X0 that sends each class of homotopy n-loops at a vertex x0 to x0 ∈ X0 , for each n ≥ 0. Then, for any morphism of simplicial sets f ∶ X → Y , we have the commutative square of sets (104)

πnTop (X)

f∗ ↺

pX

 X0

f0

/ π Top (Y ) n pY

 / Y0

The commutativity of the above squares implied the existence of natural transformation p, given object-wise above (105)

⋅

p− ∶ πnTop → (−)0 ∶ sSet → Set

for each n ≥ 0. In order to examine the relation between the base-free topological homotopy sets and topological homotopy groups, we need to recall the following set-theoretical facts.

197

Given a map of sets i ∶ U → X, then i−1 (x) the pre-image of x ∈ X along i fits in the below pull-back square  i−1 (x) ⌜

/U

  ∗

 / X.

i

ιx

If no confusion is feared it would be convenient to denote i−1 (x) by Ux . Also, having a commutative diagram U

g

i

 X

f

/V p

 /Y

Then, ∀x ∈ X, γ ∈ Ux , we have pg(γ) = f i(γ) = f (x), i.e.g(γ) ∈ Vf (x) , and g can be restricted to a map

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gx ∶ Ux → Vf (x) . Notice that gx the the unique morphism, defined by the universal property of pull-backs, that makes the following diagram commutative / Vf (x) p

gx

>

>

Ux  o

g

U

i

 ∗p

D

ιx

!

/V p

 ∗ q

ιf (x)

!  X

f

"  /Y

Based on the above notation, we recall the lemma with encodes the link between these different homotopy ’groups’ Lemma 10.4. Let U i

 X

g

D

f

/V  /Y

p

be a commutative square of maps of sets. Then, the following statements are equivalents ● D is a cartesian square. ● ∀x ∈ X, the induces map gx is bijection. Proof. Assume that D is a cartesian square, i.e. U ≅ {(x, v)∣x ∈ X, v ∈ V, p(v) = f (x)}. Then, ∀x ∈ X, Ux ≅ {(x, v)∣v ∈ V, p(v) = f (x)} ≅ {(x, v)∣v ∈ V, p(v) = f (x)} ≅ Vf (x) . using the above identifications (bijections), we see that gx sends (x, v) to v, hence it also a bijection. On the other hand, assume that ∀x ∈ X, gx is bijection. By the definition of Ux we have U ≅ ⊔ Ux x∈X

198

hence in order to show that U ≅ X ×Y V , it is enough to show that Ux ≅ (X ×Y V )x for every x ∈ X. let x ∈ X, and consider the commutative diagram U g h i

# X ×Y V

/) V

pr2

pr1

  X

 /Y

f

p

where h is the unique morphism induced by the universal property of X ×Y V . ∀x ∈ X consider the induced diagram Ux gx

$ (X ×Y V )x

(pr2 )x

/+ Vf (x)

T

hx

DR AF

by the first part of the proof, we have (pr2 )x is a bijection, and by the assumption, we have gx is a bijection, hence hx is a bijection, and Ux ≅ (X ×Y V )x . Therefore, U ≅ X ×Y V and D is a cartesian square.  The above lemma explain to some extent the name of fibre product, that fiber product of sets induces bijections on its fibres. Corollary 10.5. Let f ∶ X → Y be a morphism of simplicial sets n ≥ 0. Then, the following statements are equivalent (1) The induced maps f∗ ∶ πnT op (X, x0 ) → πnT op (Y, f0 (x0 )) is a bijection of sets for every x0 ∈ X0 . (2) The induced square, constructed in (104), is a cartesian square. πnTop (X) pX

 X0

/ π Top (Y ) n

f∗

↺

pY

 / Y0

f0

Proof. It is a direct result of the previous lemma 10.4 for the choice ′

′

′

U ′ ∶= πnTop (X) X ′ ∶= X0 ′ ′ g ∶= f∗

′

V ′ ∶= πnTop (Y ) Y ′ ∶= Y0 ′ ′ f ∶= f0

and ′ x′ ∶= x0 , where ′ ′ are used to distinguish the corresponding symbols of lemma 10.4. That, Ux = πnT op (X, x0 ), Vf (x) = πnT op (Y, f0 (x0 )), and gx = f∗ .  11. Simplicial Homotopy Recall that ∣ ∆1 ∣≅ [0, 1], the unit interval. Then, a natural question arise weather ∆1 can be used, in a similar fashion of how [0, 1] is used to develop homotopy theory, in order to develop the theory of homotopy of simplicial sets. The answer is positive, and recalled below: Let f ∶ L → X be a morphism of simplicial set. Since, we have the canonical isomorphism L × ∆0 ≅ L, we abuse notation, and we denote by f the canonically induced morphism of simplicial sets L × ∆0 → X. Definition 11.1 (Simplicial Homotopy). Let f, f ′ ∶ L → X be morphisms of simplicial sets, a simplicial homotopy from f to f ′ is defined to be morphism of simplicial sets: H ∶ L × ∆1 → X

199

such that the following diagram commute: i0

L ≅ L × ∆0

/ L × ∆1 o

i1

L × ∆0 ≅ L

H

 +Xs

f

f′

i0 = idL × ∆∂1 , and i1 = idL × ∆∂1 . If there is a simplicial homotopy from f to f ′ , we say that f is simplicial ≅ homotopic to f ′ , a we write H ∶ f → f ′ . 1

0

∂11 is often denoted by 0, and ∂10 by 1, that the first is realised by the face that consists only of the vertex v0 , whereas the later is realised by v1 . We also use the same notation for their pure degeneracy. Hence, the above condition can be written is a more familiar notation: fn = Hn (−, 0) , and fn′ = Hn (−, 1) Since morphisms of simplicial sets ∆n → X are identified with n-simplices of X, then for xn , x′n ∈ X0 we say that xn is homotopic to x′n if ιxn is homotopic to ιx′N , and we write x′n ∼ xn .

RA FT

Notice that f, and f ′ , above, are 0-simplices of Hom(L, X), and the simplicial homotopy H is a 1-simplices of Hom(L, X). Then, considering the face maps: d10 , d11 ∶ Hom(L, X)1 → Hom(L, X)0

we notice that d1i = (idL × ∆∂1 )∗ , i ∈ {0, 1}. Hence, the above conditions could be read as: i

d11 (H) = f , and d10 (H) = f ′ .

Thus, a 1- simplex H of Hom(X, Y ) is a simplicial homotopy from d11 (H) to d10 (H). Definition 11.2 (Constant Homotopy). We call a simplicial homotopy H ∶ L×∆1 → X a constant homotopy at f ∶ L → X if it factories as H = f pr1 : pr1

f

L × ∆1 → L → X

D

for some f ∶ L → X

Such f is unique if it exists for a given simplicial homotopy H, that the above factorisation implies that f is given by the formula: fn (l) = Hn (l, α), ∀l ∈ Ln , α ∈ ∆1n . Lemma 11.3. Let H ∈ Hom(L, X)1 be a constant homotopy at f ∶ L → X, then it is a simplicial homotopy from f to f . 0

Proof. Notice that the projection pr1 coincides with idL × ∆σ0 . Since H is a constant homotopy H at f , 0 then it factorises as H = f (idL × ∆σ0 ). Then, for i ∈ {0, 1}: d1i (H)

= (idL × ∆∂1 )∗ (f (idL × ∆σ0 )) = (f )(idL × ∆σ0 )(idL × ∆∂1 ) 0 i = (f )(idL × ∆σ0 ∂1 ) = (f )(idL × ∆id[0] ) = f idL×∆0 = f. i

0

0

i

 Notice that the converse of the above lemma does not hold, i.e. a simplicial homotopy from f to f is not necessarily a constant homotopy, as shown in the counter example below: Lemma 11.4. A simplicial homotopy H ∈ Hom(L, X)1 is a constant homotopy iff either of the following diagrams commutes, and as a result both: L × ∆1

id 0 σ0 ∂1 0

// L × ∆ 1

H

/ X.

// L × ∆ 1

H

/ X.

idL ×∆

L × ∆1

id 1 σ0 ∂1 0

idL ×∆

200

Proof. Assume that either of the diagrams above commutes, which implies that H is a constant homotopy i at (H)(idL × ∆∂1 ). On the other hand, let H be a constant homotopy at f , then we have the below commutative diagram. That the left upper triangle is due to having H to be constant homotopy at f , and the commutativity of the right lower tiangle due to having H a simplicial homotopy from f to itself, as we saw earlier. L × ∆1

H

= pr1

f

0 idL ×∆σ0



/9 X O

.

H

/ L × ∆1

L × ∆0

i

idL ×∆∂1

 ∂10

∂11

The precomposition of either of above diagrams with idL ×∆ , and idL ×∆ , shows that following diagram commutes: 1

L × ∆0

idL ×∆∂1

0 idL ×∆∂1

// L × ∆ 1

/ X.

H

1

0

DR AF

T

Hence, H factorises through the coequaliser of such idL × ∆∂1 and idL × ∆∂1 . Some books use one of the above diagrams to define the fibre-wise simplicial homotopy, but as we see above the choice of either diagrams is not essential, and does not change the meaning. Let i ∶ K ↪ L be an inclusion of simplicial sets, and consider i∗ ∶ Hom(L, X) → Hom(K, X)

Definition 11.5 (Relative homotopy). Let i ∶ K ↪ L be an inclusion of simplicial sets, we say that a ≅ simplicial homotopy H ∶ f → f ′ ∶ L → X is a simplicial homotopy relative to K, denoted (rel K), if i∗1 (H) ∈ Hom(K, X)1 is a constant homotopy. ≅

Then, homotopy H ∶ f → f ′ (rel K) is such simplicial homotopy that makes the below diagram commutes: 1 K × ∆ _

i×id∆1

d11 (i∗1 (H))

d10 (i∗1 (H))

/ K × ∆0 ≅ K

i∗0 (f )=i∗0 (f ′ )



L × ∆1

i∗0 (d10 (H)) ′

P r1

H

 /X

i∗0 (d10 (H)). Hence, iif i∗0 (f ) = i∗0 (f ′ ).

. Then, = = = d10 (H), and d11 (H) coincide on K. That, the restriction of f, f ∈ Hom(L, X)0 to K coincide The simplicial homotopy is not an equivalence relation on the class of morphisms of simplicial sets. In particular it is not a transitive relation, as shown in the below counter example: However, restriction attention to special morphisms, namely Kan fibrations - to be studied later in the model structure on the category of simplicial sets, one find that simplicial sets defines an equivalence relation. A natural generalisation of the simplicial homotopy is provided below Definition 11.6 (Fibre-wise Homotopy). Let H ∈ Hom(L, X)1 be a simplicial homotopy, and p ∶ X → Y a morphism of simplicial sets, we say that H is a fibre-wise homotopy with respect to p if p∗ (H) is a constant ≅ homotopy. If there exists a fibre-wise homotopy with respect to p, H ∶ f → f ′ ∶ L → X, then we rite f ∼p f ′ , i.e. if the below diagram commutes: L × ∆1

id 0 0

idL ×∆∂1 σ0

// L × ∆ 1

H

/X

p

/ Y.

In particular, we write xn ∼p x′n for xn , x′n ∈ Xn if ιxn ∼p ιxn . Hence, xn ∼p x′n iff xn ∼ x′n and pn (xn ) = pn (x′n ). Notice that every simplicial homotopy H ∈ Hom(L, X)1 is is a fibre-wise homotopy with respect to X → ∗. Hence, simplicial homotopy with respect to a morphism of simplicial sets is not necessary an equivalence relation. However, we say later that restricting ourselves to Kan fibrations guaranteed having an equivalence relation. Moreover, fibre-wise homotopy with respects to p introduce homotopy on the fibres, which justify the name, as we see below:

201

Lemma 11.7. Let p ∶ X → Y be a morphism of simplicial set, and let f, f ′ ∶ L → X, ∀y0 ∈ Y0 a vertex of Y , fy0 , f ′ y0 the canonical induced morphisms on fibres. Then, f ∼p f ′ iff fy0 ∼ f ′ y0 for ∀y0 ∈ Y0 . Proof. TBC.



Definition 11.8. Let H ∈ Hom(L, X)1 be a simplicial homotopy, p ∶ X → Y a morphism of simplicial sets, and i ∶ K ↪ L be an inclusion of simplicial sets. Then, we say that H is a fibre-wise homotopy with respect to p relative to K if H is a fibre-wise homotopy with respect to p and H is a simplicial homotopy relative to K, i.e. if the below diagram commutes: 0

1 K × ∆ _ i×id∆1

L × ∆1

id 0 0 idL ×∆∂1 σ0

idK ×∆σ0

/ K × ∆0 ≅ K

d10 (i∗1 (H)) = d11 (i∗1 (H))



// L × ∆ 1

H

 /X

p

/ Y.

≅

Let H ∶ f → f ′ ∶ L → X be a fibre-wise homotopy with respect to p relative to K, then we denote f ∼p f ′ (rel K).

RA FT

Remark 11.9. One can define a right simplicial homotopy, in an analogue way of right homotopy of topological spaces, as a morphism of simplicial sets L → Hom(∆1 , X). However, it does not provide added informations due to the adjunction − × ∆1 ⊣ Hom(∆1 , −). That, one has a one-to-one correspondence between simplicial homotopy and right simplicial homotopy. 12. Homotopy Category

Let H be a homotopy category of topological spaces with respect to weak equivalences, and Hs the homotopy category of simplicial sets with respect to simplicial weak equivalences. Since simplicial weak equivalences of simplicial sets are topological weak equivalences of simplicial sets, the realisation functor ∣ − ∣∶ sSet → Top

extends to a functor

D

∣ − ∣∶ Hs → H Also, by [Mor03, p.15-16], [May92, p.65], and [Qui67], the singular functor extends to a functor S ∶ H → Hs

In fact the induced functors are equivalences of homotopy categories, and they are quasi-inverse of each other. TBC...

D RA FT

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