Institute of Physics University of Silesia
Krzysztof Bielas
Elements of model theory and category theory in quantum physics
Master’s thesis
Supervisor: dr Jerzy Król
Katowice 2012
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Krzysztof Bielas dr Jerzy Król Wydział Matematyki, Fizyki i Chemii/Instytut Fizyki Fizyka teoretyczna Elements of model theory and category theory in quantum physics models, categories, forcing, QM
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Podpis autora pracy
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Contents 1 Introduction 1.1 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Model theory 2.1 What is model theory? . . . . . . . 2.2 Relational systems and language . 2.3 Models and theories . . . . . . . . 2.4 Forcing . . . . . . . . . . . . . . . 2.5 Boolean valued models and forcing
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3 Category theory 3.1 What is category theory? . . . . . . . . . 3.2 Preliminaries . . . . . . . . . . . . . . . . 3.2.1 Basics . . . . . . . . . . . . . . . . 3.2.2 Limits . . . . . . . . . . . . . . . . 3.2.3 Functor categories . . . . . . . . . 3.2.4 Subobjects and subobject classifier 3.3 Topoi . . . . . . . . . . . . . . . . . . . . 3.4 Sheaf topos and forcing . . . . . . . . . .
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4 Quantum mechanics 4.1 Preliminaries . . . . . . . . 4.2 Postulates of QM . . . . . . 4.3 Spectral theorem . . . . . . 4.4 Classical logic and quantum 4.5 Kochen–Specker theorem .
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5 Categories and models in QM 5.1 Presheaves on C ∗ -algebras in QM . 5.2 The topos V B in QM . . . . . . . 5.2.1 Preliminaries . . . . . . . . 5.2.2 Model construction and real 5.2.3 QM paradoxes revisited . . 5.3 Forcing in QM . . . . . . . . . . .
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6 Conclusions and perspectives A Model theory A.1 Lattices and filters . . . . . . A.2 Heyting and Boolean algebras A.3 Logic . . . . . . . . . . . . . . A.4 Set theory . . . . . . . . . . . A.5 Ordinals and cardinals . . . .
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56 56 57 59 60 61
B Category theory 63 B.1 Presheaves, sites and sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 B.2 Locales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 Bibliography
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1
Introduction
1.1
Preface
In the present master’s thesis a description of possible applications of certain aspects of model theory (MT) and category theory (CT) to quantum mechanics (QM) is given. This work covers mainly an attempt to describe difficulties in fundamental issues related to quantum physics from the point of view of MT and CT. The examples of such difficulties are: logical structure, probabilistic interpretation, noncommutativity of observables and nature of quantum physics in general. It is my goal to combine MT and CT (already united in mathematics) for application to physics. The appearance of these tools in physics, even considered separately, is rather rare and exceptional. One obvious reason is that they seem to be too abstract and thus maybe difficult to handle for physicists. Apart from that, it is interesting to consider whether some concepts and ideas in physics were abandoned once and for all; they were not brought to life even by the emergence of models and categories in mathematics. That is why I will argue that there are many promising perspectives and possibilities of application of these structures (not only) to quantum physics. It is not a purpose of this work to give full and comprehensive survey on the subject (which is impossible anyway due to enormous growth of these fields over past decades). I try to keep the description in the thesis in the language of model theory and categories. These two areas of mathematics have become very rich and of great significance to other fields, let me mention only foundations and philosophy of mathematics. Despite the above richness the text is intended to be self-contained in the sense that a basic course of both physics (QM) and mathematics (CT, MT), that is required to understand the thesis, is provided. Moreover, two important remarks are in order: 1. The elements of MT and CT used in this thesis are elementary and standard in mathematics. 2. MT is considered as a first order model theory. The higher order languages and theories can be mutually interpreted in topoi, however I do not make any direct use of this possibility which, anyway, is very attractive from the point of view of the applications to physics. The merge of MT and CT in this thesis is of different character and preserves individuality of some first order MT constructions. The more complete analysis of higher order model theory and its relation to first order one is out of the scope of the thesis and would require at least separate work.
1.2
Motivations
QM seems to be the most accurate field of contemporary physics; in fact, quantum field theory (QFT) and its Standard Model of elementary particles and their interactions (SM), based on QM and special theory of relativity (SR), gives results with enormous and fantastic precision in comparison to experimental data (omitting problems such as anomalous magnetic moment of muon, which reveals quite big divergence between measurement and calculations). The first motivation to use models and categories in QM is that mathematical structure of quantum theory is not a satisfactory theory. Firstly, it is not a realistic theory, i.e. it only describes the results of experiments and not ”what is really going on”. Subsequently, what follows, there are many interpretations of quantum theory, attempting to explain what reality is hidden behind Hilbert spaces, a random collapse of wave function, decoherence etc. However, I do not assume that such totally realistic formulation is possible. Instead, I try to find aspects of mathematical formalism which help in exceeding usual Copenhagen approach.
7
The second reason for adapting new tools to QM is that it can not be coherently combined with another great physical theory, namely general theory of relativity (GR). One could of course admit that QM is the most fundamental theory and try to adapt formalism of GR (differentiable manifolds) to formalism of QM. However, it might be useful to think over QM again and try to reformulate the basic concepts in alternative approach. As mentioned, MT and CT may serve as different environments in which QM might be easier to handle. The last argument comes from the earlier experience of adapting apparently abstract structures (non-euclidean geometries) to actual models of spacetime (GR). The point is that one does not have to make unnecessary classical assertions about so-called reality, for example accepting that spacetime is modeled by a simple R4 ; instead, it can be valuable to admit other, possibly more abstract structures. As will be discussed further, such classical prejudices about foundations and logic of QM may also be superfluous and misleading.
1.3
Glossary MT CT QM H (·, ·) [·, ·] B (H) σ (A) P (X) h·, ·, ..., ·i | {z }
— — — — — — — — — —
model theory category theory quantum mechanics separable, complex Hilbert space scalar product commutator algebra of operators (densely) defined on a Hilbert space H spectrum of an operator A powerset of a set X, i.e. the set of all subsets of X n-tuple
ran (f ) C (X) dim (V ) fun (f ) J·K iff YX ,→ On dom (f ) cod (f ) CH AC ZF(C)
— — — — — — — — — — — — — —
range (image) of a function f C (X, C), the set of all complex-valued functions on X, dimension of space V f is a function truth-value function if and only if a set (an object) {f | f : X → Y } an inclusion map a class of all ordinals domain of a function f codomain of a function f continuum hypothesis axiom of choice axioms of Zermelo–Fraenkel set theory (with axiom of choice)
n
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2
Model theory
2.1
What is model theory?
Usual way to practice mathematics in a set-theoretic view is to become familiar with some concepts from a particular branch (or branches) of mathematics and then to prove new statements about them, called theorems. Then one introduces, if necessary, new definitions to enrich (or explain) the given structure. However, during such a procedure one might come across questions like: what is the role of a language in the reasoning? how do we know that our reasoning is consistent? what does it mean to prove something? what is the relationship between syntactics (language) and semantics (meaning)? are there any mathematical ”backgrounds” (models) different from the usual one (as far as it exists)? These questions have given rise to new field of mathematics, namely model theory; roughly, as stated, it is the study of models or structures. Such general structures, defined as sets with particular operations on them, which among others include groups, groupoids, vector spaces, etc. are subject to universal algebra. However, combining such general structures with language and logic through the notion of truth, i.e. deciding if particular sentence is satisfied in some structure, leads to model theory. The first famous example of existence of various models could be the invention of non-euclidean geometries in the 19th century.1 These geometries, obtained by Lobachevsky, Bolyai and Gauss, found their models in Riemannian surfaces, also rejecting the axiom of parallel lines. Surprisingly, even this early instance of MT, although purely mathematical, found its further physical realization in SR and GR. If one is concerned especially about foundations of set theory, it is of importance (since set theory is a basis for almost every contemporary field in mathematics) to ask about possible models of axioms of set theory. Here, as is the usual case, I deal with Zermelo–Fraenkel (ZF) axioms of set theory, possibly with the Axiom of Choice (AC), that is ZFC (for a full list of axioms of ZFC see Appendix A.4). Since these sentences are formulated in the so-called first-order (predicate) language, the resulting theories are first-order theories and very interesting and often unexpected facts are met here. Let me mention just two [1]: 1. in ZFC one can not prove the consistency of ZFC, 2. (assuming consistency of ZFC) there is a countable model of ZFC. By the works of L¨ owenheim, Tarski and G¨odel in 1930’s, 40’s and 50’s it has been recognized that some important concepts in mathematics were specific (relative) to particular models, while others were not. Moreover model theory has given an important contribution to many unsolved issues, especially independence results, such as Continuum Hypothesis (CH)2 or independence of AC. It was Cohen in 1963 who proved that CH is independent from ZFC axioms [18], so one can take CH into account (or not) without falling in contradiction.3 What Cohen did was to take the model of ZFC and to extend it to another model of ZFC in such a way that CH or ¬CH is forced to hold in new models. This method is called forcing. By that time model theory was already very rich, nevertheless Cohen’s result was a milestone in this branch of mathematics. A new light on above subject was shed by Dana Scott, Robert Solovay and independently by Petr Vopenka, who developed so called Boolean valued models. These works have given better understanding of Cohen forcing from different perspective. As the result, forcing procedure was substantially simplified and clarified. 1
Recall that they were invented by denial of the Euclid’s fifth axiom about parallel lines. In 1878 Georg Cantor put a hypothesis, called Continuum Hypothesis, that there is no cardinality between ω and C, i.e. the set P (ω) of all subsets of N is the next cardinality after ω (see Appendix A.5). 3 Note that consistency of ZFC with CH was already proved by K. G¨ odel in 1930’s. 2
9
By the time of development of model-theoretic tools, category theory was born. It turned out that these two have met in a very surprising way, to mention only a relation between forcing and intuitionism as inherent logic in categorical domain of discourse, called a topos. However, in this section only models in set-theoretic environment (Set) are considered. In Sec. 3 a wider perspective is given. In the following subsections I introduce basic concepts underlying models and their constructions. However, in order to make the text readable, the fundamentals about partial orders, filters, Boolean and Heyting algebras, set theory and some other details are put in Appendix A.
2.2
Relational systems and language
In the beginning I define a generalization of an algebraic structure; such a structure encloses groups, fields, etc. Definition 2.2.1. A relational system is a set A = hA, R, F, Ci, (2.1) wheren A is a setocalled universe of a system, R = riA | i ∈ I is a family of relations on A, F = fjA | j ∈ J is a family of operations on A and C = cA k | k ∈ K is a subset of A consisting of distinguished elements (constants). The characteristic property of any relational system is the arity of relations and operations and number of constants. Definition 2.2.2. Let τ1 (i) denote a number of arguments of relation ri for i ∈ I, τ2 (j) denote a number of arguments of operation fj for j ∈ J and τ3 (k) = 0 for k ∈ K. Then a signature of a system is a triple hτ1 , τ2 , τ3 i. Example 2.2.3. Any Boolean algebra B = hB, ∨, ∧, ¬, 0, 1i is a relational system with signature τ1 = ∅ (there is no relation), τ2 = h2, 2, 1i (two binary relations and one unary relation), τ3 = h0, 0i (two constants). Subsequently, it is desirable to introduce suitable language in order to express relations between elements of the universe. It is important to notice that all definitions, theorems and remarks in this section are formulated in the so-called first-order language. Thus, as will be stated, all variables range over elements of the universe. On the other hand, in higher-order logic variables are allowed to range over subsets of the universe, e.g. over predicates. This apparently harmless assumption in the logical structure leads to unexpected results; I will return to this issue in the context of topoi in Sec. 3.3. Definition 2.2.4. A language of a signature τ = hτ1 , τ2 , τ3 i is a set L (τ ) = hR, F, C, X, Si where R = {ri | i ∈ I} is a family of relational symbols (predicates), F = {fj | j ∈ J} is a set of function symbols (functors), C = {ck | k ∈ K} is a set of constants, X = {xn | n ∈ N } is a set of variables, S = {=, ¬, →, ∀} is a set of logical symbols. 10
(2.2)
Remark 2.2.5. It is clear that predicates ri are to be interpreted as relations riA , functors fj are to be interpreted as operations fjA , constants ck are to be interpreted as distinguished elements cA k , variables xn are to be interpreted as elements a of the universe A, while logical symbols get the usual interpretation of equality, negation, implication and universal quantifier. Other logical symbols are obtained by the use of elements of S: φ∨ψ φ∧ψ φ↔ψ ∃xn (φ)
is is is is
¬φ → ψ, ¬ (φ → ¬ψ), (φ → ψ) ∧ (ψ → φ), ¬∀xn (¬φ).
It is possible to build expressions in a language L that are not elements of X ∪C. The operations fj can do that. This case is generalized by Definition 2.2.6. Let Tm0 = X ∪ C, Tml+1 = Tml ∪ {fj (t1 , ...tm ) | j ∈ J, t1 , ..., tm ∈ Tml }.
(2.3)
Then terms are elements of the set Tm =
[
Tml .
(2.4)
l∈N
Thus terms arise by picking up variables and constants and applying any operation fj to them. From (2.3) it follows that each term t corresponds to some elements of the set Tm0 = X ∪ C, called free variables of t and denoted V (t). Definition 2.2.7. Let t be a term and define V (t) by V (xn ) = {xn }, V (c) = ∅ for any xn ∈ X, c ∈ C,
V (f (t1 , ..., tm )) = V (t1 )∪...∪V (tm ) . (2.5)
Now it is necessary to state the way of assigning elements of the universe to terms, such that variables are mapped to elements of the universe, constants are mapped to constants and operations preserve that mapping. Definition 2.2.8. An (interpretation) in A is a function i : X ⊇ Y → A; by conditions i (ck ) = cA k,
i (fj (t1 , ..., tm )) = fjA (i (t1 ) , ..., i (tm ))
(2.6)
it is extended to i : Tm ⊇ Y → A on all terms t such V (t) ⊆ Y . Subsequently, it is desirable to combine such into longer expressions, called formulas. Definition 2.2.9. Let Fm0 = {t = s | t, s ∈ Tm} ∪ {ri (t1 , ..., t2 ) | i ∈ I, t1 , ..., tm ∈ Tm}, Fml+1 = Fml ∪ {¬φ | φ ∈ Fml } ∪ {φ → ψ | φ, ψ ∈ Fml } ∪ {∀xn (φ) | φ ∈ Fml , n ∈ N }.
(2.7)
Then formulas are elements of the set Fm =
[ l∈N
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Fml
(2.8)
Thus one formulates basic statements about equality and relations between terms and then extend these statements with use of logical symbols. Moreover, it is possible again to speak about free variables of a formula. It can be done by extending the function V (see Definition 2.2.7) as follows: V (t = s) = V (t) ∪ V (s) , V (ri (t1 , ..., tm )) = V (t1 ) ∪ ... ∪ V (tm ) , V (¬φ) = V (φ) ,
(2.9)
V (φ → ψ) = V (φ) ∪ V (ψ) , V (∀xn (φ)) = V (φ) \ {xn }. In case of V (φ) = ∅, φ is said to be a sentence. Example 2.2.10. Consider the formula φ (x) = ”x = 0”; then V (φ) = V (x) ∪ V (0) = {x}. Define i, j : x 7→ R such that i (x) = 0 and j (x) = 1. Clearly, i gives the true sentence and j gives the false one, hence interpretation carries the truth of formulas, which is given below in a precise way. Remark 2.2.11. The notation φ means that φ could be either sentence or a formula (when free variables of φ are not important). The expression φ (x1 , ..., xn ) always denotes a formula with free variables x1 , ..., xn . To say that a system A is a model of some set of sentences, it is required to state what it means, that a formula φ is true in a relational system A (A satisfies φ, written A |= φ). Definition 2.2.12. The relation A |= F [i] is defined inductively: (A |= (t = s) [i]) ≡ (t [i] = s [i]) , (A |= ri (t1 , ..., tm ) [i]) ≡ ri (t1 [i] , ..., tm [i]) , (A |= ¬φ [i]) ≡ (not A |= φ [i]) ,
(2.10)
(A |= (φ → ψ) [i]) ≡ (not A |= φ [i] or A |= ψ [i]) , (A |= ∀xn (φ [i])) ≡ (for every a ∈ A, A |= φ [i (a/xn )]) . where i (a/xn ) means that a is a value of xn under given substitution i. Above definition means that when given a formula, the truth of this formula in a system is evaluated recursively. For the purpose of further results it is interesting to consider how to extend given language L. Definition 2.2.13. Let τ = hτ1 , τ2 , τ3 i,
σ = hσ1 , σ2 , σ3 i
(2.11)
be signatures. The signature σ is an enrichment of the signature τ if σi is an extension of τi for i = 1, 2, 3. Since such an extension implies new relations, operations and constants, also the language L (σ) enriches the language L (τ ) and it holds that Tm (L (τ )) ⊆ Tm (L (σ)) ,
Fm (L (τ )) ⊆ Fm (L (σ))
12
(2.12)
2.3
Models and theories
Definition 2.3.1. We say that a system A is a model of a set T of sentences in a language L, if A |= φ for any φ ∈ T . For example, the universe of mathematical discourse Set (described in the next subsection) is a model of the axioms of ZFC (if it is consistent). A class consisting of all models (in Set) of a given set of sentences T is denoted Mod (T ). Let K be a class of systems of a given signature. K is axiomatizable, if K = Mod (T ) for some set of sentences T ; such T is called an axiomatics of K. Finally, a set of sentences Th (K) = {φ | ∀A ∈ K (A |= φ)} is called a theory of a class K. It is easy to see that if K is axiomatizable, then K = Mod (Th (K)). Remark 2.3.2. Observe that any set of formulas T can be seen as a set of axioms for some theory. Example 2.3.3. The class of all groups Grp is axiomatizable since Grp = Mod (T ) for T consisting of group axioms, i.e. associativity, identity and invertibility. On the other hand, the class of all topological spaces Top is not axiomatizable (in the first-order language), i.e. for any set T there will always be some model of T that is not a topological space. More generally, it is also possible to consider models of sets of formulas, although it is necessary to take into account an interpretation of this formulas in some relational system. What follows, let T be a set of formulas; a pair hA, ii is a model for T if A |= φ [i] for every φ ∈ T . Example 2.3.4 (Peano arithmetic). Let N = hN, +, ·, 0, 1i, where N = N, + and · are arithmetical addition and multiplication, 0 and 1 are constants. Then a language of the system N obviously contains two binary operations and two constants ∆0 , ∆1 . Axioms of arithmetics (denoted P) are the following: 1. (commutativity) x + y = y + x,
x · y = y · x, (x · y) · z = x · (y · z),
2. (associativity) (x + y) + z = x + (y + z) , 3. (distributivity) (x + y) · z = (x · y) + (x · z), 4. (neutral elements) x + ∆0 = x,
x · ∆1 = x,
5. (cancellation) x + z = y + z → x = y, 6. x 6= ∆0 iff ∃y (x = y + ∆1 ), 7. (induction) for any formula φ and variable x (φ (∆0 /x) ∧ ∀x [φ → φ ((x + ∆1 ) /x)]) → ∀x (φ). As will be stated, Peano arithmetic is crucial in investigation of consistency and completeness. Given set T of formulas, it is possible to prove other statements, called theorems, by so-called inference rules, such as detachment (modus ponendo ponens): from φ and φ → ψ it is concluded that ψ.
(2.13)
Moreover, there are sentences which are universally valid, i.e. satisfied under any interpretation in any system, such as φ ∨ ψ → ψ ∨ φ. These sentences are called axioms of logic and denoted by LOG. The set LOG with rules of inference form the so-called deductive system (cf. axioms for intuitionistic and classical first-order logic in Appendix A.3). It is possible now to define what a theorem in a theory T is. 13
Definition 2.3.5. Let T0 = T ∪ LOG, Tn+1 = Tn ∪ {ψ | ∃φ (φ ∈ Tn and φ → ψ ∈ Tn )}. Then a theorem in a theory T is an element of the set [ T∗ = Tn .
(2.14)
(2.15)
n∈N
One writes T ` φ for φ ∈ T ∗ . It turns out that T ` φ iff A |= φ in every A ∈ Mod (T ). If T = ∅, then it is written ` φ for T ` φ and φ is universally valid (called a tautology), as it is proven only from axioms of logic and rules of inference. Definition 2.3.6. A theory T is called inconsistent, if for some formula φ it holds T ` φ and T ` ¬φ; otherwise, T is consistent, written Con (T ). A striking fact is that if T is inconsistent, then T ` ψ for any formula ψ, hence A |= ψ in any A ∈ Mod (T ). Moreover, if T is consistent, then a set of its theorems {φ | T ` φ} is also consistent. Definition 2.3.7. A formula φ is said to be decidable in T if either T ` φ or T ` ¬φ holds. Otherwise, φ is independent of T and it is equivalent to a statement that both T ∪ {φ} and T ∪ {¬φ} (written T + φ and T + ¬φ) are consistent. A theory T is complete if there are no formulas independent from T ; otherwise, T is called incomplete. For the purpose of the next section the following theorem is given [1]. Theorem 2.3.1. Let T be a theory. If T ` φ, then there exists finite sequence of formulas hφ0 , ..., φn i such that φn = φ and for every i 6 n it holds that φi either belongs to T or LOG or arises by modus ponens from φi−2 and φi−1 . Then each such sequence hφ0 , ..., φn i is called a proof of φi in T . As already mentioned in introduction to this section, some properties are relative to a particular model chosen. For the sake of further investigation, the notion of relativization requires some explanation. Definition 2.3.8. Let M be any class;4 for any formula φ we define φM , the relativization of φ to M , by induction: let x, y ∈ M and (x = y)M (x ∈ y)M (φ ∧ ψ)M (¬φ)M [∃x (φ)]M
is is is is is
x = y, x ∈ y, φM ∧ ψ M , ¬ φM , ∃x x ∈ M ∧ φM .
4
Note that classes can be used interchangeably with formulas, since for any formula φ there is a class M (written M (x) if x is among free variables of φ) such that φ holds for every member of C; for example, the universal class V of all sets is a class that corresponds to the formula x = x.
14
Hence φM is obtained by simply restricting quantifiers to M . Accordingly, φ is said to be true in M , i.e. M |= φ iff φM holds. What follows, φ is said to be absolute for M if ”φM holds iff φ does”.5 It will be shown that some objects and notions happen to be absolute and some do not. Since in further analysis it will be stated many times that there is a model of ZFC, such a statement needs some explanation. First of all, there are three results of K. G¨odel that are the milestones in model theory [1]: Theorem 2.3.2 (G¨ odel’s completeness theorem). Every consistent set of sentences has a model. Theorem 2.3.3 (G¨ odel’s incompleteness theorems). A theory T that includes Peano arithmetic can not be both consistent and complete. Moreover, such a theory can not prove its own consistency, i.e. T ` Con (T ) does not hold. Remark 2.3.9. We say that a theory T includes Peano arithmetics P if for some formula M (x) defining the class M it holds that T ` ∃x [M (x)] and P M = {φM | φ ∈ P } ⊆ T , i.e. any axiom of P restricted to M (x) is an axiom of T . ZFC obviously satisfies above, hence these theorems also adapt to set theory. Thus there are two possibilities: either ZFC is consistent or it is not. If ZFC is consistent, then by G¨odel’s second incompleteness theorem a statement ”ZFC is consistent” is not provable in ZFC, i.e. ”ZFC is consistent” is not a theorem of ZFC. If ZFC is inconsistent, then every statement is a theorem of ZFC, also ”ZFC is consistent”. Nevertheless, when there is a statement ”assume M is a model of ZFC” in further text, it always includes the condition ”if ZFC is consistent” implicitly. One of the most important facts about models and cardinalities of the first-order languages comes from [1] Theorem 2.3.4 (L¨ owenheim–Skolem). If a set of sentences T of a language L has an infinite model, then it has a model of arbitrary cardinality > card L. Since a language of set theory is countable6 and it has an infinite model, it also has a countable model. This might seem strange, because set theory speaks about uncountable objects, such as R; on the other hand, it has access only to countable many elements. This situation is an example of the so-called Skolem paradox and is simply an observation, that cardinality can be relative to the model (bijections that give equinumerosity could be present in one model and absent in another, see Appendix A.5), so it is not absolute. Nevertheless, L¨owenheim–Skolem theorem allows to make use of countable models in context of set theory. For instance, as already suggested, countability is relative to the model used, as there are countable models M of ZFC that contain real numbers, that are provable to be uncountable in M . On the other hand, notions like A × B or ”R is a relation” are absolute.
2.4
Forcing
As mentioned in the beginning of this section, forcing has been used to prove some independence results in set-theory. Remark 2.4.1. The strategy of forcing is the following: 1. take a transitive model M = hM, ∈i7 of ZFC (the so-called ground model ), 5
The sentence ”φ holds” is understood as ”V |= φ”. The set theory language contains a binary predicate ∈, countable set of variables and the set S of logical symbols (cf. Definition 2.2.1). 7 A model M is said to be transitive if the universe M is a transitive set. 6
15
2. find some set G (called generic) such that G ∈ / M, 3. create a larger transitive model M [G] of ZFC, called a generic extension, where already G ∈ M [G]. The first point of Remark 2.4.1 about existence of transitive model needs to be justified [25]. Theorem 2.4.1 (Mostowski collapsing theorem). Let A be a model of an axiom of extensionality with a well-founded relation8 R on A. There exists an isomorphism G from A to a transitive ∈structure M, called Mostowski collapse, where G (x) = {G (y) | y ∈ A and yRx} and M = ran (G). Moreover, G and M are unique. Hence given any model A of ZFC, there is a unique transitive model M of ZFC that is isomorphic (as a relational structure) to A. Let M be a ground model. Let P = hP, 6, 1i ∈ M be a partially ordered set. In this subsection, a poset P will be called a notion of forcing, while the elements of P are forcing conditions. If p 6 q then p is said to be stronger than q, so it is inversed with respect to an ordering. If p, q have common lower bound in P , namely ∃r (r 6 p ∧ r 6 q) then p and q are said to be compatible (written p k q); otherwise they are incompatible (written p ⊥ q). A set D ⊂ P is said to be dense if ∀p ∈ P ∃q ∈ D (q 6 p). A set of conditions G ⊂ P is (P -)generic over M if G is a filter on P and it intersects every D ∈ M that is dense in P . A poset is called separative, if x y → ∃z (z < x ∧ z ⊥ y). The second point of Remark 2.4.1 about existence of a generic set (over poset P ) is satisfied by the following [15] Lemma 2.4.2 (Rasiowa–Sikorski). If M is countable and P is a poset in M (P ∈ M and P is a poset), then for every p ∈ P there is a set G that is P -generic over M and p ∈ G. In general, such generic set over the poset does not necessarily exist. However, by L¨owenheim– Skolem theorem, there exists a countable model M of ZFC, hence by Rasiowa–Sikorski theorem there always exists a P -generic set G over M. This shows why it is comfortable to start with a countable ground model. Although existence of a generic G is satisfied, G is also required not to belong to M, since G has to carry new information about the generic extension [25]. Lemma 2.4.3. If M is a transitive model of ZFC and P ∈ M is a poset such that ∀p ∈ P ∃q, r ∈ P [(q 6 p) ∧ (r 6 p) ∧ (q ⊥ r)]
(2.16)
and G is P -generic over M, then G ∈ / M. It can be shown that if P is separative and atomless then it satisfies (2.16), hence, as will be explained below, it leads to nontrivial forcing. The fact that P is separative and atomless is especially significant in Boolean valued model setting, described in the next subsection. The third point of Remark 2.4.1 follows from [15] Theorem 2.4.4 (The Generic Model Theorem). Let M be a transitive model of ZFC and let P be a notion of forcing in M. If G ⊂ P is generic over M, then there exists a transitive model M [G] such that 1. M [G] is a model of ZFC, 2. M ⊂ M [G] and G ∈ M [G], 8
A relation is well-founded, if for each set x the class Rx = {y | yRx} is a set, and each nonempty set x has an / for any z ∈ x. element y such that z Ry
16
3. OnM[G] = OnM , 4. if N is a transitive model of ZFC such that M ⊂ N and G ∈ N , then M [G] ⊂ N . Hence one needs only to find appropriate generic G over P in a transitive model of ZFC. Then a new model arises; it is larger in the sense that it contains new set G, if P satisfies (2.16). Also, this operation does not change the class On. It is clear that M [G] is the least model that fulfills desired conditions. i.e. the minimal model of ZFC that arises by adjoining some generic set G; for that reason M [G] is called generic extension of M. However, it is not yet clear how such M [G] is constructed. Before introducing the forcing language, two examples of application of above notions give a foretaste of how powerful forcing is [15]. Example 2.4.2. Let P = {hp (0) , ..., p (n − 1)i | p : n → {0, 1} and n ∈ ω} be ordered such that p < q if q ⊂ p. It is obvious that p k q if p ⊂ q or q ⊂ p. Let M be the ground model and let G ⊂ P be a generic set over M. S S p. It is now desired to obtain from G a specific object that is not in M. Let f = G = p∈G
We show that f is not in M. Since G is a filter, for any p, q ∈ G their common lower bound is in G, so p k q and such a set of compatible sequences makes f to be a function, for if there would be p (i) 6= q (i) for some i ∈ n then p ⊥ q. Now define Dn = {p ∈ P | n ∈ dom (p)}; for any n ∈ ω the set Dn is dense in P since any element of Dn can be extended by extending its domain. Thus, since G is generic, it holds G ∩ Dn 6= ∅ for any n ∈ ω, so dom (f ) = ω. We can identify above function f : ω → {0, 1} with some subset A ⊂ ω (see Sec. 3.2.4 for explanation of this). It is now claimed that f is not present in the ground model M, so it is added in some sense by the generic set G. Let Dg = {p ∈ P | p 6⊂ g} for any g ∈ M; of course, Dg is dense in P since any p ∈ P can be extended by adding a value that differs from g at some point. Now suppose f = g for some g ∈ M. Then G∩Dg = ∅, since G contains only compatible extensions of g, while Dg does not. But Dg is dense in P and G is generic. Hence G ∩ Dg 6= ∅ and f 6= g for any g ∈ M and f ∈ / M. Thus the whole operation ended with adjoining the new subset of natural numbers, that was absent in the ground model M. Since every subset of ω corresponds to some real number, such subsets are called Cohen generic reals. Example 2.4.3. Let M be a ground model, P = {hα0 , ..., αn−1 i | αi is a countable S ordinal∧n ∈ ω} with analogous ordering and similarly we let G ⊂ P to be generic over M and f = G. By similar reasoning, f is a function and dom (f ) = ω. Define Eα = {p ∈ P | α ∈ ran (p)}; it is again dense, thus ran (f ) = ω1M . Hence, it defines surjection f : ω → ω1M , thus f collapses ω1M to countable ω. It can be shown that in both examples notions of forcing are atomless and separative [15]. Now, the generic extension M [G] is to be defined. What is surprising is that it is possible to speak about M [G] entirely in M, i.e. to introduce such a language that in some way can express assertions about M [G] in terms of M. This language is called the forcing language and is defined by an addition a set of constants (names) to the language of M, one for each element of M [G]. Through this, every element n of M [G] will have a name τ in M (which is often identified with corresponding constant in the language). The generic set G will also have a name G˙ in M , since G ∈ M [G]. The point is that there is no knowledge in M about deciding an object named, until the interpretation of any name by a choice of particular G is given. More precisely, the construction goes as follows. Definition 2.4.4. τ is a name iff τ is a binary relation and ∀hσ, pi ∈ τ (σ is a name and p ∈ P ). 17
Clearly, names are defined by recursion; however, the definition is not ”circular” by the axiom of extensionality (cf. Appendix A.4). That is, one starts with an empty set ∅, takes subsets of the powerset of ∅ × P and so on. Hence a name τ is a collection of names σ with certain elements of P . As will be seen in Sec. 2.5, elements of P assigned to names can be seen as probabilities that the names are actual elements of other names. Let V P denote the class of all names and let MP = V P ∩ M; then MP is the class of the names that are in M. Define now an interpretation of a name by generic G. Such interpretations are also called G-names. Definition 2.4.5. τ G = {σ G | hσ, pi ∈ τ and p ∈ G}. Again, G-names are defined recursively. The generic extension M [G] of M is the class of all G-names in MP . Definition 2.4.6. M [G] = {τ G | τ ∈ MP }. There is a simple way to show that indeed M ⊂ M [G] and G ∈ M [G]. For the former, take x ∈ M and define x˙ = {hy, ˙ 1i | y ∈ x}.9 Thus ∅˙ = ∅, ˙ = {hy, 1˙ = {∅} ˙ 1i | y ∈ 1} = {h∅, 1i}
(2.17)
and so on. By induction, assume that y˙ G = y. Since 1 ∈ G for any ultrafilter G, it is clear that x˙ G = {y˙ G | y ∈ x}. Since, by induction assumption y˙ G = y, we have x˙ G = {y | y ∈ x} = x, hence every element x ∈ M has a name x˙ and M ⊂ M [G]. For the latter, let Γ˙ = {hp, ˙ pi | p ∈ P }. Then Γ˙ G = {p˙G | hp, ˙ pi ∈ Γ˙ and p ∈ G}. It holds p˙G = p and hp, ˙ pi ∈ Γ˙ iff p ∈ P , hence Γ˙ G = {p | p ∈ P and p ∈ G} = G. Moreover, other results from Theorem 2.4.4 applie to M [G] just defined. Coming back to forcing, note that one can construct sentence about M [G] in the forcing language entirely in M, as provided by names that are accessible in M; for example, such a statement could be φ (τ1 , ..., τn ). However, one can not decide the truth of φ until a specific G is given, since G decides what an interpretation of this names comes. Let define the forcing relation
in the following way: Definition 2.4.7. p φ (τ1 , ..., τn ) iff ∀G (G is generic over M ∧ p ∈ G) → M [G] |= φ τ1G , ..., τnG .
(2.18)
The relation satisfies some conditions which indicate that it is indeed correct description of forcing some property to hold in the generic extension [15]. Theorem 2.4.5 (Properties of forcing). Let P be a notion of forcing in the ground model M; let p, q ∈ P and φ, ψ be any formulas in the forcing language. Then following statements hold true: 1. [p φ ∧ q 6 p] → q φ, 2. there is no p such that both p φ and p ¬φ, 3. ∀p ∈ P ∃q 6 p (q φ ∨ q ¬φ), 4. p ¬φ iff there is no q 6 p such that q φ, 9
Such a way of assigning a name to the element of M is called canonical.
18
5. p φ ∧ ψ ↔ p φ ∧ p ψ, 6. p ∀x (φ) ↔ ∀τ ∈ MP [φ (τ )], 7. p φ ∨ ψ ↔ ∀q 6 p ∃r 6 q (r φ ∨ r ψ), 8. p ∃x (φ) ↔ ∀q 6 p ∃r 6 q ∃τ ∈ MP (r φ (τ ), 9. p ∃x (φ) → ∃τ ∈ MP [p φ (τ )]. Above theorem justifies calling elements of P conditions, since they behave like conditions with strength given by an order 6 on P . The consistency of ZFC + ¬CH as an example of forcing is shown in Appendix A.4. The forcing relation gains wonderfully simple description in terms of Boolean valued models, since it will be clear that creating the generic extension M [G] by giving names to all its elements is essentially the same procedure as creating Boolean valued model.
2.5
Boolean valued models and forcing
In 1920’s John von Neumann gave, along with his definition of natural numbers, the construction of the universe of sets V : V0 = ∅, Vα+1 = P (Vα ) , [ V = Vα .
(2.19)
α∈On
Firstly, notice that by the A 7→ χA (subset 7→ characteristic function) isomorphism, von Neumann’s universe can be set up in the following way: V02 = ∅, Vα2 = {x | x : dom (x) → 2, dom (x) ⊆
[ γ<α
V2 =
[
Vγ2 },
(2.20)
Vα2 .
α∈On
Hence, the operation of powerset is replaced by taking two-valued sets, which are known to be ordinary subsets of a given set. What would happen, if 2 was replaced by another object? It turns out a true generalization of the above procedure to admit different Boolean algebra B as a codomain of characteristic functions and receive so-called B-valued sets. This step reminds a little the concept of fuzzy set10 , but it is not necessary to restrict to the [0, 1] interval (it is customary in case of Boolean valued models to speak about probabilities, though). More precisely, a Boolean valued model arises as follows. Definition 2.5.1. Let B be a complete Boolean algebra; a Boolean valued model (of the language L of set theory) A consists of an universe (called Boolean) A and functions J∈KB , J=KB : A×A → B 10
A fuzzy set is the pair (X, m), where X is an ordinary set and m : X → [0, 1] assigns any element x ∈ X a grade of membership m (x) of x to fuzzy set (X, m)
19
such that for any x, y, z, v, w ∈ A Jx = xKB = 1,
Jx = yKB = Jy = xKB ,
(2.21)
Jx = yKB · Jy = zKB 6 Jx = zKB ,
Jx ∈ yKB · Jv = xKB · Jw = yKB 6 Jv = wKB .
Where no confusion is likely, J∈K, J=K is written for J∈KB , J=KB . It is desired to assign a Boolean truth value JφK to every sentence φ in L; such an assignment should fulfill a couple of natural conditions, such as φ and ψ is true iff φ is true and ψ is true. Thus define J¬φK = ¬JφK,
Jφ → ψK = JφK ⇒ JψK, ^ J∀x (φ (x))K = Jφ (u)K.
(2.22)
u∈A
According to footnote Remark 2.2.5, other Boolean truth values are defined as follows: Jφ ∨ ψK = JφK ∨ JψK,
Jφ ∧ ψK = JφK ∧ JψK,
(2.23)
Jφ ↔ ψK = JφK ⇔ JψK, _ J∃x (φ (x))K = Jφ (u)K. u∈A
Obviously, on the left hand side there are logical symbols, i.e. elements of S in L, whereas on the right hand side there are operations on Boolean algebra B. By (2.21)-(2.23), it is possible to assign a Boolean truth value to any sentence in L, as demanded. A formula φ is valid or holds with probability 1 in A iff JφK = 1. Note that the assignment φ → JφK is a generalization of the satisfaction predicate |=, since A |= φ iff JφK = 1.
(2.24)
Now it is possible to construct a true generalization of the universum of sets, namely an universum of B-valued sets. Let B be a complete Boolean algebra and define V0B = ∅, VαB = {x | x : dom (x) → B, dom (x) ⊆
[
VγB },
γ<α
VB =
[
(2.25)
VαB .
α∈On B
To check that V is a Boolean valued model, one needs to assign truth values to atomic sentences, i.e. u = v and u ∈ v, and defines them again by recursion: _ Ju ∈ vK = (v (y) ∧ Ju = yK) , (2.26) y∈dom(v)
Ju = vK =
^ x∈dom(u)
(v (y) → Jx ∈ vK) ∧
B
^ y∈dom(v)
(u (x) → Jy ∈ uK) .
(2.27)
The elements of V are called B-valued sets. It turns out that V B satisfies (2.21), so it is a Boolean valued model. What is more important, the following theorem holds [5]. 20
Theorem 2.5.1. All of the ZFC axioms are valid in V B . Hence V B is a Boolean valued model of ZFC. Note that since there are sentences which are assigned a truth-value neither equal to 0 or 1, V B is not an ordinary model of ZFC. However, it was already said that the classical universe V and universe of two-valued sets V 2 are isomorphic. It is done by an embedding V 3 x 7→ x ˇ ∈ V 2 defined as follows. Definition 2.5.2. Let x ˇ = {hˇ y , 1i | y ∈ x} for every x ∈ V . Hence x ˇ are constant functions on dom (ˇ x) = {ˇ y | y ∈ x} such that x ˇ (ˇ y ) = 1. Elements of V of the form x ˇ for some x ∈ V are called standard. In other words, standard elements of V B are these elements which consist of other standard elements with probability 1, hence they can be viewed as representatives for elements of V . For standard elements it holds [5] B
Theorem 2.5.2. Let x, y ∈ V , u ∈ V B . Then 1. x ∈ y ↔ Jˇ x ∈ yˇKB = 1, 2. x = y ↔ Jˇ x = yˇKB = 1, 3. ˇ: V → V 2 is bijective in the sense that it is one-to-one and given u ∈ V 2 there is exactly one x ∈ V such that Jˇ x = uKB = 1, 4. if φ (x1 , ..., xn ) is a formula, then φ (x1 , ..., xn ) ↔ Jφ (x1 , ..., xn )K2 = 1.
(2.28)
Now it is possible to repeat the whole procedure (2.25) for some transitive model M of ZFC instead of the universe V and a particular complete Boolean algebra B corresponding to notion of forcing P . It is necessary to introduce such connections between partial orders and Boolean algebras in order to make the Boolean valued model method of forcing manageable. Boolean algebras are partial orders in the natural way, since hB + , 6i is a poset for any Boolean algebra B with x 6 y iff x ∧ y = x for any x, y ∈ B + , where B + means B \ {0}. The crucial step toward transferring forcing to this setup is taken by following [15] Theorem 2.5.3. For every poset P there is a unique (up to isomorphism) complete Boolean algebra B (P ) and a function e : P → B (P )+ such that 1. if p 6 q, then e (p) 6 e (q), 2. p k q iff e (p) · e (q) 6= 0, 3. {e (p) | p ∈ P } is dense in B.11 Moreover, if P is separative, then e is a 1 − 1 mapping. Since P -generic set G plays a key role in forcing, it is necessary to translate genericity to boolean framework. It turns out that it holds [15] Lemma 2.5.4. A set G is B + -generic iff G is an ultrafilter on B and if A ⊆ G and A ∈ M, then V A ∈ G. 11
A set D is dense in Boolean algebra B if D is dense in poset B + .
21
Recall that in previous subsection P was assumed to be separative and atomless in order to receive a nontrivial forcing. Since any Boolean algebra B is separative as a poset hB + , 6i, the following theorem holds [14]. Theorem 2.5.5. The generic ultrafilter G on B is not in the ground model M iff B is atomless. Hence, analogically to Lemma 2.4.3, B has to be atomless to provide nontrivial forcing. Subsequently, it can be shown how forcing arises in the context of Boolean valued models. Again, let M be a transitive model of ZFC and P ∈ M a notion of forcing. By Theorem 2.5.3, there exists particular Boolean algebra B = B (P ). Define MB analogically to (2.25): MB 0 = ∅, MB α = {x | x : dom (x) → B, dom (x) ⊆
[
MB γ , x ∈ M},
γ<α
MB =
[
(2.29)
MB α.
α∈On
Let MP = M B(P ) = MB and call the elements of MP by names. Clearly, names are just B-valued sets defined inside M. Let LB denote the language L of set theory with additional constant symbols (again identified with names in MB ). Now define the forcing relation by Definition 2.5.3. p φ (x1 , ..., xn ) iff e (p) 6 Jφ (x1 , ..., xn )K,
(2.30)
where φ is a formula of L and x1 , ..., xn ∈ MB .
It turns out that such defined relation satisfies all properties from Theorem 2.4.5, hence it is correct analogue to Definition 2.4.7. To get the generic extension of M one introduces an interpretation by a generic ultrafilter G on B. Definition 2.5.4. For every x ∈ MB , the interpretation xG is defined inductively: ∅G = ∅, xG = {y G | x (y) ∈ G}.
(2.31)
Finally define M [G] = {xG | x ∈ MB }. Such M [G] satisfies Theorem 2.4.4, hence it is the generic extension of M. Also, M [G] is a model of ZFC [15].
22
3
Category theory
3.1
What is category theory?
Category theory (CT) was created in order to formalize and explore the notion of natural tranformation. CT started in the mid 1940’s with publication of the paper ”The general theory of natural equivalence” by Samuel Eilenberg and Saunders MacLane. The next idea was to axiomatize categories without basing it on set theory and finding out, whether these new structures give rise to new basis for mathematics. The answer turned out to be extremely positive — almost every field of mathematics has felt an impact of category theory such that categories became something more than just branch of mathematics. In the next subsection there are given definitions and theorems needed by further constructions in the text. The strategy of using categories needs not to be justified by searching for completely new and abstract universes; as a matter of fact, categories give an opportunity to take a look at some already familiar structures from a different point of view. Namely, as mentioned in the previous section, classical mathematics tries to describe all objects by sets, e.g. every algebraic structure is of the form of an ordered tuple (which is a set) of some set, family of sets and some functions, which are also sets. The particular aim of CT was to shift this point of view to more functional one, i.e. to emphasize the role of functions or, in other words, the importance of dynamics in mathematics. The role of category theory in mathematics is wonderfully described by J. L. Bell in [3]. It is stated that category theory relates to an abstract algebra in the same way as an abstract algebra relates to elementary algebra. That is, while elementary algebra replaces numbers by variables and keeps operations fixed, an abstract algebra varies over operations keeping the structures fixed (e.g. groups, ring, vector spaces etc.). Then, CT is about changing also the structure, hence it allows to make mathematics dynamical at higher level. In the next subsections I present rather standard material which fixes the notation on the one hand, and is necessary for speaking about topoi on the other.
3.2 3.2.1
Preliminaries Basics
Definition 3.2.1. A category C consists of collection Ob (C) of objects A, B, C, ... and collection Arr (C) of morphisms (also called arrows) f , g, h, ... such that 1. for every arrow f there exist objects A, B called domain and codomain of f which are denoted f
dom (f ) and cod (f ), respectively; it also is denoted f : A → B or A − → B, f
g
g◦f
2. composition: if A − → B and B → − C, then there exists an arrow A −−→ C called composition of f and g; 1
f
g
A 3. identity: for every object A there exists an arrow A −→ A such that for any A − → B, C → − A the following hold f ◦ 1A = f, 1A ◦ g = g; (3.1)
f
g
h
4. associativity: for any A − → B, B → − C, C − → D it holds (h ◦ g) ◦ f = h ◦ (g ◦ f ).
23
(3.2)
Hence it is not defined what really objects and morphisms are; instead, some properties of these are assumed. As will be shown, notions like being an element, an empty set, a product and various other constructions take a very simple and elegant form in category theory. Moreover, they are all defined in terms of arrows rather than objects. Objects and arrows in C are also called C-objects and C-arrows. If A, B are C-objects then a collection of C-arrows between A, B is written C (A, B) or HomC (A, B).12 Examples 3.2.2. There could be given numerous examples of categories (objects and arrows), to name only a few: 1. sets and functions (Set), ordinals and functions (On), 2. groups and group homomorphisms (Grp); 3. vector spaces and linear maps (Vect); 4. above categories in finite case, written as Finset, Finon, Fingrp, Finvect; 5. topological spaces and continuous functions (Top); Hausdorff spaces and continuous functions Haus; 6. partially ordered sets and order-preserving functions (Pos), lattices and lattice homomorphisms (Lat), Boolean algebras and homomorphisms (Bool), Heyting algebras and homomorphisms (Heyt); 7. unital C ∗ -algebras and ∗-homomorphisms CStar, unital commutative C ∗ -algebras and ∗homomorphisms cCStar. 8. the matrix category MatR , where objects are positive integers and each MatR (m, n) consists of m × n matrices with entries in unital commutative ring R such that an arrow composition is matrix multiplication. There are also peculiar examples, such as so-called abstract categories, whose arrows need not be the functions; for example 1. the monoid M with one object M and arrows as elements of M ; in this case the identity 1M element is the identity arrow M −− → M , while associativity is fulfilled by the axioms of monoid (notice that a group G turns out to be such a one-object category that for any arrow f : G → G there exists the arrow f −1 : G → G such that f ◦ f −1 = f −1 ◦ f = 1G ); 2. the single poset hP, 6i forms a category P with elements of P as P-objects such that there is an arrow between p, q ∈ P iff p 6 q. Remark 3.2.3. Any category-theoretic structure is defined firstly by observing some properties of such a structure in familiar category like Set and then abstracting these properties to categorical form, i.e. in terms of arrows. Before giving a categorical versions of basic mathematical notions, consider the following definition of isomorphism, which would be one of the main tools in further investigation. 12
Note, however, that C (A, B) does not necessarily form a set; if it does for any C-objects A, B, then C is called locally small. Furthermore, if Arr (C) is a set, then also Ob (C) is a set and C is called small.
24
Definition 3.2.4. Fix a category C; an arrow f : A → B is called monomorphism or monic iff for any g1 , g2 : C → A, f ◦ g1 = f ◦ g2 implies g1 = g2 , i.e. f is left-cancellable. An arrow f : A → B is called epimorphism or epic iff for any g1 , g2 : B → C, g1 ◦ f = g2 ◦ f implies g1 = g2 , i.e. f is right cancellable. Monic and epic arrows are written f : A B, f : A B, respectively. An arrow f : A → B is called an isomorphism or iso iff there exists an arrow g : B → A such that f ◦ g = 1B and g ◦ f = 1A .13 If there exists an isomorphism f : A → B then A, B are said to be isomorphic and it is denoted as A ∼ = B. Examples 3.2.5. 1. In Set monomorphisms are injective, epimorphisms are surjective and isomorphisms are bijective. 2. In Top isomorphisms are homeomorphisms. 3. In the poset P every arrow is monic and epic, although the only iso are identity arrows. An important feature of category theory, that is heavily used (both in category theory and model theory) is the idea of duality. As one will surely find it out during the text, the role of duality can not be overestimated. As could be seen, many mathematical structures and objects are dual to other. Definition 3.2.6. Let C be a category; define C op to be a category with the same objects as C and whose arrows are the arrows of C in opposite direction. These conditions mean that Ob (C op ) = Ob (C) and f op ∈ Arr (C op ) iff f ∈ Arr (C). Hence, every statement S expressed in terms of objects and arrows in C can be formulated in C op by simply reversing arrows to obtain statement S op . Example 3.2.7. Consider the notion of a filter and an ideal in the category Lat (see Appendix A.1). Since partial order is rewritten in terms of arrows, closedness upward and downwards are dual to each other. Moreover, as will be stated, the lattice operations of meet and join are dual to each other, hence filters and ideals are essentially dual notions. As concluded in Sec. 2, the language of ZFC contains a binary relation symbol ∈, that ”speaks of” being an element. We should begin with redefinition and enrichment of this basic notion. Firstly, note that empty set in Set is distinguished by the property that for any set X there is exactly one function ∅ → X, namely the empty one, and exactly one function X → {∗}, namely the constant one. Definition 3.2.8. An object A is called initial if for any object B there exists exactly one arrow A → B. By duality, an object A is called terminal, if for any object B there is exactly one arrow B → A. Initial and terminal objects are unique up to isomorphism, thus it is written 1 for a terminal object and 0 for initial object. Examples 3.2.9. 1. In Set the initial object is an empty set ∅ and the terminal object is any one-element set (singleton) {∗}.14 2. In Grp the trivial group {e} is both initial and terminal. 13
Note that there is at most one such g and g is also iso. Moreover, every arrow that is iso is also monic and epic. This justifies the notation 1, 0 for terminal and initial object, since ∅ and {∗} ∼ = {∅} are conventionally denoted respectively 0, 1 in the class On of ordinals. 14
25
3. In a poset, considered as a category, the least element is initial and the largest element is terminal. Observe now that in Set, functions of the form {∗} → X correspond to elements of X, since for any x ∈ X there is unique function fx : {∗} → X such that f (∗) = x; conversely, any such function fy : {∗} → X gives rise to specific element y ∈ X such that y = f (∗). Definition 3.2.10. Let C be a category with terminal object. A (global) element a of a C-object A is an arrow a : 1 → A. What is more important, such a reformulation gives one the possibility to consider elements B → A of an object A that are not global, in the sense that B 1 (such elements are called generalized or local ). Remark 3.2.11. Let S be a state space of some physical system. Then global elements correspond to points of X, i.e. states in X. Observe that given two collections X, Y ⊆ S of such states, a function f : X → Y might be interpreted as a process (transition) between X and Y , since moving a state x : 1 → X in X by f gives a state f ◦ x : 1 → Y in Y . Remark 3.2.12. The collection of local elements of some object A gives rise to another category, called slice (or comma) category C/A, where C/A-objects are local elements f : B → A of some fixed C-object A and C/A-arrows between local elements f : B → A and g : C → A are C–arrows k : B → C such that g ◦ k = f . It is not manageable to encompass as a whole the impact of the notion of local elements on mathematics; in fact, they are crucial elements of connection between forcing and some special categories, called topoi (see Sec. 3.4). In order to represent reasonings in CT by graphical manipulations, the notion of a diagram is introduced. The fact that such graphical calculus is possible is highly nontrivial and allowed for substantial progress of CT. Definition 3.2.13. A diagram D in a category C is a set of C-objects (called vertices), together with a set of C-arrows between these objects. An example of such a diagram is A
g
B
f h
C A diagram is called finite if it consists of finite number of vertices and finite number of arrows. A path in a diagram D is a sequence hf1 , ..., fn i of arrows of D such that dom (fi+1 ) = cod (fi ) for i = 1, ..., n − 1; then a diagram D is said to commute (or to be commutative) iff for any two paths hf1 , ..., fn i, hg1 , ..., gm i in D such that dom (f1 ) = dom (g1 ), cod (fn ) = cod (gm ) it holds fn ◦ ... ◦ f1 = gm ◦ ... ◦ g1 .
(3.3)
For example, above diagram commutes iff f = h ◦ g. Notice that any category C can be seen as a collection of diagrams (possibly empty, arrowless, etc.) in C. Recall that a substructure of some structure is generally defined as some subset that contains distinguished constants and that is closed under operations, for example a subgroup of a group G is a set H ⊆ G that is itself a group, hence it is closed under group operation, contains neutral element and the inverse for any element of H. 26
Definition 3.2.14. A subcategory of a category C is a diagram D in C such that if D contains a C-object A, it also contains a C-arrow 1A ; if D contains a C-arrow f then it contains dom (f ) and cod (f ) and whenever D contains composable C-arrows f , g, then it also contains a composition g ◦ f. The subcategory D of a category C defined in such way is obviously a category itself; it is called full if D (A, B) = C (A, B) for any two D-objects A, B. Examples 3.2.15.
1. The category Finord is a full subcategory of Set.
2. The category Lat is a subcategory of Pos that is not full (there are arrows in Pos that need not preserve lattice operations). For any two sets X, Y ∈ Ob (Set) it is possible to construct a product X × Y ∈ Ob (Set) f
f
Y equipped with two projections X × Y −−X → X, X × Y −−→ Y , such that for any Z with maps gX gY Z −−→ X, Z −−→ Y there is unique map h : Z → X × Y such that fX ◦ h = gX and gY ◦ h = gY , namely h = hfX , fY i.
Definition 3.2.16. Define a product A × B of C-objects to be a C-object P with a pair of arrows π
π0
π
π0
B B (called projections A ←−A− P −−→ B such that for any C-object C and arrows A ←−A− C −−→ B there exists unique arrow f : C → P such that
C π10
A
π20
f
P
π1
π2
B
commutes. The product and coproduct can be extended to arbitrarily many factors, namely Q let {Ai | i ∈ I} be a set of C-objects.Q A product of such a family is a C-object, denoted by Ai , together with a set of arrows {πi : Ai → Ai | i ∈ I}, such that for any C-object B and any set of Q arrows {fi : B → Ai | i ∈ I} there is a unique arrow f : B → Ai such that a diagram B
f
Q
fi
Ai πi
Ai commutes for any i ∈ I. A coproduct is the dual to product. Examples 3.2.17. 1. In Set the product and coproduct of X, Y are the usual product X × Y and disjoint sum X t Y . 2. In poset P, the product and coproduct are given by an infimum ∧ and a supremum ∨, respectively. A poset which has binary products and coproducts is a lattice (see Appendix A.1).
27
Remark 3.2.18. In classical physics, the joint system of two systems S, T is given by an ordinary product of underlying sets S × T . Thus given states s ∈ T and t ∈ T , the joint state is (s, t) ∈ S × T and conversely, given a state of the joint system (u, w) ∈ S × T , by projections pS : S × T → S, pT : S × T → T one gets the states u ∈ S and w ∈ T by u = pS (u, w) ,
w = pT (u, w) .
(3.4)
However, in QM the situation is different: given two quantum systems S1 , S2 and their Hilbert spaces H1 , H2 , the joint system is no longer described by the cartesian product H1 × H2 , but by a tensor product H1 ⊗ H2 . Since there are no linear mappings p1 : H1 ⊗ H2 → H1 , p2 : H1 ⊗ H2 → H2 that would satisfy (3.4), it is not possible to extract pure states of systems from a pure state of the joint system. Clearly, such a property violates locality, e.g. in EPR setting, where two electrons are given the joint state that can not be separated to independent parts. Also, there is famous result in QM, which is also important in quantum information, that cloning of quantum state is prohibited. Recall that in the setting of classical physics one constructs a unique function ∆ : S → S × S, called diagonal, such that f (s) = (s, s), i.e. the state s is cloned. In the case of QM there is no such diagonal map ∆ : H → H ⊗ H, hence cloning is impossible. Note that both properties are perfectly describable in the category Hilb of Hilbert spaces and unitary operators [2]. 3.2.2
Limits
There exists an object that generalizes many categorical constructions, including initial and terminal objects, products and so on, i.e. a limit. Definition 3.2.19. Let D be a diagram in C with vertices {Di | i ∈ I}. A cone is a family {fi : A → Di | i ∈ I} from some fixed C-object A such that for any d : Di → Dj the diagram A fj
fi
Di
d
Dj
commutes. The object A is then called a vertex of the cone. An arrow from a cone {fi : A → Di | i ∈ I} to a cone {fi0 : A0 → Di | i ∈ I} is a C-arrow g : A → A0 such that the diagram A fi
g
A0
fi0
Di commutes. We say that {fi : A → Di | i ∈ I} factors through {fi0 : A0 → Di | i ∈ I} in case there exists such g : A → A0 . Note that cones over D form a category Con (D) with cones over D as Con (D)-objects and C-arrows that fulfill above diagram as Con (D)-arrows. A limit for the diagram D is a terminal object in the category Con (D), i.e. each cone over D factors uniquely through a limit for D; it is written limD for a limit for D. Dually, define a cocone over a diagram D in a category C to be a cone over D in a category C op . Thus a colimit for a diagram D in C is a limit for D in C op ; it is written colimD for a colimit for D. 28
Examples 3.2.20. Some examples of limits: 1. (terminal and initial object) Consider D as an empty diagram in C; the limit and colimit for D are the terminal and initial objects, respectively; 2. (product and coproduct) Consider the limit for an arrowless diagram A
B
which is the product A × B; dually, the colimit for an above is the coproduct. More generally, Q let D be an arrowless diagram {Ai | i ∈ I}; the limit and colimit for D are the product Ai ` and coproduct Ai , respectively; 3. (equalizer and coequalizer ) Let D be a diagram f
A
B
g
in Set. Define E = {x ∈ A | f (x) = g (x)} and i : E ,→ A. Then i equalizes f and g, i.e. f ◦ i = g ◦ i. It can be now abstracted categorically, thus an equalizer is an arrow e : C → A such that f ◦ e = g ◦ e and for any e0 : C 0 → A such that f ◦ e0 = g ◦ e0 there exists unique h : C 0 → C such that the following diagram f
e
C h
A
g
B
e0
C0 commutes. f
g
In Set, the pullback of X − → Z ← − Y is defined by putting W = {hx, yi | x ∈ X, y ∈ Y and f (x) = g (y)} ⊆ X × Y } and defining f , g to be projections such that f 0 (x, y) = y and g 0 (x, y) = x. W is also denoted A ×Z B. Definition 3.2.21. A pullback (or fibered product) is a limit for the diagram A f
B
g
C
Note that a cone for the above diagram consists of three arrows f 0 , g 0 , i such that D
g0
A i
f0
B
g
29
f
C
commutes. Since f ◦ g 0 = i = f 0 ◦ g, an arrow i can be dropped and hence a pullback for a diagram f
g0
g
f0
A− →C← − B can be defined as a pair of arrows A ← − D −→ B such that f ◦ g 0 = g ◦ f 0 and for any j h pair of arrows A ← −E→ − B with f ◦ h = g ◦ j there exists unique arrow k : E → D such that E k
j
h
D
g0
A
f0
B
f
g
C
commutes. The square f , g, f 0 , g 0 is called a pullback square; f 0 is said to be f pulled back along g. Examples 3.2.22. simply A × B.
f
g
1. If A − →C ← − and C = 1 (C is a terminal object), then the pullback is
2. Inverse image: given a function f : X → Y and W ⊆ Y , the diagram f −1 (W )
X
f | W
W
f
Y
is a pullback in Set and hence an inverse image of W arises by pulling back W along f . Existence of notions like terminal object, products etc. in some category can possibly point out how far from Set is the given one. Definition 3.2.23. A category C is called (finitely) complete if every (finite) diagram in C has a limit; dually, C is called (finitely) cocomplete if every (finite) diagram has a colimit in C. Surprisingly, C is (finitely) complete iff it has (finite) products and equalizers; dually, C is (finitely) cocomplete iff it has (finite) coproducts and coequalizers [4]. Examples 3.2.24.
1. Set, Grp, Top are a (co)complete categories.
2. Finset, Fingrp, Finvect are finitely (co)complete, but not (co)complete. 3. Complete lattices form a complete category that is not cocomplete. 4. Any poset that has no largest element, e.g. On, is not complete as a category since it has no terminal object. Remark 3.2.25. If a category C has terminal object and a pullback for any pair of arrows with common codomain, then it is finitely complete, i.e. the limit for any finite diagram D in finitely complete category C is computable from terminal object and pullbacks [9].
30
Let X, Y ∈ Ob (Set) and Y X ∈ Ob (Set) be the set of all functions from X to Y ; define the evaluation function ev : Y X × X → Y by ev (f, x) = f (x). Taking this concept categorically, we have Definition 3.2.26. A category C has exponentiation if it has binary products and for any C-objects A, B there exists a C-object B A and a C-arrow ev : B A × A → B, called an evaluation arrow, such that for any C-object C and any C-arrow f : C × A → B there exists a unique C-arrow fˇ : C → B A such that BA × A ev fˇ×1A
B f
C ×A commutes. Examples 3.2.27. 1. In Finord, for any m, n an exponential mn is exactly the ordinal that has mn elements. 2. In Heyting algebra H seen as a poset category, for any p, q ∈ H the exponential q p is given by implication p ⇒ q. The above assignment f 7→ fˇ is a bijection between C (C × A, B) and C C, B A , since it is injective by fˇ, gˇ : C → B A and fˇ = gˇ implying f = ev ◦ fˇ × 1A = ev ◦ gˇ × 1A = g, and surjective by taking any h : C → B A and defining g = ev ◦ h × 1A so that h makes above diagram commuting and by uniqueness of gˇ it holds gˇ = h. Thus it is obtained Theorem 3.2.1. C (C × A, B) ∼ = C C, B A .
(3.5)
Observe that in general C (A, B) needs not to be a C-object in some category C. If for any Cobjects A, B the collection C (A, B) is a C-object, then C is said to have exponentiation. Moreover, a finitely complete categories that have an exponentiation constitute an important class of categories called cartesian closed. 3.2.3
Functor categories
Since any group hG, ◦i is itself a category G, a natural and important question arises whether a map between categories could be categorically defined. Definition 3.2.28. A covariant functor F : C → D between categories C, D is a function that assigns to each C-object A a D-object F (A) and to each C-arrow f : A → B a D-arrow F (f ) : f
g
F (A) → F (B) such that if A − →B→ − C, then F (g ◦ f ) = F (g) ◦ F (f ) and F (1A ) = 1F (A) for any C-object A. Functors F : C → Set are called set-valued functors; especially, functors F : C op → Set are called a presheaves on C (see Appendix B.1 for a more detailed discussion on this subject). Examples 3.2.29. Some examples of covariant functors: 1. the identity functor 1C : C → C which has 1C (A) = A and 1C (f ) = f ; 31
2. the forgetful functor is a set-valued functor that ”forgets” the structure of its domain, e.g. U : Top → Set that maps each topological space hX, Oi to its underlying set X and each continuous function f to itself. 3. the power set functor P : Set → Set that maps each set X to its powerset P (X) and each function f : X → Y to the function P (f ) : P (X) → P (Y ) that assigns to each A ⊆ X its f -image f (A) ⊆ Y ; 4. the general linear group functor GLn : cRng → Grp that maps each commutative ring R to the general linear group GLn (R) of non-singular n × n matrices with entries from R; each homomorphism h : R → R0 induces a homomorphism GLn (h) : GLn (R) → GLn (R); 5. the tensor product (bi-)functor Vect → Vect that maps each two vector spaces V , W to the tensor product V ⊗ W and each two homomorphisms h : V1 → W1 , k : V2 → W2 a homomorphism h ⊗ k : V1 ⊗ W1 → V2 ⊗ W2 such that h ⊗ k (v ⊗ w) = h (v) ⊗ k (w); 6. the representation functors G → Set for permutation representation and G → MatR for matrix representation with group operation inducing the composition of permutations or matrix multiplication; 7. given a C-object A, define the hom-functor C (A, −) : C → Set that maps each C-object B to the set C (A, B) and each C-arrow f : B → C to the function C (A, f ) : C (A, B) → C (A, C) such that C (A, f ) (g) = f ◦ g; the hom-functor C (A, −) is also denoted HA . By duality, a contravariant functor F : C → D is the covariant functor from C op to D, hence for contravariant functor F it holds F (g ◦ f ) = F (f ) ◦ F (g) and F (1A ) = 1F (A) , i.e. the contravariant functors preserve the composition in the reversed way.15 Examples 3.2.30. Some examples of contravariant functors: 1. the (−)op functor, that maps each category C to its opposite category C op in such a way that (−)op (A) = A and (−)op (f ) = f op , so it formalizes the operation of getting from C to C op ; 2. the contravariant power set functor P : Set → Set that maps each set X to its powerset P (X) and each function f : X → Y to the function P (f ) : P (Y ) → P (X) that assigns to each S ⊆ Y its inverse image f −1 (S) ⊆ X; 3. let C be the category of finite-dimensional vector spaces over a field K and linear maps; define a functor (−)∗ : C → C op that maps each vector space V to its dual space V ∗ (the space of linear functionals from V to K) and each linear map f : V → W to the linear map f ∗ : W → V defined by f ∗ (g) = g ◦ f ; 4. given a C-object A, define the contravariant hom-functor C (−, A) : C op → Set that maps each C-object B to the set C (B, A) and each C-arrow f : B → C to the function C (f, A) : C (C, A) → C (B, A) such that C (f, A) (g) = g ◦ f ; the hom-functor C (A, −) is also denoted H A. Remark 3.2.31. Functors are important in some approaches to quantum field theories (QFT’s), where one tries to connect in a functorial way structures from different physical theories. 15
Note that it is only a matter of convenience to use contravariant functors, since they can be replaced by covariant functors with suitable change of codomain.
32
Example 3.2.32. 1. Let Mink be a poset category where open subsets of a Minkowski spacetime M are Mink-objects and inclusion maps are Mink-arrows. A functor M : Mink → CStar that satisfies following rules [37]: (a) for any open O ⊆ M, a C ∗ -algebra M (O) is a von Neumann algebra,16 (b) (isotony) O1 ⊆ O2 → M (O1 ) ⊆ M (O2 ), (c) (local commutativity) if O1 is space-like separated from O2 , then M (O1 ) ⊆ M (O2 )0 , !00 S (d) (additivity) ∀ open O0 ⊆ M M (O0 + x) = B (H). x∈R4
is called algebraic (local) quantum field theory (AQFT) [10]. 2. Recall that second quantization is about many-particle systems and involves passage from Hilbert spaces to Fock spaces, which are particular sort of Hilbert spaces. What follows, given Hilbert space H and an operator a : H → H0 , one has symmetric/antisymmetric Fock space Γ (H) and an operator Γ (a) : Γ (H) → Γ (H0 ) given by Γ (H) =
∞ M
H⊗N ,
Γ (a)
N =0
∞ M
ψN =
N =0
∞ M
a⊗N ψN .
(3.6)
N =0
Thus second quantization is a functor Γ : Hilb → Hilb [36]. Thus following quotation after E. Nelson became famous: ”First quantization is a mystery, but second quantization is a functor!” 3. Indeed, there is no functor from the category Symplec of symplectic manifolds and canonical transformations to the category Hilb, that would satisfy some required properties. This might be the explanation for the fact, that quantization is still so mysterious and unnatural operation. However, there are attempts to use this point of view in order to combine QFT and GR. An example is the topological quantum field theory (TQFT), i.e. a functor T : nCob → Vect that takes n-cobordisms (some special, n-dimensional manifolds with boundary representing space) to vector spaces (particularly Hilbert spaces) and (n + 1)-cobordisms (representing spacetime) to homomorphisms (particularly unitary maps) [2]. Then, interesting correspondence is given by n-cobordism (n + 1)-cobordism composition of cobordisms identity cobordism time-reversal
←→ ←→ ←→ ←→ ←→
Hilbert space operator between Hilbert spaces composition of operators identity operator adjoint operator
In the context of Remark 3.2.18, it is worthwhile to mention that the category nCob, just as the category Hilb, is not cartesian, i.e. the product in nCob, given by disjoint union X t Y , does not have the projections of the form (3.4). Generally, categories that have a ”tensor” product instead a cartesian product are called monoidal.17 Then a remarkable conclusion is that categories of GR and QM, (intentionally nCob and Hilb) have much more in common than either of them with the category Set. 0
A von Neumann algebra A is a ∗-subalgebra of B (H) such that (A0 ) = A, where A0 = {a ∈ B (H) | ∀a ∈ A ([a, b] = 0)} is the so-called commutant of A consisting of all operators from B (H) that commute with elements of A. 17 A monoidal category C is the one equipped with the bifunctor ⊗ : C × C → C that is associative and has an identity. Thus monoidal category is a generalization of a monoid (cf. Examples 3.2.2). 16
33
Definition 3.2.33. A functor F : C → D is said to be full (faithful ), when for any C-objects A, B the induced map C (A, B) → D (F (A) , F (B)) is surjective (injective). Moreover, a functor F : C → D is said to be dense if it is surjective on objects, i.e. for any D-objects B there is a C-object A such that F (A) = B. Remark 3.2.34. It is clear that if F is an isomorphism, then it is full, faithfull and dense. Since requiring isomorphism between categories can be too strong demand, one introduces weaker correspondence. Definition 3.2.35. Two categories C, D are said to be equivalent if there exists a full, faithfull and dense functor F : C → D. In particular, if categories C, Dop are equivalent, then C, D are said to be dually equivalent. Example 3.2.36. (Stone duality [16]) Let Ston be a category of compact, Hausdorff and totally disconnected spaces and continuous maps. Then the category Bool is dual to the category Ston. Example 3.2.37. (Gelfand duality [16]) Let kHaus be a category of compact Hausdorff spaces and continuous functions. Then the category cCStar is dual to the category kHaus (for a categorical generalization see Appendix B.2). As could be seen, functors may be thought of as arrows between categories. Remark 3.2.38. There exists a category Cat, whose objects are small categories and arrows are functors between them. However, it is also possible (and in fact very fruitful) to form a category such that its objects are functors. Firstly, it is required to define what an arrow between functors is. Definition 3.2.39. A natural transformation between functors F, G : C → D is a map η : Ob (C) → Arr (D) such that η (A) : F (A) → G (A), also written ηA , is a D-arrow for any C-object A; also for each C-arrow f : A → B the following diagram F (A)
ηA
G (A)
F (f )
G(f )
F (B)
ηB
G (B)
commutes. The arrows ηA are called components of a natural transformation η. If each component ηA is an isomorphism in D, then η is called a natural isomorphism. It is obvious that one can compose natural transformations η, σ between F, G : C → D such that a diagram F (A)
ηA
F (f )
F (B)
G (A)
σA
G(f )
ηB
G (B)
34
H (A) H(f )
σB
H (B)
also commutes, hence σ ◦ η is again a natural transformation with (σ ◦ η)A = σA ◦ ηA . Examples 3.2.40. Some examples of natural transformations: 1. an obvious example is the identity 1F : F → F of F : C → D, which assigns to each C-object a D-arrow 1F (A) : F (A) → F (A); 2. let Inv : CRng → Grp be a functor that assigns to each commutative ring R a group R∗ of its invertible elements and to each ring homomorphism a group homomorphism. Let detR : GLn (R) → R∗ be an arrow in Grp by detR (M ) denoting a determinant of a n × n matrix M with entries in R. Each arrow f : R → R0 leads to a commutative diagram detR
GLn (R) GLn (f )
Inv (R) Inv(f )
GLn (R0 )
detR0
Inv (R0 )
Hence a determinant is a natural transformation. It is now possible to construct categories of functors, since both objects and arrows of such categories are introduced. Definition 3.2.41. A functor category DC is a category with functors F : C → D as DC -objects and natural transformations between them as DC -arrows.18 The identity transformation 1F : F → F plays the role of identity arrow 1F for any DC -object F . Notice that such a category can be seen as a category of diagrams indexed by C-objects. The functors F , G are naturally isomorphic (written F ∼ = G) if there exists a natural isomorphism between F , G. Write Nat (F, G) for a collection of all natural transformations between functors F , G. 3.2.4
Subobjects and subobject classifier
Consider a set X; for any Y ⊆ X there is a canonical injection i : Y ,→ X given by i (y) = y for y ∈ Y . Conversely, any injective (monic) i : Z X gives some subset of X, namely ran (i) ∼ = Z = dom (i). Thus, subsets of X are given by injective (monic) maps with given domain, hence categorically a subobject of a C-object A would be a monic C-arrow f : B A. Now it is clear that the powerset P (A) can be constructed categorically as a collection of all monic arrows with codomain A. Moreover, it can be ordered by inclusion like P (X). Definition 3.2.42. For f : B A, g : C A let f ⊆ g if there exists h : B → C such that the diagram B
f
A
h g
C 18
The base category C is also required to be small due to further results.
35
commutes. It is easy to see that such an inclusion is reflexive and transitive, but f ⊆ g and g ⊆ f give only equivalence f ∼ = g defined by B ∼ = C, not exactly f = g. In order to reconstruct a partial order on subobjects, define for f : B A, g : C A a relation f ∼ = g if there is an isomorphism h : B → C such that h ◦ f = g. Then the order on monics is partial and it is possible to define a categorical version of subsets of a given set. Definition 3.2.43. Let [f ] = {g | g ∼ = f } on above order. A subobject of a C-object A is an equivalence class [f ] = {f | f is monic and cod (f ) = A}; (3.7) furthermore, define a collection of subobjects Sub (A) = {[f ] | f is monic ∧ cod (f ) = A}.
(3.8)
For example, under such identification it holds that Sub (X) ∼ = P (X) in Set. As mentioned in Sec. 2, in Set there is an isomorphism between subsets of a given set and characteristic functions, since for any X ⊆ Y there is unique χX : Y → {0, 1} defined by 1, x ∈ X (3.9) χX (x) = 0, x ∈ /X and conversely, any χX : Y → {0, 1} gives rise to unique subset of Y , namely X = {x | x ∈ Y ∧ χX (x) = 1}. Hence there is an isomorphism P (Y ) ∼ = 2Y . It follows that characteristic functions fit into particular pullback diagram: !
X
{∗}
f
> χX
Y
{0, 1}
Note that {0, 1} plays the role of truth-value set, while {∗} is usually written 1 = {∅} in this context, which follows from that singletons are terminal objects in Set. Categorically, the concept of subobject classifier is constructed as follows. Definition 3.2.44. Let C be a category with terminal object. A subobject classifier (truth-value object) in C is a C-object Ω with monic 1 → Ω called the truth arrow such that for each monic f : A B there is unique arrow χf : B → Ω such that the diagram A
!
f
B
1 >
χf
Ω χg
>
is a pullback and conversely, any diagram of the form C −→ Ω ← − 1 has a pullback. Since subobject classifier Ω acts as truth-value set, it is natural to define a categorical analogue of a powerset P (X) (equivalently the set 2X ). 36
Definition 3.2.45. Let C be a category with a subobject classifier Ω. A power-object of a C-object A is defined to be a C-object ΩA . If for any C-object A the power-object ΩA exists, then C is said to have power-objects. Remark 3.2.46. Clearly, if a category has exponentiation and a subobject classifier then it has power-objects. Also, there is an obvious correspondence between power-object and collection of subobjects [30]. Theorem 3.2.2. Let C be a category with a subobject classifier Ω. Then for any C-object A it holds Sub (A) ∼ = ΩA .
(3.10)
All these informations are sufficient to introduce the notion of a topos.
3.3
Topoi
It was already said that the category Set can be seen as most ”basic”19 category to deal with, i.e. it is complete, cocomplete, and cartesian closed. From the model-theoretic point of view, Set is a model for ZFC. However, category theory gives an opportunity to change a point of view on axiomatics of set theory and give it a more arrow-like shape. Definition 3.3.1 (Lawvere–Tierney). An elementary topos is a category that is finitely complete, has exponentials and subobject classifier. What follows, elementary topoi are also finitely cocomplete. Equivalently, since finite completeness and having exponentials characterize cartesian closedness, another possible definition is Definition 3.3.2. An elementary topos is a category that is cartesian closed and has a subobject classifier. To sum up, it follows that a topos in particular contains 1. initial and terminal objects, hence the analogues of ∅ and {∗} in Set, 2. products and coproducts, hence the analogues of X × Y and X t Y in Set, 3. exponentials, hence the analogues of Y X in Set, 4. subobject classifier, hence the analogue of Boolean algebra {0, 1} in Set, 5. power-object ΩA , hence the analogue of the powerset P (X) in Set. On account of these properties it is clear that a concept of a topos is an attempt to give an axiomatics for set-like universe in purely categorical way. However, the content of the notion of topoi is much greater. As pointed out in [17], topoi exhibit different features dependent on perspective, to name only a few: 1. a topos is a category with finite limits and power-objects, 2. a topos is a generalized space, 3. a topos is a semantics for intuitionistic formal systems, 19
Not to confuse with word ”trivial”, but rather ”the most common”.
37
4. a topos is a category of sheaves on a site. Indeed, it will be shown that topoi satisfy this characteristics. At the moment, consider following examples of topoi. Examples 3.3.3.
1. Set, Finord are topoi.
2. If C is a small category, then the category of presheaves SetC
op
is a topos (see Appendix B.1).
3. If ε is a topos, then for any C-object A comma category C/A is also a topos. It is clear that, by Definition 3.3.1, topoi generalize universes of sets. However, one may wonder what are more precise properties that point out this generality [30]. Theorem 3.3.1. For any object A in a topos E, the poset Sub (A) (equivalently, the power-object P (A)) is a Heyting algebra. By Theorem 3.2.2 it holds Sub (1) ∼ = Ω1 ∼ = Ω,
(3.11)
hence subobject classifier of a topos is also a Heyting algebra. Obviously Ω is equipped with logical binary operators (morphisms) ∨, ∧, ⇒: Ω × Ω → Ω and a negation operator ¬ : Ω → Ω. Since Heyting algebras are Lindenbaum-Tarski algebras for intuitionistic logic (see Appendix A.3), topoi may be seen as models for intuitionistic logic. (Note that in the case of a topos Set the subobject classifier is a Boolean algebra, hence the universum of sets is a model for classical logic.) Actually, the following theorem holds [30]. Theorem 3.3.2. Let E be a topos. The following conditions are equivalent: 1. E is Boolean; 2. the negation operator ¬ : Ω → Ω satisfies ¬¬ = 1Ω ; 3. for any E-object A, the Heyting algebra Sub (A) is Boolean; 4. for every subobject B A in E it holds that ¬B ∨ B = A. Clearly, Set is a Boolean topos. As generalized universe of sets, a topos also might have a natural numbers object. 0
s
Definition 3.3.4. If E is a topos, then E-object N with arrows 1 → − N → − N is called natural a
f
numbers object if for any E-object A and any arrows 1 − →A− → A the following diagram 0
1
N
s
N
a h
A
h f
A
commutes. 0
s
Clearly, the two arrows 1 → − N → − N are the analogues of 0 ∈ N and the succesor function n 7→ n + 1. The natural number object in a topos allows to construct many structures internal to these topoi (cf. Remark 3.3.6). It turns out that categories, especially topoi, naturally arise in model theory. An elementary example of such interaction is the following. 38
Example 3.3.5 (syntactic category). Let L be a (classical or intuitionistic) propositional language and let S be a set of sentences in L. Let CS be a category such that CS -objects are formulas in L, while CS -arrows π : φ → ψ are pairs hP, φi, where P is a proof of ψ from S + φ = S ∪ {φ} (hence identity arrows 1φ : φ → φ are of the form h∅, pi, since a proof of φ from S ∪ {φ} is empty). Clearly the composition of proofs hφ1 , ...φi i ◦ hφi , ...φn i = hφ1 , ..., φn i gives the composition σ ◦ π in CS . Moreover Conversely, any category can be seen as deductive system with objects given by sentences and arrows given by proofs, such that the composition law plays the role of rule of inference. Remark 3.3.6. The very special place of topoi in model-theory is given by the language-topos correspondence. This correspondence concerns higher order (typed) logic, namely the one in which quantification over types (which are interpreted as objects in topoi) is allowed, where objects can be power-objects (and higher powers) obviously. As mentioned in introduction, there are first order theories which are considered in the thesis as a source for various model theoretic results. Nevertheless, it is remarkable that topoi are able to interpret the logic of any order. That is, given any type theory T , there is a syntactic topos ET constructed similarly to Example 3.3.5. Conversely, given a topos E there is a type theory T for which ET is equivalent to E [4]. Moreover, in this way topoi are models for intuitionistic logic, hence, in general, neither law of exluded middle nor AC holds in topoi. Thus all constructions and reasoning in a topos have to be constructive, e.g. they can not use the proofs using reductio ad absurdum and AC. Moreover, the objects constructed in these way are called internal to the topos given.
3.4
Sheaf topos and forcing
As an example of placing topoi in the context of model theory, the connection between sheaf topoi and forcing is shown (following [31] and [30]). For all notions concerning sheaves, see Appendix B.1. Let hC, Ji ≡ C be a site, where C has finite limits. Let X be a sheaf over C, i.e. X ∈ Ob (Sh (C)). Then, since sheaves are special kind of presheaves, the set X (A) is to be regarded as a set varying with the C-object A. Also, say that x ∈ X (A) is an element of X at stage A.
(3.12)
Since Sh (C) is a topos [30], all constructions from Sec. 3.3 are available. In particular, for any collection X, Y, X1 , ..., Xn ∈ Ob (Sh (C)), one can take the powersheaf P (X), the product X1 × ... × Xn and the exponential Y X . Recall that the language consists of properties (relations), functions (operations) and elements (variables or constants) (see Sec. 2.2). Hence, properties P of type X1 × ... × Xn are subsheaves of X1 × ... × Xn , functions f from X1 × ... × Xn to Y are morphisms f
b
X1 × ... × Xn − → Y , while elements b of X are global elements of X, i.e. morphisms 1 → − X. For any sheaf X there are free variables ranging over X. Now, following the reasoning in Sec. 2.2, one builds terms (by operations) and formulas (by logical connectives). Particularly, if t (x1 , ..., xn ) is a term, where x1 , ..., xn range over X1 ×...×Xn , then t (x1 , ..., xn ) is to be interpreted as morphism (natural tranformation) X1 × ... × Xn → Y for some sheaf Y . The crucial part is that the interpretation of terms and formulas is given by the so-called Kripke– Joyal semantics, that is relative to the C-objects. Namely, the morphism X1 ×...×Xn → Y , being an interpretation of a term t (x1 , ..., xn ), is specified by its components tA on C-objects. If a1 ∈ X1 (A), ..., an ∈ Xn (A), then tA (a1 , ..., an ) is defined inductively, as in Sec. 2.2.
39
The most important part is to define the forcing relation. Let φ (x1 , ..., xn ); then, for a1 ∈ X1 (C), ..., an ∈ Xn (A), write A φ (a1 , ..., an ) (3.13) if φ (a1 , ..., an ) (at stage A). A is then said to force φ (a1 , ..., an ) to hold. It follows that the relation corresponds exactly to relation (2.30), since all properties listed in Theorem 2.4.5 hold. In particular, one obtains [30] Theorem 3.4.1. Let X be a sheaf on a site (C, J), let φ, ψ be formulas in the language of the topos Sh (C) and let a ∈ X (A). Then 1. A [φ (a) ∧ ψ (a)] iff A φ (a) and A ψ (a); f
2. A [φ (a) ∨ ψ (a)] iff there is R ∈ J (C) such that for each B − → A ∈ R either B [φ (X (f ) (a))] or B [ψ (X (f ) (a))] holds; f
3. A ¬φ (a) iff for every B − → A, if B [φ (X (f ) (a))] then 0 ∈ J (B); f
4. A [φ (a) → ψ (a)] iff for every B − → A, if B [φ (X (f ) (a))] then B [ψ (X (f ) (a))]; By these facts one can say that forcing is inherent element of sheaf topoi. Also, note that the following holds [24]. Remark 3.4.1. For any complete Boolean algebra B, the Boolean valued model V B is a topos of sheaves of sets on B. This sets a remarkable relationship between MT and CT, since constructing Boolean valued models is of the same nature as constructing sheaves on Boolean algebra B.
40
4
Quantum mechanics
4.1
Preliminaries
QM is a physical theory that describes world in the microscale (. 10−9 m), e.g. nanostructures, atoms, elementary particles etc. It can be also important in cosmological scales, when dealing with highly condensed objects, e.g. black holes, neutron stars etc., but in general quantum effects are negligible small when it comes to classical area & 1 m. The beginning of quantum theory dates back to 1900 when Max Planck guessed the correct formula explaining an experimental black-body radiation curve.20 Then Niels Bohr came up in 1913 with his model of a hydrogen atom, where he introduced discrete electron orbits and quantum jumps. In 1925, while studying hydrogen spectral lines, Werner Heisenberg has set some rules between possible frequencies of emitted radiation; these rules were soon identified with matrix calculus by G¨ottingen group (Born, Jordan), giving rise to so-called matrix mechanics. Whereas Heisenberg’s approach has relied on observables, almost the same time Erwin Schr¨ odinger has invented the state approach to quantum physics by differential operators and their eigenfunctions (so-called wave-functions), which has culminated with one of the most recognizable equations of physics, namely the Schr¨ odinger equation. Later, the squared modulus of wave-function has been interpreted as probability density.21 The equivalence of these two approaches has been shown with use of Hilbert spaces by no other than von Neumann, who also wrote in 1932 the first rigorous textbook on foundations of QM. In fact, the general formalism that is commonly used in QM has not changed since then. As will be shown later, a logical structure of QM is not so well-behaved, hence it is not hard to notice that differences on such fundamental level could make the quest of combining QM and GR a difficult task. The situation goes even more dramatic, if the full list of ”axioms” of QM is given.
4.2
Postulates of QM
Postulates of QM have been brought to life in order to formalize the attribution to physical systems in micro-scale appropriate structures and procedures that provide predictions about these systems [12]. 1. For any quantum system S there is a corresponding complex, separable Hilbert space of states H. 2. For any measured physical quantity (observable) of the quantum system S under measurement there is corresponding self-adjoint operator a : H → H defined on the Hilbert space related to S. 3. Time evolution of a quantum system S is described by one-parameter group of unitary operators defined on the Hilbert space. 4. The result of measurement of an observable a in a state ψ belongs to a set ∆ ⊂ σ(a) with probability paψ (∆) = (ψ, Ea (∆) ψ) , (4.1) where Ea is the spectral measure corresponding to an operator a. 20
However, Planck himself did not aim to prove the existence of quanta; in fact, he attributed the correct quantum interpretation of his formula to Einstein. 21 Again, note that it was not the goal of Schr¨ odinger to interpret his equation probabilistically, since he wanted it to be fully classical.
41
As mentioned earlier, this collection of rules guarantees great precision and agreement with results of measurements. However, despite of pros, one should also be aware of cons of such theory. Remarks 4.2.1. 1. Hilbert space H that is assigned to a quantum system is a collection of vectors, among which there are vectors called pure states. Each such vector ψ corresponds to some one-dimensional subspace of H, namely {λψ | λ ∈ C} and hence ψ corresponds to a projection Pψ defined by Pψ (φ) = (ψ, φ) ψ. However, if some uncertainty of state is introduced (for example, light that goes through horizontal-vertical polarizer), P the state is in pi (·, ψi ) ψi , general represented by a density matrix, i.e. a linear operator ρ of the form ρ = P i where pi is a probability that a system will be in the state ψi ; hence also pi = 1. i
2. Self-adjoint operators from B (H) do not commute in general, which leads to many peculiar features of QM such as uncertainty principle. This non-commutativity is the cornerstone of Kochen–Specker theorem, discussed further. 3. It follows that by Stone theorem22 that there is unique self-adjoint operator that generates the evolution of a quantum system. Through Schr¨odinger equation, this operator is the hamiltonian H of the system. 4. A key role in the postulates plays the last one. Note that it does not claim that a physical quantity possesses some real value; instead, the postulate asserts only that the real emerges as the result of a measurement. Hence the last postulate does not speak about independent reality. In fact, it was one of the main reasons to bring topoi to QM. This will be discussed later in context of Landsman approach.
4.3
Spectral theorem
Definition 4.3.1. The set {Eλ }λ∈R of projections is called a spectral family if V Eλ = 0, 1. λ∈R
2.
W
Eλ = 1,
λ∈R
3. Eλ =
V
Eµ for any λ ∈ R.
µ>λ
Theorem 4.3.1 (Spectral theorem). For any self-adjoint a ∈ B(H) there is a spectral measure E : B (σ (a)) → B (H), defined on Borel σ-algebra of subsets of σ(a), such that Z a = λdEλ . (4.2) In other words, since Eλ is a projection for any λ ∈ σ (a), for any self-adjoint a there exists a spectral family (called a spectral resolution) {Eλ }λ∈R of projections such that (4.2) holds. Thus spectral theorem makes the last postulate of QM meaningful, since for any observable it is possible to calculate probability of obtaining a real value from a given interval. It is interesting that commutativity of self-adjoint operators can be translated to commutativity of their spectral resolutions. Namely, let {Eλ }λ∈R and {Eµ0 }µ∈R be spectral resolutions of selfadjoint operators a and b, respectively. Then a and b are said to commute when Eλ Eµ0 = Eµ0 Eλ for any λ, µ ∈ R.23 22
The Stone theorem gives a 1-1 correspondence between self-adjoint operators and one-parameter groups of unitary operators on Hilbert spaces in the form a 7→ eita . 23 When a, b are bounded, it is reduced to ordinary commutativity ab = ba.
42
The spectral resolutions are crucial in developing the correspondence between self-adjoint operators from B (H) and real numbers in Boolean valued model V B , where B will be a Boolean algebra of projections from L (H). In the end, note that there is another formulation of spectral theorem, involving real measurable functions on spaces L2 (X, µ). Since it is a beautiful example of application of Boolean valued analysis, this version of the theorem is given in Sec. 5.2.
4.4
Classical logic and quantum logic
Before going deeper inside QM and its logic, consider in general how logic in physics arises. Recall from Sec. 2.2 that formulas of a language under given interpretation are associated some truth values. The truth value of a proposition about physical system is to be verified experimentally, hence propositions of particular interest are of the form: ”a quantity F has a value a ∈ R” or, more generally, ”a quantity F has a value inside a measurable set ∆ ⊆ R”.24 Following a spatial logic (see Appendix B.2), recall that a logical structure of classical physics is built upon following ingredients. A state space of the system is a topological space (manifold) X. Define observables to be measurable, real-valued functions on X; then C (X) denotes the algebra of observables with multiplication and addition defined pointwise. Since observables f are identified with physical quantities F , such as momentum or energy, the propositions are of the form ”f has a value inside a measurable set ∆ ⊆ R”, written f ∈ ∆. These are atomic formulas in the language of classical physics. Let F be a set of all atomic formulas (propositions). Say that f ∈ ∆ is true in a state x ∈ X
f (x) ∈ ∆
iff
iff
x ∈ f −1 (∆) ,
(4.3)
hence one can identify the formula f ∈ ∆ and measurable set f −1 (∆) ⊆ X. Analogously to Sec. 2.2, every (pure) state x ∈ X gives a truth-value of propositions under an interpretation h : F → X, where h (f ∈ ∆) ⊆ X is a set of states x ∈ X that satisfy x ∈ f −1 (∆), i.e. that make the proposition f ∈ ∆ true. Then one states (4.3) by f ∈ ∆ is true in a state x ∈ X
iff
x ∈ h (f ∈ ∆)
(4.4)
and h (f ∈ ∆) can be identified with f −1 (∆) . As in Appendix A.3, one can construct Lindenbaum– Tarski algebra of equivalence classes by f ∈∆≈g∈Γ
iff
h (f ∈ ∆) = h (g ∈ Γ) .
(4.5)
Such an algebra corresponds to the Boolean algebra B (X) of measurable subsets of X by an interpretation of logical connectives ∨, ∧, → by set-theoretic operations ∩, ∪, ⊆ in B (X). Moreover, X and ∅ are top 1 and bottom 0 elements in B (X), respectively. With interpretation of ¬ by X \(−) in B (X) it follows that also the law of excluded middle holds. Remark 4.4.1. As summarized in [33], such a formalism is characterized by the following features. For any (f ∈ ∆) , (g ∈ Γ) ∈ F it holds that 1. f ∈ ∆ is true for x ∈ X iff x ∈ h (f ∈ ∆); 2. ¬ is an exclusion negation, i.e. ¬f ∈ ∆ is true in x ∈ X iff f ∈ ∆ is not true in x, i.e. if f ∈ X \ ∆ is true in x; 3. (f ∈ ∆) ∧ (g ∈ Γ) is true in x ∈ X iff f ∈ ∆ is true in x and g ∈ Γ is true in x; 24
One could also take under consideration propositions like: ”the system is in a state x ∈ X”, but such a proposition regarding state is fully characterized by propositions concerning observables and their values on a state x.
43
4. (f ∈ ∆) ∨ (g ∈ Γ) is true in x ∈ X iff f ∈ ∆ is true in x or g ∈ Γ is true in x; 5. there is a standard implication h (f ∈ ∆) ⇒ h (g ∈ Γ) of the form (A.7). In 1936, John von Neumann and Garrett Birkhoff proposed to consider a lattice of projections L (H) on a Hilbert space H as a logical structure, upon which one could logically reason about quantum systems. Unfortunately (though profitably, I believe), the lattice lacks some properties that would allow to speak strictly about quantum logic. Definition 4.4.2. A projection p ∈ B (H) is a self-adjoint and idempotent operator, i.e. p∗ = p2 = p. Remark 4.4.3. Projections can be identified with one-dimensional, closed linear subspaces of H: this is because ran (p) = {ψ | p (ψ) = ψ} is closed linear subspace for any projection p, and conversely, for each closed linear subspace S there is a projection pS defined by condition ran (pS ) = S. One writes 0, 1 for projections on empty subspace and whole H, respectively; also, p⊥ denotes the orthocomplement of p. The lattice L (H) is introduced shortly in the Example A.1.4. By postulates of QM (especially the last one) quantum logic arises as follows in [33] and [7]. Firstly, notice that QM, contrary to classical physics, is essentially a probabilistic theory. It follows from (4.1) that even for pure states the probability is not negligible. Thus, the truth of a proposition of the form a ∈ ∆ in a state ψ can not be reduced to the set {0, 1}, since it necessarily involves also a probability (4.1). For example, in QM it is not said that the value of a physical quantity A lies in interval ∆ ⊆ R; instead, it is said that the value of a physical quantity A lies in interval ∆ ⊆ R with some probability r. For simplicity, take only those propositions a ∈ ∆, which state that a ∈ ∆ holds with probability 1, and let F Q be a set of these propositions. Again, say a ∈ ∆ is true in a state ψ ∈ H
ψ ∈ Ea (∆) H,
iff
(4.6)
i.e. when ψ belongs to closed subspace corresponding to the projection Ea (∆). Analogically, one defines a function hQ : F Q → L (H) on propositions such that hQ (a ∈ ∆) = Ea (∆) .
(4.7)
It is possible to repeat all steps from Sec. 4.1 upon above definitions. In particular, as in the (4.4), an interpretation is given by a state ψ ∈ H. However, the logical structure that arises in the quantum case is different from the classical one, described in Remark 4.4.1. We say that a ∈ ∆ is true in a state ψ ∈ H
iff
while a ∈ ∆ is false in a state ψ ∈ H
iff
ψ ∈ hQ (a ∈ ∆) ,
(4.8)
⊥ ψ ∈ hQ (a ∈ ∆) .
(4.9)
Despite the truth of a proposition is defined in the same way as (4.4), the negation is different. Namely, say that ¬ (a ∈ ∆) is true in a state ψ ∈ H
iff
a ∈ ∆ is false in ψ.
Accordingly 1. (a ∈ ∆) ∧ (b ∈ Γ) is true in ψ ∈ H iff a ∈ ∆ is true in ψ and b ∈ Γ is true in ψ; 44
(4.10)
2. by De Morgan’s law, (a ∈ ∆) ∨ (b ∈ Γ) is true in ψ iff ¬ [¬ (a ∈ ∆) ∧ ¬ (g ∈ Γ)] is true in ψ. Then, as in the classical case (4.3), one can take the equivalence class [a ∈ ∆] to be Ea (∆); then the equivalence classes form a structure similar to the Lindenbaum–Tarski algebra. However, this structure is neither Boolean nor Heyting algebra (see Appendix A.2). Firstly, L (H) is not a distributive lattice (see Example A.1.4), hence there is no way to model a classical or intuitionistic logic by L (H) in the sense of Lindenbaum–Tarski algebra. Secondly, note that (4.10) defines a choice negation, i.e. ¬ (a ∈ ∆) is true whenever a ∈ ∆ is false. Obviously, it is not equivalent to the statement that ¬ (a ∈ ∆) is true whenever a ∈ ∆ is not true. Contrary to exclusion negation, which stated that (see Remark 4.4.1) h (f ∈ ∆) ∪ X \ h (f ∈ ∆) = X,
(4.11)
h (a ∈ ∆) ∪ [h (a ∈ ∆)]⊥ 6= H.25
(4.12)
now it is the case that Hence, it is not true that for any h (a ∈ ∆) and for any state ψ ∈ H either ψ ∈ h (a ∈ ∆) or ψ ∈ [h (a ∈ ∆)]⊥ holds. On the other hand, for any a ∈ ∆ it holds (a ∈ ∆) ∨ [¬ (a ∈ ∆)] = 1,
(4.13)
hence the law of excluded middle in QM is purely syntactical.26 (As will be shown, despite of non-distributivity, it is a deep philosophical reason for concerning topoi in QM.) Thirdly, consider subspaces p = h (a ∈ ∆), q = h (b ∈ Γ) and a subspace p ∨ q = h (a ∈ ∆) ∨ h (b ∈ Γ) spanned by these two. Obviously, by taking ψ = aφ + bχ where φ ∈ p and χ ∈ q, it is possible that p ∨ q is true although neither p not q is true. Fourthly, there is a famous implication problem [7], i.e. there is no implication in L (H). It is interesting to elaborate a little on this point. In Sec. 2.3 an implication → (a syntactic prototype, in fact) plays a crucial role in the rule of inference, called detachment. On the other hand (semantically), an implication ⇒ in a Heyting algebra H is an operator sending two elements a, b to the largest element c such that a ∧ c 6 b. Regarding first point, it is necessary to state what properties an implication is required to have [7]. Definition 4.4.4. A binary logical symbol → is called an implication-connective if for any formulas α, β it holds that 1. α → α is true, 2. if α is true and α → β is true then β is true. In the logic of classical physics, an implication α → β = ¬α ∨ β
(4.14)
satisfies above definition, while in quantum logic it does not. Going to second point, one can show that any lattice with pseudocomplementation (see Appendix A.2) is distributive, hence L (H) can not have an implication with property from Definition A.2.1. Instead, there are various other proposals for the implication operator. The closest to the classical one, called Sasaki arrow, is given by α →S β = α⊥ ∨ (α ∧ β) . (4.15) Obviously, there are some laws that are violated by this operator, e.g. α → (β → α) (see [7], cf. Appendix A.3). 25
It follows from (A.4) that an ordinary sum ∪ is not a supremum ∨ in L (H). A semantic analogue of the excluded middle is called a bivalence principle, which obviously does not hold by (4.12). 26
45
Remark 4.4.5. It is not hard to see where the source of above problems lies. Recall that in classical physics logical operations ∧, ∨, ¬ are interpreted as ordinary set-theoretic operations ∩, ∪, X \ (−). However, due to superposition principle in QM there is deviation from these rules. Namely, one can not interpret ∨ as an ordinary sum ∪, because outcoming subspace is not closed in general. Also, it is apparently plausible to define a negation ¬ by set-theoretic complement H \ (−) to get an exclusion negation. It would cause a semantic bivalence on the one hand; on the other hand, under such reformulation quantum logic becomes incomplete, i.e. for any state ψ ∈ H there is a proposition a ∈ ∆ such that neither ψ ∈ h (a ∈ ∆) nor ψ ∈ H \ h (a ∈ ∆). Above features of the lattice L (H) could cast doubts on adequacy of the Hilbert space formalism to quantum logic. On the one hand, it is clear that L (H) is not a proper ”algebra of logic”, since it is not a Lindenbaum–Tarski algebra of neither intuitionistic nor classical logic (cf. Appendix A.3). On the other hand, it is indicated that one should abandon global Boolean description and possibly ”weaken” logic. This point of view is continued in Sec. 5.1 by use of topoi of presheaves, while the model-theoretic view on this subject is presented in Sec. 5.2.
4.5
Kochen–Specker theorem
In this subsection, an insight into the one of the most essential ingredients of quantum theory, namely the impossibility of assigning real values to physical quantities (which could be understood as a part of realistic description of the world), is given. An importance of this assignment comes from the belief that quantities such as position, momentum, energy etc. should have definite values no matter they are observed (interacted) or not. Suprisingly, QM is not the case. In Sec. 5.1 it is elaborated on the Kochen–Specker theorem in the sense of Landsman approach, which sheds a completely new light on this subject. Definition 4.5.1. A (global) valuation on B (H) is a function λ : B (H) → R that satisfies 1. λ (A) ∈ σ (A), 2. for any pair A, B of self-adjoint operators such that B = h (A) for some h : R → R, there is λ (B) = h (λ (A)). These two conditions ensure that a valuation is a proper tool for assigning real values to physical quantities in the realistic way: although independent from measurement, they should lie in σ (A), i.e. they are among possible values of A, as stated in postulate 4. In addition, the assignment λ should preserve functional structure in B (H) and actually it does. We are ready to formulate the famous Kochen–Specker theorem, developed in 1967 by P. Kochen and E. Specker. Theorem 4.5.1 (Kochen–Specker). If dim (H) > 2, then there exists no valuation on B (H). The conclusion is the following: given quantum system with Hilbert space such that dim (H) > 2, it is impossible to assign real values to whole B (H). Remark 4.5.2. Recall that a (non-contextual)27 hidden-variable theory states that there are quantities of which existence eliminates probability from a description of the system, namely they determinate all values of given physical quantities in the sense of global valuation. What follows, the Kochen–Specker is a no-go theorem, since it rejects such theories; however, one can take into consideration local valuations that give rise to contextual (dependent on measurement) theories. 27
A physical theory is called non-contextual if properties revealed by experiment of a system in this theory are independent from any kind of measurement or interaction.
46
5
Categories and models in QM
Firstly, one should ask the question: do we need in QM the abstract mathematical tools such as model theory or category theory? Is there a reasonable explanation to introduce new entities into description of nature? Note that questions of this kind were always inherent not only to physics, but also to mathematics.28 For almost every new physical theory a new mathematical background has also been required, for example statistical physics requires probability theory, general relativity needs differential geometry, quantum mechanics demands Hilbert spaces and functional analysis etc. On the other hand, the non-euclidean geometries in 19th century were considered as merely abstract, non-physical beings until Einstein and GR. Apart from that, there are several reasons to regard seriously models and categories as necessary tools: 1. they give much more flexibility when thinking of possible environments for some physical theory; however, physicists tend to trivialize the situation by taking under consideration only category Set; 2. there is still lack of theory of quantum gravity (QG), that would coherently combine QM and GR; 3. there are some indications that the problem with QG does not lie in wrong tools of analysis or geometry, but in the use of wrong structures (logic, language) that underlie given physical theories. Remark 5.0.3. As pointed out in [20], one should be aware that topoi do not provide a solution for quantum logic in the sense that there is no topos of which internal logic would be quantum logic. This is due to the fact that for every topos the logic relies on Heyting algebras, which are welldefined Lindenbaum–Tarski algebras. In particular, they are distributive, they have an implication operator etc.
5.1
Presheaves on C ∗ -algebras in QM
The application of topoi of sheaves to QM dates back to the end of 1990’s with the work of C. Isham and J. Butterfield [13]. It was proposed that, in the context of QM, one should consider presheaves over von Neumann algebras from B (H) in order to handle with quantum logic. Later, it was observed by N. P. Landsman et al. that C ∗ -algebras are also suitable for the topos approach to QM [26], [27]. Since these works are precise and systematic, the exposition of application of presheaves to QM in this thesis is based on them. Namely, let be a Hilbert space of a quantum system and A ⊆ B (H) be a C ∗ -algebra of the operators on H. It is argued that the (classical) information about the system is contained in commutative C ∗ -subalgebras of A. As mentioned in [26], such an approach follows Bohr’s view on perception of QM. Indeed, the so-called Bohr’s doctrine states that since experiments necessarily involve classical elements of reality, all we can do is to look at quantum world through ”classical glasses” and to take just classical ”snapshots” of reality. Such a philosophical statement gains a strict mathematical meaning through identification of quantum and classical physics with noncommutative and commutative structures, respectively. This concept, although very apparent and natural, is not so obvious in direct usage. On the other hand, it allows to use a wide variety of techniques, both algebraic and topological, involving also MT and CT. All these tools are bring into play for one main reason: to recognize a proper logical structure in QM that reflects somehow the logical structure of classical physics, i.e. to find 28
One has to admit that mathematics is much more tolerant in such case, though.
47
1. a quantum phase space Σ, 2. a spatial form of propositions, 3. a spatial form of states and observables, Since C ∗ -algebra A in question is in general noncommutative, the search for spatial perspective should make use of some generalized geometry (see Appendix B.2). More precisely, the construction goes as follows. Let A ⊆ B (H) be a unital C ∗ -algebra and let C (A) be a set of unital, commutative C ∗ -subalgebras of A. Then C (A) is a poset, hence a category, under inclusion. Remark 5.1.1. The operation of assigning such poset to any C ∗ -algebra is a functor C : CStar → Pos such that C (A) = {C ⊆ A | C ∈ Ob (cCStar)} C (f ) : C 7→ f (C)
(5.1)
for any CStar-object A and any CStar-arrow f : A → B. The realization of Bohr’s doctrine would be to describe a quantum system, characterized by A, by a poset C (A). This is possible through the notion of a presheaf (see Appendix B.1). Recall that presheaves on any category C (especially on a poset C) can be seen as variable sets indexed by the C-objects. What is needed here is to index sets by classical perspectives from C (A), i.e. to form presheaves C (A) → Set. Given presheaves define a category (topos) T (A) = SetC(A) .
(5.2)
Now, the suggestion is that the system appears to be quantum because an observer is confined to the topos T (A), while the system is in the topos Set. This statement will be justified further. As said in Sec. 3, topoi are categories that allow to reason internally, i.e. they enable all set-theoretic constructions if only provided to be constructive (see Remark 3.3.6). Since T (A) is a topos, above remark applies also to T (A). In particular, provided that a topos T has a natural number object (see Definition 3.3.4), one can define frames, locales, C ∗ -algebras etc. internally. Such internal objects will be underlined in further text. The Bohr’s doctrine finds its incarnation in the following Definition 5.1.2. Let A be a C ∗ -algebra. Define the functor A : C (A) → Set that is forgetful on objects, i.e. A (C) = C (5.3) and determines an inclusion arrow A (f ) : A (C) ,→ A (D) for any C (A)-arrow f : C → D. The functor A plays crucial role, since the following fact holds [26]. Theorem 5.1.1. For any C ∗ -algebra A, the functor A is a commutative C ∗ -algebra in T (A). The process of assigning such commutative C ∗ -algebra A inside T (A) to any C ∗ -algebra A is called Bohrification. Hence, roughly speaking, following Bohr’s doctrine leads to making the quantum (noncommutative) algebra of observables a classical (commutative) one. The strength of this picture relies on naturalness of the construction and further possibilities. Now, given a commutative algebra A, it is possible to construct the Gelfand spectrum of A, since there is a compact, regular locale Σ (A) associated to A. (see Theorem B.2.2). Since locales can 48
be considered as generalized spaces (see Appendix B.2), the following passage is obtained: given a (possibly noncommutative) C ∗ -algebra A, by Bohrification of A one gets a commutative C ∗ -algebra A and through the Gelfand spectrum Σ (A) of A one ends with a locale (space) corresponding to A. Furthermore, one is given whole model-theoretic and categorical machinery associated to locales. Firstly, consider how elements of quantum logic arise in topos-theoretic setting. Definition 5.1.3. Let A be a C ∗ -algebra corresponding to quantum system S. Then a phase space of S is, under above notation, a locale Σ (A). Thus Hilbert space H as phase space is replaced by the locale Σ (A). Since it is possible to apply many topological constructions to locales (see Appendix B.2), they allow to get a more spatial view of quantum logic (cf. Sec. 4.4). Since the locale Σ (A) is defined inside topos T (A), all properties from Sec. 3.3 apply. In particular, both subobject classifier in T (A) and the power-object of Σ (A) are Heyting algebras. Hence, one gets a view on QM by intuitionistic logic. Also, without going into details, it is possible to attach to every self-adjoint operator a a locale map (called after C. Isham and A. D¨ oring a Daseinisation of a) δ (a) : Σ → IR, where IR is a specific poset on R. Then, one has a frame map δ −1 (a) : O (IR) → O (Σ) and propositions a ∈ ∆ (see Sec. 4.4) again correspond to opens O (Σ) in a phase space Σ. In the end, one can give a topos-theoretical version of Kochen–Specker theorem (see Sec. 4.5) [26]. Theorem 5.1.2. Let H be a Hilbert space such that dim (H) > 2. If A is a C ∗ -algebra of bounded operators on H, then the locale Σ (A) has no points. Since points of a locale correspond to models of some propositional theory (see Appendix B.2), the following interesting conclusion holds. Corollary 5.1.4. Under postulates of Theorem 5.1.2, if T is a propositional theory interpreted on the locale Σ (A), then Σ (A) lacks of standard models of T . Then one can see that in the case of classical physics, the locale given by a topological phase X space contains pure states as standard models for classical logic. On the other hand, although the quantum system has pure states, it has no standard models. It can be shown that Theorems 4.5.1 and 5.1.2 are in some sense equivalent. [29] Theorem 5.1.3. Let A be a C ∗ -algebra of bounded operators on a Hilbert space H. Then there is a bijective correspondence between valuations on A and points of Σ (A).
5.2
The topos V B in QM
It was discussed in the Sec. 5.1 that quantum logic, based on lattice L (H), is rather peculiar way to reason about QM, since there are problems with deduction and interpretation of logical operations. However, using methods of Boolean valued models, there are at least two strategies of handling this situation: 1. take whole L (H) and try to build a model V L(H) , 2. pick different (maximal, complete) Boolean algebras B from L (H) and build different Boolean valued models V B .
49
The first point was already discussed by G. Takeuti in [35] and summarized briefly in [32]. Despite of the fact that V L(H) is not a model of ZFC, it is a model of slightly different axiomatics, which can be seen as quantum counterpart of ZFC. Namely, since there are problems with implication in L (H), it is not clear how to define analogues of (2.26) and (2.27), because they involve an implication operator. Nonetheless, one can use the Sasaki implication (see (4.15)) and build relations (2.26) and (2.27) upon it, giving rise to the so-called quantum set theory. Nevertheless, the fact that V L(H) is not a model of ZFC can be seen as a hint towards understanding the nature of quantum world. The second point is also interesting and much easier to apply, since Boolean algebras from L (H) fit well into methods of Boolean valued models. As stated in Sec. 3.4, the Boolean valued model V B is a topos of sheaves over B. Thus all reasonings from Sec. 3.3 applies also to Boolean valued models built upon complete Boolean projection algebras. It is the subject of discussion in Sec. 5.3. The choice of the algebra of projections is strictly related to possible applications of the topos V B in QM. This construction gives a 1 − 1 correspondence between R (real numbers in the universe V ) and self-adjoint operators on H [34]. What is more important, this correspondence might be the next incarnation of relativity principle in QM [8]. 5.2.1
Preliminaries
Definition 5.2.1. A complete Boolean projection algebra B is a set B ⊆ L (H) of pairwise commutting projections, such that 1. 0, 1 ∈ B, 2. if p ∈ B then p⊥ ∈ B, V W pi ∈ B. pi ∈ B and 3. if {pi }i∈I ⊆ B, then i∈I
i∈I
A basic fact about complete Boolean projection algebras is Lemma 5.2.1. For any set A of pairwise commuting projections, there is a complete Boolean algebra that contains A. Recall that every self-adjoint operator a corresponds to some set of projections, i.e. the spectral resolution of a (see Theorem 4.3.1). Definition 5.2.2. Let a be a self-adjoint operator and {pi }i∈I a spectral resolution of a. Then a is said to belong to a complete Boolean projection algebra B if {pi }i∈I ⊆ B. The closure of a complete Boolean projection algebra B is a set B of all self-adjoint operators that belong to B. Clearly, since any p ∈ B is self-adjoint and its spectral resolution is {p, p⊥ } ⊆ B, it holds B ⊆ B. Since any self-adjoint operator is represented by its spectral resolution, from Lemma 5.2.1 one concludes Corollary 5.2.3. For any set A of self-adjoint pairwise commutable operators there exists a complete Boolean projection algebra B such that A ⊆ B. 5.2.2
Model construction and real numbers in V B
Let B be a complete Boolean projection algebra. The construction of Boolean valued model V B goes exactly as in (2.25) in Sec. 2.5, i.e. by taking an empty set ∅ and then form B-valued sets as functions defined on B-valued sets with values in B. It follows from Theorem 2.5.1 that V B is 50
a model of ZFC for any complete Boolean projection algebra B. Comparing it to the statement that V L(H) is not a model of ZFC gives a hypothesis, that complete Boolean projection algebras B (equivalently models V B ) are some kind of ”local descriptions” of a Hilbert space H describing a quantum system, whereas the global description is given by the universe V L(H) of quantum set theory. Now, the point of Takeuti’s work [34] was to develop analysis of set theory in V B . In order to do that, one has to construct natural numbers N in V B firstly, then go on with the construction of integers Z, rational numbers Q, real numbers R (and complex numbers C, in addition). Generally, numbers arise in V B as so-called mixtures. P Definition 5.2.4. Let {ai }i∈I ⊆ B and {ui }i∈I ⊆ V B . The mixture i∈I ai · ui of {ui }i∈I with respect to {ai }i∈I is the element u ∈ V B such that [ dom (u) = dom (ui ) (5.4) and for any z ∈ dom (u) u (z) =
_
(ai ∧ Jz ∈ ui K) .
(5.5)
As a result, a mixture is set of B-valued sets ”mixed” with some probabilities from B. Then, the following question arises: what are naturals, integers, rationals and reals in V B ? Generally, say that u is a natural number in V B , if Ju ∈ ωKB = 1.
(5.6)
Furthermore, any set of numbers in V B is characterized by condition (5.6). Now one may examine when (5.6) is the case. It turns out, that every natural number u in V B is equal to a mixture of some natural numbers from ω, i.e. X u= ni · Pi , (5.7) i
where each ni ∈ ω and {Pi } is a partition of unity. Conversely, any mixture of the form (5.7) gives rise to a natural number in V B . Moreover, the same goes for integers as pairs of naturals and rationals as pairs of integers [34]. In addition, such a definition and rationals P P of naturals, integers ri · Pi0 in V B the ri · Pi and v = yields simple operational rules, e.g. for two rationals u = i
i
following holds: u+v =
X
ri + rj0 Pi · Pj0 ,
i,j
u·v =
X
ri · rj0 Pi · Pj0 .
(5.8)
i,j
However, in the case of real numbers an interesting result is obtained. Namely, recall that real numbers can be defined by Dedekind cuts, i.e. by partitioning rational numbers into two blocks A and B, such that any element of B is greater than every element in A, and A contains no greatest element. Then the rationals are defined by the smallest elements of B. Formally, the sentence ”a is a real number” can be stated as a ⊆ Q ∧ ∃s ∈ Q (s ∈ a) ∧ ∃s ∈ Q (s ∈ / a) ∧ ∀s ∈ Q [s ∈ a ↔ ∀t ∈ Q (s < t → t ∈ a)] .
(5.9)
Then, as in (5.6), one defines real numbers in V B to be those u for which Ju ∈ RKB = 1 51
(5.10)
holds true. The set of real numbers in V B is denoted R(B) . V Pr . Now, let u ∈ R(B) , r ∈ Q and define Pr = Jr ∈ uK. For any real number λ define Eλ = r>λ
Then one can show that {Eλ } is a resolution of the identity in B. Hence for any real u one has a resolution of identity in B. Conversely, given a resolution of identity {Eλ } ⊆ B one can define Pr = Er and define the function u such that dom (u) = {ˇ r | r ∈ Q} and u (ˇ r) = Pr . Then u ∈ R(B) , (B) hence for any resolution of identity in B one has a real number in R [34]. (B) Hence there is a bijective correspondence between real numbers in R and spectral resolutions in B. Note, that there is also a bijective correspondence between spectral resolutions in B and elements of B. Thus one comes to the following conclusion. Corollary 5.2.5. There is bijective correspondence between R(B) and B. In other words, an interpretation of the self-adjoint operator in V B is a real number [5]. In fact, this bijection is an algebra isomorphism, hence Takeuti was able to show that many theorems from analysis transfer to Boolean valued analysis. Remark 5.2.6. It is common practice to quantize by substituting real objects, such as real-valued functions, by operators on some Hilbert space. For example, given a relation between energy and momentum one gets p2 ∂ψ E= ←→ i} = Hψ. (5.11) 2m ∂t Then the conclusion is somewhat similar to the conclusions of Landsman (see Sec. 5.1): world gets quantum by changing the topos that describes it. In the topos of sheaves on C ∗ -algebra, the other context gives a commutative C ∗ -algebra instead of non-commutative one. Here, by passing from Set to the topos of sheaves over complete Boolean projection algebras, self-adjoint operators on the Hilbert space become real numbers, hence description becomes classical. Remark 5.2.7. It is shown in [34] that given a measure space (X, µ), one can also build Boolean valued model V B upon Boolean algebra of measurable subsets of X. Then, analogically, real numbers in V B are in 1 − 1 correspondence with measurable functions on X. Hence one gets a correspondence between self-adjoint operators on some Hilbert space H and measurable functions on some measure space (X, µ). In other words, every self-adjoint operator is unitarily equivalent to a multiplication. This fact is given by following theorem [11]. Theorem 5.2.2. Let H be a separable Hilbert space. If A is a self-adjoint operator on H, then there is a real, measurable function φ on some measure space (X, µ) and a unitary operator U : H → L2 (X, µ) such that U AU −1 f (x) = φ (x) f (x) for each f ∈ L2 (X, µ). Conversely, given any real measurable function φ on a measure space (X, µ), there exists a Hilbert space H and a unitary operator U : L2 (X, µ) → H such that U T U −1 is a self-adjoint operator on H, where T is defined by (T f ) (x) = φ (x) f (x). The point is that it is possible to prove this theorem by use of Boolean valued analysis [34]. Later, M. Davis put a suggestion that Boolean valued models can be important in QM [8]. Namely, such models can serve as possible interpretation of quantum phenomena. The first idea is that, while self-adjoint operators representing observables need not be commutative in general, by picking up complete Boolean algebra B from L (H) one restricts himself to commutative, (classical) perspective. Such a description closely resembles that of special relativity. Indeed, now it is a common knowledge that the results of a measurement of stick’s length may be different due to Lorentz contraction, i.e. the relative motion of the stick and an observer. In the present interpretation of QM one has Boolean frames of references instead of inertial frames of reference. 52
Hence no one is surprised that results of a measurement of electron’s both position and momentum differ from the results of a measurement, when only position is taken under consideration. In Boolean-frame interpretation it is justified by the fact that there is no Boolean algebra that contains both position and momentum operators, since they do not commute. Also, Boolean frames can be connected by Fourier transforms, as are position and momentum operators. The relationship between SR and QM: is summarized in the following table from [8]. inertial frames Lorentz transformations Lorentz group space-time vector xµ Minkowski metric ds2
←→ ←→ ←→ ←→ ←→
Boolean frames Fourier transformations unitary group wave function ψ Hilbert space metric ψ ∗ ψ
The first three rows may indeed reveal some properties of QM, that have been unseen earlier. Apart from the first point, the second point seems to be very interesting. Coming back to the logical structure of QM, one may wonder how to relate experiments with Boolean valuation of propositions. Generally, if V B |= α, i.e. JαKB = 1 for any complete Boolean projection algebra, then α expresses some property of reality. On the other hand, since measurements correspond to projections (during the measurement one projects the state on some subspace of H), they will be described in terms of particular Boolean frames. For example, one takes propositions α, β such that α → β claims about some experiment. The propositions α, β may be of the form ”the counter clicks” and ”the particle hit the screen”, respectively. Then, the proposition α → β obviously holds, i.e. V B |= α → β, in any Boolean frame B, but truth values JαKB and JβKB are dependent on particular Boolean frames. Hence, if one wants to get JβKB = 1, it is necessary to set the experiment in such a way that gives JαKB = 1. 5.2.3
QM paradoxes revisited
A description of QM by Boolean frames of reference gives a very simple and natural explanation of conceptual difficulties in QM, i.e. [8] 1. the two-split experiment, 2. the Schr¨ odinger’s cat, 3. the EPR paradox. As to the first one, recall that the experiment consists of the source (e.g. of electrons), a barrier with two parallel slits and a screen. Now, the peculiarity of this experiment comes from the fact, that the image of particles on the screen forms an interference pattern. However, this pattern is present as long as no detection of particles in the slits is conducted. In particular, this interference pattern is not a combination of patterns obtained by closing separately the first slit and then the second one. The usual explanation of this result is that, while measuring of interference pattern one the screen corresponds to the momentum measurement, the detection of particle in a slit corresponds to the position measurement. Since these observables are complementary, it is impossible to get a knowledge about which slit was passed without disturbing the interference pattern. The obvious Boolean frame explanation is that these measurements, since concerning noncommuting operators, correspond to two different Boolean frames of reference, hence the joint measurement is not possible. As to the second one, a cat is connected with some apparatus that causes his death or survival by the collapse of wave function. Such an apparatus might be any two-stated, mixed system. The conclusion is that, due to measurement, a microscopic collapse causes macroscopic result on a cat. 53
Recall that, by Copenhagen interpretation, before measurement a cat is neither dead nor alive. Again, considering Boolean frames of reference, one is led to conclusion that the state of a cat depends on the Boolean frame of reference, namely the one in which cat is alive and the other, where cat is dead. No collapse of wave function holds; also, there is no splitting of the universe in the sense of Everett interpretation. As to the last one, it concerns entangled states, for example the system of two electrons A and B, moving in opposite directions, such that the total momentum of the system is 0. Then it is obvious that a measurement of the momentum MA of A yields immediately the momentum MB of B such that MA + MB = 0. Thus, after Einstein, the ”spooky action at a distance” occurs, since A and B might be very far away from each other when MA is measured. However, by Boolen-frame formalism, the equality MA + MB = 0 is inferred. Indeed, take pA and pB to be the operators representing momenta MA and MB of A and B, respectively, at some time t. One then finds a Boolean algebra B such that pA , pB ∈ B and it holds V B |= (pA + pB = 0)
(5.12)
Hence, by Theorem 2.5.2 one infers that also MA + MB = 0 holds. The result is obtained due to a choice of the proper Boolean frame B which contains both momenta.
5.3
Forcing in QM
Note that since forcing, intuitionism and sheaves are fundamentally connected, it is clear that one has to wonder what possible applications of forcing to QM are. There are several various propositions in [19] on treating forcing in the context of QM, especially problems with contextuality of Kochen–Specker theorem (see Sec. 4.5). Firstly, the correct description of reality might be that, while doing measurements and interacting with other objects, each object (element of reality) undergoes the model change. Roughly, the quantum system S may be described by some model M and, by interaction with another system T , the system S undergoes model change from M to N . Now, recall that for quantum systems there are no global valuations on B (H) (see Theorem 4.5.1), hence there is ”lack” of values (real numbers) of these valuations. If one admits interpretation of changing the models, the following straightforward (but great) conclusion may hold [19]: Corollary 5.3.1. A measurement performed on quantum system S described in the model M forces M to add some real values to the model. Hence, under such interpretation, forcing is fundamentally connected with a measurement in QM, or, more generally, with every interaction incident. Obviously, an interpretation involving passing between models and forcing should be more obvious in the case of Boolean frames of reference (see Sec. 5.2.2), since Boolean valued models correspond to forcing essentially. However, note that all complete Boolean projection algebras from L (H) are atomic [21], hence by Theorem 2.5.5 the forcing upon such Boolean algebras is trivial. In order to obtain nontrivial forcing one can make the Boolean projection algebras atomless or embed them in some atomless Boolean algebra. However, there has not been any successful realization of this idea so far. On the other hand, it might appear more interesting to consider passages between different models V B . Following [8], one may wonder what maps on the level of models are assigned to unitary operators on the level of Boolean algebras. It might be promising to use tools of CT (e.g. sheaves) to obtain interesting results. There are also other various proposals of application of forcing to physics in general; these are listed in the Sec. 6. 54
6
Conclusions and perspectives
Model-theoretic and categorical description of certain issues related to QM (or quantum physics, in general) guides to many important conclusions. It is also necessary to remind that classical (Tarski) first-order MT and CT find their full power in combining them together. That is, it seems that considering one of them alone leads to the loss of some perspective in the other. Indeed, CT without MT considers categories as particular algebraic structures, without noticing the huge variety of categorical universes for theories. On the other hand, MT without CT is limited to set theory. Nevertheless, both MT and CT provide a very rich basis for exploring mathematics in general. Above remarks on mathematics are underestimated in the case of physics. In fact, physics (particularly QM) is commonly considered to be totally disconnected from the MT and CT issues. It could be justified e.g. in the case of quantum logic of von Neumann (1936), since MT was in its early stage and it was no CT at the time. However, the most fruitful time for MT and CT has started in about 1960’s; by the time, theoretical and mathematical physics has been faced many new and abstract theories. As the result, while CT has been used sometimes, e.g. in the context of string theories, MT has been nearly invisible in physics. As I mentioned in introduction, it might be an effect of the abstractness of MT. Moreover, as stated in [19], MT reasoning is avoided due to assumption of the language (logic) constancy (or transparency). Namely, despite various differences between QM and classical physics, physics in general is formulated as if in the topos Set. Since this category relies on set theory and has a fixed, classical logic, it is not very surprising that QM can not be combined with GR easily, as long as both are not allowed to be expressed entirely in some different category. Accordingly, it was shown in the present thesis that introduction MT and CT machinery into physics (QM) initiates vastness of new methods and ideas. Especially, of great importance are topoi, of which generalization of set theory seems to be an appropriate answer to many difficulties. However, even now the majority of the content of MT an CT is still absent in the mathematical language of physics. Nevertheless, there are some present attempts to apply: 1. synthetic differential geometry (SDG), that places differential geometry in the context of topoi. It is a wonderful return to the ideas of Leibniz and his infinitesimals. There are some works that concern application of SDG in GR (see the Siberian toposes group at http://users.univer.omsk.su/topoi/); moreover, such infinitesimals may arise in the model by the forcing procedure (see [31]); 2. as mentioned in Sec. 5.3, there are attempts to apply forcing to QM in principle, i.e. to prove that forcing not only arises by measurement, but also that it is an inherent operation on models in physics. In particular, there is a proposition to interpret forcing as adding to every point of spacetime a continuum of reals, i.e. a string [22]. Secondly, in the context of Kochen–Specker theorem, forcing may serve for model-theoretic description of getting real numbers as a result of measurement, even if there is no global valuation [19]. 3. using categorical methods one can investigate relationship between various Boolean valued models V B inside V L(H) (see [34]). 4. by correspondence between measurable functions and real numbers in Boolean valued models one can investigate the possibility of formalization functional integrals that occur in Feynman’s path integral approach [23]. 5. divergent expressions in QFT might be finite when formulated inside some toposes [23].
55
A
Model theory
A.1
Lattices and filters
Definition A.1.1. A partially ordered set (poset) is a pair hP, 6i (written simply P if confusion is not likely) such that P is a non-empty set and 6 is a binary relation that is reflexive, antisymmetric and transitive, i.e. 1. ∀p ∈ P (p 6 p); 2. ∀p, q ∈ P [(p 6 q and q 6 p) → p = q]; 3. ∀p, q, r ∈ P [(p 6 q and q 6 r) → p 6 r]. It is written p < q for p 6 q, p 6= q. Definition A.1.2. A lattice is a poset hL, 6i such that for any p, q ∈ L there exist least upper bound (supremum or join) p ∨ q and greatest lower bound (infimum or meet) p ∧ q in L. A lattice L is called 1. bounded if there is a minimal (bottom) element 0 ∈ L such that ∀p ∈ L (0 6 p) and a maximal (top) element 1 ∈ L such that ∀p ∈ L (p 6 1), 2. distributive if p ∨ (q ∧ r) = (p ∨ q) ∧ (p ∨ r) for any p, q, r ∈ L,29 W V 3. complete if every subset X of L has a supremum X and an infimum X, 4. orthocomplemented (an ortholattice) if L is bounded and equipped with an operation ⊥ : L → L that satisfies ⊥ = p, p ∧ p⊥ = 0, p 6 q → q ⊥ 6 p⊥ . (A.1) p⊥ 5. orthomodular if p 6 q → p = q ∧ p ∨ q ⊥ for any p, q ∈ L. Remark A.1.3. Equationally, a bounded lattice L is defined by a ∨ 0 = a, a ∨ a = a, a ∨ b = b ∨ a, (a ∨ b) ∨ c = a ∨ (b ∨ c), (a ∨ b) ∧ b = b,
a ∧ 1 = a, a ∧ a = a, a ∧ b = b ∧ a, (a ∧ b) ∧ c = a ∧ (b ∧ c), (a ∧ b) ∨ b = b
(A.2)
for any a, b, c ∈ L. Example A.1.4. Let L be a set of projections on a Hilbert space H. Then hL, 6i is a poset with p 6 q ↔ ran (p) ⊆ ran (q) ↔ pq = qp = p.
(A.3)
Clearly, this order is reflexive, antisymmetric and transitive. L is bounded with 1 and 0 defined as projections on whole H and ∅, V respectively, since for any p ∈ L it holds that ∅ ⊆ ran (p) ⊆ H. W Define supremum and infimum as follows: _ [ pi = pS , where S = closed linear span of pi , 30 (A.4) i∈I
i∈I
29
Hence also p ∧ (q ∨ r) = (p ∧ q) ∨ (p ∧ r) for any p, q, r ∈ L. One can not take an ordinary sum of closed subspaces to be join, because the outcoming set is not a closed subspace in general. 30
56
^
pi = pS , where S =
i∈I
\
ran (pi ) .
(A.5)
i∈I
If projections p, q ∈ L (H) commute, i.e. pq = qp, the join (A.4) and meet (A.5) are given by p ∨ q = p + q − pq,
p ∧ q = pq.
(A.6)
Notice that p⊥ = 1 − p defines an orthocomplementation in L, hence L is an ortholattice. However, p ∧ (q ∨ r) 6= (p ∧ q) ∨ (p ∧ r) in general, hence L is not distributive. Moreover, L is orthomodular, thus L is complete orthomodular ortholattice denoted by L (H). Definition A.1.5. Let L be a bounded distributive lattice. A filter on L is a set F ⊂ L that is closed upwards and closed with respect to meet, i.e. 1. if p ∈ F and q > p, then q ∈ F , 2. if p, q ∈ F , then p ∧ q ∈ F . An ideal is dual to the filter31 , hence any statement about filters can be dualised to statement about ideals (and vice versa). The closedness of a filter with respect to ordering and being a proper subset imply 1 ∈ F, 0 ∈ / F for any filter F .32 Hence, if p ∈ F , then ¬p ∈ / F .33 Example A.1.6. Any non-zero element p ∈ L generate a particular filter {p}+ = {q | p 6 q}, called principal; it is easy to see that {p}+ is the least filter containing p. Moreover, a filter F is called 1. prime, if ∀p, q ∈ L [p ∨ q ∈ F → (p ∈ F or q ∈ F )], 2. an ultrafilter, if ∀p ∈ L (p ∈ F or ¬p ∈ F ).
A.2
Heyting and Boolean algebras
Definition A.2.1. A Heyting algebra is a bounded lattice H such that for any a, b ∈ H a set {c | c ∈ H and a ∧ c 6 b}
(A.7)
has a largest element, denoted by a ⇒ b; an operator that assigns a ⇒ b to a, b ∈ H is called an implication. Example A.2.2. Let X be a topological space and let O (X) be a collection of open sets in X. Then O (X) is a Heyting algebra with join, meet and A ⇒ B given by sum, intersection and Int AC ∪ B respectively, where AC = X \ A. Moreover, it holds that O (X) is infinitely distributive in the sense of ! [ [ O∩ Oi = (O ∩ Oi ) (A.8) i∈I
i∈I
for any O ∈ O (X) and {Oi }i∈I ⊆ O (X). 31
Given a bounded lattice L, the set I ⊂ L is an ideal if it is closed downwards and closed with respect to join. 1 > p holds for any p ∈ F ; on the other hand, if 0 ∈ F , then F = A. 33 If p, ¬p ∈ F , then p ∧ (¬p) = 0 ∈ F , which is not the case. 32
57
Define the equivalence to be such an operation ⇔ that assigns to each a, b ∈ H an element a ⇔ b = (a ⇒ b) ∧ (b ⇒ a) .
(A.9)
The pseudocomplementation is an operation that sends each a ∈ H to the element a∗ = a ⇒ 0. A pseudocomplement a∗ needs not in general be the complement of a.34 ∀a ∈ H [(a∗ )∗ = a]
∀a ∈ H (a ∨ a∗ = 1) .
iff
(A.10)
Remark A.2.3. Heyting algebras can be defined equivalently by equational characterization of implication. Then a Heyting algebra is a distributive bounded lattice L with binary operation ⇒: L × L → L such that a ⇒ a = 1, a ∧ (a ⇒ b) = a ∧ b, b ∧ (a ⇒ b) = b,
(A.11)
a ⇒ (b ∧ c) = (a ⇒ b) ∧ (a ⇒ c) . for all a, b, c ∈ L. Definition A.2.4. A Boolean algebra is a Heyting algebra that satisfies one (and hence both) of the equalities (A.10); equivalently, a Boolean algebra is an orthocomplemented distributive lattice. Accordingly, a Boolean algebra is completely characterized as algebraic structure with binary operations ∧, ∨, one unary operation ¬ and constants 0, 1 such that a∨b=b∨a (a ∨ b) ∨ c = a ∨ (b ∨ c) (a ∨ b) ∧ b = (a ∧ c) ∨ (b ∧ c) a∨0=a a ∨ (¬a) = 1
a ∧ b = b ∧ a, (a ∧ b) ∧ c = a ∧ (b ∧ c), (a ∧ b) ∨ b = (a ∨ c) ∧ (b ∨ c), a ∧ 1 = a, a ∧ (¬a) = 0.
(A.12)
The implication in any Boolean algebra can be defined by x ⇒ y = ¬x ∨ y. Example A.2.5. Let X be a set; then the set P (X) is a Boolean algebra with meet, join and complementation given by ∩, ∪ and (−)C , respectively. Remark A.2.6. Every Boolean algebra B is a commutative ring hB, ⊗, ⊕i with p ⊕ q = (p ∧ (¬q)) ∨ (q ∧ (¬p)) , p ⊗ q = p ∧ q,
(A.13)
∀p (p ⊗ p = p) . By (A.12) every Boolean algebra B is a poset hB \ {0}, 6i such that p 6 q iff p ∧ q = p (equivalently p ∨ q = q). Definition A.2.7. Let B be a Boolean algebra. An element p > 0 is called an atom if ∀q ∈ B [q 6 p → (q = 0 or q = p)] .
(A.14)
If B contains at least one atom, it is called atomic. A Boolean algebra that contains no atom is called atomless (more generally, one defines an atom of a poset in analogous way). 34
For example, consider O (X) and note that a pseudocomplementation Int (X \ −) by taking an interior of settheoretic complement does not fulfill O ∪ Int (X \ O) 6= X in general.
58
Example A.2.8. The Boolean algebra P (X) is atomic, since one-elements sets are atoms in P (X). The following definitions are important in forcing procedure. Definition A.2.9. Let B be a Boolean algebra. The set A ⊆ B is an antichain in B if a ∧W b = 0 for any a, b ∈ A such that a 6= b. A partition of unity in B is an antichain A in B such that A = 1. Definition A.2.10. A a Boolean algebra B is said to satisfy countable chain condition (ccc) if every antichain in B is countable. In the following subsection it is explained why above algebras can be regarded as algebras of logic.
A.3
Logic
Definition A.3.1. Intuitionistic logic is characterized by the list of axioms: 1. φ → φ ∧ φ, 2. (φ ∧ ψ) → (ψ ∧ φ), 3. (φ → ψ) → [(φ → χ) → (ψ → χ)], 4. [(φ → ψ) ∧ (ψ → χ)] → (φ → χ), 5. ψ → (φ → ψ), 6. φ → (φ ∨ ψ), 7. (φ ∨ ψ) → (ψ ∨ φ), 8. [(φ → χ) ∧ (ψ → χ)] → [(φ ∨ ψ) → χ], 9. ¬φ → (φ → ψ), 10. [(φ → ψ) ∧ (ψ → ¬ψ)] → ¬φ and the single rule of inference φ, φ → ψ ψ
(A.15)
called detachment (modus ponendo ponens). Definition A.3.2. Classical logic arises by adding to axioms of intuitionistic logic a single rule of inference to above list: ¬¬φ . (A.16) φ The difference can be illustrated in the fact that the law of excluded middle φ ∨ ¬φ is not derivable in intuitionistic logic. What follows, given a consistent theory T , define the relation ≈ on T such that φ ≈ ψ iff T ` (φ ↔ ψ).35 It is easy to see that ≈ is an equivalence relation. Let H (T ) denote the quotient Fm /≈ and [φ] 6 [ψ] iff T ` φ → ψ; surprisingly, if a theory T is given in intuitionistic language (i.e. axioms of logic are intuitionistic), then hH (T ) , 6i is a Heyting algebra. 35
φ ↔ ψ is defined by φ → ψ ∧ ψ → φ.
59
However, if (A.16) is added to the axioms of logic resulting in classical logic, the quotient algebra hH (T ) , 6i is Boolean (written hB (T ) , 6i) and is called Lindenbaum–Tarski algebra. This conclusion shows why Heyting and Boolean algebras are called algebras of logic, since properties of logical operations correspond exactly to operations on the algebras, while elements of the algebras are to be interpreted as truth-values. Thus Heyting algebras are considered to be algebras of intuitionistic logic, while Boolean algebras are algebras of classical logic.
A.4
Set theory
For the completeness of an exposure, ZFC axioms are given below: 1. (extensionality) ∀x ∀y [∀z (z ∈ x ↔ z ∈ y) → x = y], 2. (separation) ∀u ∃v ∀x [x ∈ v ↔ x ∈ u ∧ ϕ (x)], 3. (replacement) ∀u [∀x ∈ u ∃y ϕ (x, y) → ∃v ∀x ∈ u ∃y ∈ v ϕ (x, y)], 4. (union) ∀u ∃v ∀x [x ∈ v ↔ ∃y ∈ u (x ∈ y)], 5. (power set) ∀u ∃v ∀x [x ∈ v ↔ ∀y ∈ x (y ∈ u)], 6. (infinity) ∃u [∅ ∈ u ∧ ∀x ∈ u ∃y ∈ u (x ∈ y)], 7. (foundation) ∀x [∀y ∈ x ϕ (y) → ϕ (x)] → ∀xϕ (x), 8. (choice) ∀u ∃f [fun (f ) ∧ dom (f ) = u ∧ ∀x ∈ u [x 6= ∅ → f (x) ∈ x]]. Consider now an example of application of forcing to show an independence of CH from ZFC. Although it will be only shown that there exists a model of ZFC+¬CH, the method of obtaining a model of ZFC+CH is quite similar to the presented one, hence CH must be independent from ZFC. Example A.4.1. Start with countable, transitive model M of ZFC. Recall that CH states that | R | = ℵ1 , i.e. P (ω) = ℵ1 , so there is no cardinality between ℵ0 and | R | . It is shown that there is a generic extension M [G] such that | R | > ℵ1 , so that M [G] is a model of ZFC+¬CH. Let P be the set of all functions p such that 1. dom (p) is a finite subset of ω2M × ω, 2. ran (p) = {0, 1}. Then P is a poset with p 6 q iff q ⊆ p, i.e. iff p extends q. Lemma A.4.1. Let P be the poset from the Example A.4.1. Then P is atomless and separative. Proof. It is obvious that any finite partial function f can be extended to another function g by adding 0 or 1 to the array of values, i.e. by adding a pair (x, 0) or (x, 1) such that x ∈ / dom (f ); then f ⊆ g and g 6 f . To show the separativity, observe that if f g, then g f , hence f is not an extension of g. It it now necessary to find h such that h < f and f ⊥ g. Note that there is a location in the array that g admits and f does not; switching this value and agreeing everywhere else with f gives rise to h; thus h properly extends f and has no common lower bound with g, hence h < f and f ⊥ g. S Let G be a generic ultrafilter. Define f = G; then 60
1. f is a function, since f is a set of ordered pairs and G is a filter, hence for any x ∈ ω2M × ω there is at most one y ∈ {0, 1} such that (x, y) ∈ f (for if (x, y1 ) , (x, y2 ) ∈ f with y1 6= y2 , there would be p, q ∈ G such that p (x) = y1 and q (x) = y2 , thus no r ∈ G such that p ⊆ r, q ⊆ r, i.e. r 6 p, r 6 q, which gives a contradiction, because G is a filter), 2. dom (f ) = ω2M × ω; the sets Dx,n = {p ∈ P | (x, n) ∈ dom (p)} are dense, since it is always possible to extend element of Dx,n by extending its domain; G is generic, hence G ∩ Dx,n 6= ∅ and (x, n) ∈ dom (f ) ∈ ω2M × ω. Let fα : ω → {0, 1} be defined by fx (n) = f (x, n). To show that for any x 6= y it holds fx 6= fy , suppose given a function F : ω → {0, 1} and define DF,x = {p ∈ P | ∃n (p (x, n) 6= F (x))}, which is dense, since any p can be extended to differ from F in some point. Hence G ∩ DF,x 6= ∅ and fx 6= F for any x ∈ ω2M , for if fx = F , then G ∩ DF,x = ∅. Thus, x 7→ fx gives a 1 − 1 correspondence between ω2M and P (ω) = {0, 1}ω . Observe that every fx corresponds to some subset of ω; these subsets are called Cohen generic reals. Thus we see it has been proven that there is bijection between ω2M and P (ω). However, it is very important to remember that cardinalities can be relative to the particular model, so it must be checked that M[G] ℵM ; in fact, it is a property of the poset P (Boolean algebra B (P ) that gives this equality, 2 = ℵ2 namely that P fulfills the countable chain condition (ccc). In fact, the following lemma holds [5]. Lemma A.4.2. If P satisfies ccc, then M and M [G] have the same cardinals, i.e. ˇ α = ℵα . V B |= ℵ
(A.17)
Since it can be shown that the poset P of partial functions satisfies ccc [25], it follows that M [G] |= ZF C + ¬CH.
A.5
Ordinals and cardinals
In the Sec. 2 there is a lot about the notion of infinity. One may like to formalize statements like ”there is more reals that naturals” or ”infinities are ordered”, since such problems were always present in philosophy of mathematics. A linear (total) order 6 on a set P is the partial order 6 on P such that any two elements of P are comparable, i.e. ∀p, q ∈ P (p 6 q ∨ q 6 p). A well order 6 on a set P is a total irreflexive36 order 6 on P such that any non-empty subset of P has the least element, i.e. ∀X ⊆ P ∃x ∈ X ∀y ∈ X (x 6 y). A set α is called transitive if all elements of its elements are themselves its elements, i.e. x ∈ y ∈ α implies x ∈ α. Then a set α is called an ordinal if it is transitive and well-ordered by ∈; ordinals are denoted α, β, γ, ... . Consider now the set of natural numbers ω = {0, 1, 2, 3, ...}, where for every n define a successor S (n) = n ∪ {n}, starting with 0 = ∅, 1 = {0} = {∅}, 2 = {0, 1} = {∅, {∅}} and so on. Hence every natural number n = {1, 2, ..., n − 1} is an ordinal. It turns out that all ordinals do not form a set; instead, they form a proper class denoted On. The class On is also well-ordered by ∈, that is 1. ∈ is transitive, i.e. ∀α, β, γ ∈ On (α ∈ β ∧ β ∈ γ → α ∈ γ), 2. ∈ is irreflexive, i.e. ∀α ∈ On (α ∈ / α), 3. ∈ is well founded, i.e. any non-empty subset of On has the ∈-least element. 36
An order 6 on P is irreflexive if ∀x ∈ P (x x).
61
Thus write α < β for α ∈ β and α 6 β for α ∈ β ∨ α = β. The successor operation may be applied generally to ordinals such that if α is an ordinal, then S (α) is also an ordinal. By this notion, one splits the class On into successors, i.e. ordinals β such that there exists an ordinal α and S (α) = β, and limits, i.e. ordinals β such that β 6= 0 and β is not a successor. One also defines a finite ordinal to be an ordinal β such that ∀α 6 β (α = 0 ∨ is a successor). It follows that S ω can be defined as the least limit ordinal. Moreover, if X is a nonempty set of ordinals, then X = sup (X) and T X = inf (X) are also ordinals. To get beyond ω, one may count further and obtains ω + 1, ω + 2, ..., ω + n, ... up to ω + ω = 2 · ω, but addition and multiplication in On needs to be defined. Firstly, for any R that well-orders a set X there is a unique ordinal α such that hX, Ri ∼ = hα, ∈i. Hence it is possible to create sets that have desirable properties of sum and product of ordinals and then assign ordinals (representing sum and product in On) to these sets. Formally, for ordinals α, β let α · β be the unique ordinal γ ∼ = α × β. Analogously, α + β is the unique ordinal γ ∼ = ({0} × α) ∪ ({1} × β); in both cases, ordered pairs are compared using the lexicographic order.37 Operations + and · defined on ordinals give rise to the ordinal arithmetic, which satisfies many intuitive rules of arithmetic of naturals, e.g. associativity, cancellation, subtraction and so on. However, in order to compare a size of (possibly infinite) sets, one has to introduce a notion of cardinality. Two sets X, Y are said to have the same cardinality, written | X | = | Y | , if there exists a bijective map X → Y . Then a relation ≈ defined by X ≈ Y iff | X | = | Y | is an equivalence relation. One can also introduce an order 6 defined by | X | 6 | Y | iff there exists an injective map X → Y . Such an order is obviously reflexive and transitive. By Cantor–Bernstein theorem, it is also antisymmetric.38 By AC, it is fulfilled that any two sets X, Y are comparable, i.e. either X 6 Y or Y 6 X holds. It is written X < Y if X 6 Y and Y X. A set X is said to be countable if | X | 6 ω; otherwise it is uncountable. It turns out that the powerset operation yields larger and larger cardinalities, since it can be proven that | X | < | P (X) | . On the other hand, ordinal arithmetic applied to infinite ordinals, e.g. ω + ω, does not change the cardinality, i.e. | ω + ω | = | ω | . More formally, define a cardinal number to be an ordinal α such that | α | = 6 | β | for all β < α; cardinals are denoted κ, λ, µ, ... . Recall that if a set X is well-ordered by relation R, then there exists an ordinal α that is bijective to X. Hence define | X | to be the least ordinal α such that | X | = | α | . It can be shown that ω together with any n ∈ ω are cardinals and every cardinal κ such that κ > ω is a limit ordinal. Moreover, for any ordinal α there is a cardinal κ such that κ > α; the least such κ is denoted α+ . The infinite ordinal numbers that are cardinals are called alephs. It is possible to define a sequence of alephs by transfinite recursion: ℵ0 = ω, ℵα+1 = ℵ+ α,
(A.18)
ℵα = sup{ωβ | β < α} if α is a limit ordinal. It follows that every infinite cardinal is equal to ℵα for some α and if α < β, then ℵα < ℵβ .
That is hα, βiRhα0 , β 0 i ↔ [α < α0 ∨ (α = α0 ∧ β < β 0 )]. The Cantor–Bernstein (sometimes called Schr¨ oder–Bernstein) theorem says that | X | = | Y | iff | X | 6 | Y | and | Y | 6 | X | . 37
38
62
B
Category theory
B.1
Presheaves, sites and sheaves
The notion of a sheaf comes from topology and, roughly speaking, is about stretching some structure above another, underlying structure. Such a construction, under some conditions, allows to investigate locally the underlying space. Remark B.1.1. The ”original” definition coming from topology is that, considering a topological space X and the algebra of its open subsets O (X), one assigns to each open set U ⊆ X a set of real, continuous functions on U . For V ⊂ U and f : U → R one assigns the restriction arrow of the form f 7→ f |V such that (f S|V ) |W = f |W . Moreover, the so-called gluing condition is satisfied, that is, given any cover U = Ui and continuous functions on Ui there is exactly one continuous i∈I
function on U such that f |Ui = fi and fi (x) = fj (x) for x ∈ Ui ∩ Uj , i, j ∈ I. All these can be characterized categorically. Definition B.1.2. A presheaf on a category C is (generally) a functor C op → D; more commonly, presheaves are considered to be set-valued. op
Presheaves over a category C constitute a category SetC with natural transformations as op arrows. Presheaves are also called varying sets, since SetC can be seen as category of diagrams in Set indexed contravariantly by C-objects. In other words, changing objects A in the category C, the sets (values) of functors change contravariantly with C. This idea, due to W. Lawvere, has allowed to take a different look on mathematics in general, since sets were always seen as constant, rigid objects. Moreover, the important theorem holds [4]. Theorem B.1.1. If C is a small category, then SetC
op
is a topos.
The hom-functors mentioned in Example 3.2.29 give rise to the so-called representable presheaves. Definition B.1.3. A presheaf F : C → Set is said to be covariantly (contravariantly) representable if it is naturally isomorphic to HA : C → Set (H A : C → Set) for some C-object A. Then A is called the representing object for F and if η : HA → F (η : H A → F ) is the above natural isomorphism, then hη, Ai is called a representation of F . Observe that it is possible, as in a general functor category DC , to consider generalized elements of a presheaf F from SetC , namely the natural transformations η : G → F from different presheaves G. On the other hand, for any C-object A a set F (A) can be considered as the set of elements of F at stage A. One can investigate if there is a correspondence between these elements; in fact, hom-functors as stages of generalized elements play crucial role. [4] Lemma B.1.2 (Yoneda). Let C be a category, let F be a presheaf on C and let A be a C-object. Then the map θ : Nat (HA , F ) → F (A) such that θ (η) = ηA (1A ) is a bijection. Hence there is a bijection between generalized elements of F at stage HA and elements of a set F (A). Moreover, if F = HB , then it holds Nat (HA , HB ) ∼ = HB (A) = C (A, B), hence there is a bijective correspondence between natural transformations from HA to HB and arrows between C-objects A, B. There is a categorical way to ”topologize” any category by the notion of sieve. Let C be a category. Consider the collection of C op -arrows SA = {f | cod (f ) = A}. (Note that SA is a collection of objects in comma category C/A.) 63
(B.1)
Definition B.1.4. Let C be a category. A sieve on a C-object A is a collection R of C op -arrows f
g
with codomain A such that B − → A ∈ R and C → − B imply f ◦ g ∈ R, i.e. a sieve on A is a collection R ⊆ SA that is closed under right composition. Then SA is called a maximal sieve on A. Example B.1.5. Consider a poset P seen as a category P. Clearly for any p ∈ P the principal filter Fp = {q | q > p} is a maximal sieve on p, whereas non-maximal sieves on p correspond to filters over p, i.e. sets U ⊂ Fp such that q ∈ U and r > q imply r ∈ U . Definition B.1.6. A Grothendieck topology on a category C is a function J that maps each C-object A to a collection J (A) of sieves on A such that 1. J (A) contains the maximal sieve on A; f
g
2. if R ∈ J (A) and B − → A, then f ∗ (R) = {C → − B | f ◦ g ∈ R} ∈ J (B); f
→ A ∈ R it holds that f ∗ (S) ∈ J (B), 3. if R ∈ J (C) and S is a sieve on A such that for every B − then S ∈ J (A). Definition B.1.7. A site hC, Ji (or simply C) is a category C equipped with a Grothendieck topology J. Given a Grothendieck topology on arbitrary category C with finite limits, it is possible to define sheaves categorically. Definition B.1.8. A sheaf on a site hC, Ji is a presheaf on hC, Ji that satisfies the following condition: for every R ∈ J (A) and for every family of elements {xf | f ∈ R and xf ∈ F (cod (f ))} such that xf ◦g = F (g) (x), there is a unique x ∈ F (C) such that F (f ) (x) = xf for all f ∈ R. Denote the category of sheaves on C by Sh (hC, Ji) (or simply Sh (C)). op
Remark B.1.9. Sh (C) is a full subcategory of SetC .
B.2
Locales
In example A.2.2 it was mentioned that the topology O (X) is an example of a particular kind of lattice. Such lattices are called frames. Definition B.2.1. A frame L is a complete lattice such that for any p ∈ L, {pi }i∈I ⊆ L ! _ _ p∧ pi = (p ∧ pi ) i∈I
(B.2)
i∈I
holds. Equivalently, one defines a frame to be a complete Heyting algebra. [6] Frames and frame homomorphisms (i.e. maps that preserve finite meets and arbitrary joins) constitute a category Frm. Remark B.2.2. Although the topology O (X) is a basic example for a frame, not all frames are of the form O (X). For example, let Oreg (R) be a set of regular open subsets of R, i.e. sets U such C that U C = U , where U C is the interior of R \ U . Then Oreg (R) is a frame that is not of the form O (X) for any topological space X.
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Nevertheless, the notation O (X) for a frame is commonly accepted. Now, the concept of generalized space is the following: if a topological space X is behind the topology O (X), what kind of space can possibly lie behind a general frame O (X)? The answer is purely categorical and leads to the notion of a locale. Note that any Top-arrow f : X → Y gives a Frm-arrow, namely f −1 : O (Y ) → O (X). Then reversing all arrows in Frm gives a desired generalization. Definition B.2.3. The category Loc of locales is defined to be the category Frmop . Generally, a locale associated to the frame O (X) is denoted by Ω (X) or simply X, although again Ω (X) is not necessarily built upon topological space. Moreover, if f −1 : O (Y ) → O (X) is a frame map, then it is written usually f : Ω (X) → Ω (Y ) or simply f : X → Y for the corresponding locale map. Remark B.2.4. Note that there is a contravariant functor Loc : Top → Loc such that Loc (X) = O (X)op .
(B.3)
Furthermore, one can mimic constructions from Top to topologize locales. It can be done by categorical tools in the following way. Definition B.2.5. A point (a global element) of a locale X is a morphism p : {∗} → X, i.e. p : O ({∗}) → O (X) in Frmop , hence a frame map p−1 : O (X) → O ({∗}). Since {∗} is a trivial topological space, i.e. the only open sets in {∗} are {∗} and ∅, it follows that O ({∗}) = {{∗}, ∅} ∼ = {0, 1}, which is isomorphic to Ω in Set. Thus, points of a locale X are given by frame maps O (X) → Ω. Definition B.2.6. An open of a locale X is a frame arrow 1 → O (X). Remark B.2.7. Note that there is a topological space S such that its opens O (S) determine the arrows 1 → O (X). Such a space is called Sierpinski space and it is a two-point set S = {0, 1} such that O (S) = {∅, {1}, S}. Since Frm-arrows are obliged to preserve meets and joins (in particular ∅ 7→ ∅ for any Frm-arrow), the map O (S) → O (X) are is completely determined by its value at {1} ∈ O (S). Then, by {1} ∼ = {∗} ∼ = 1, it follows that an open in a locale X is a locale map X → S. There is also the second possibility to define the opens, namely to take the set Pt (X) of points of a locale and define the opens in Pt (X). Then, the interplay between locales X and Pt (X) reveals important properties of X. Definition B.2.8. An open in Pt (X) is a set Pt (U ) = {p ∈ Pt (X) | p−1 (U ) = 1, U ∈ O (X)}.
(B.4)
By this definition, the set Pt (X) becomes a topological space under topology given above.[30] Remark B.2.9. There exists a functor pt : Loc → Top that maps each locale X to Pt (X). Moreover, for any Loc-arrow f : X → Y there is a Top-arrow pt (f ) : Pt (X) → Pt (Y ) such that pt (p) = f ◦ p. By above notions different classes of locales appear. For example, a locale is said to have enough points (or to be spatial ), if for any U, V ∈ O (X) there is a point p : 1 → X such that p−1 (U ) 6= p−1 (V ), i.e. opens in X are distinguishable by points of X. It holds that X has enough points iff X ∼ = Loc (Y ) for some topological space Y . On the other hand, a topological space X is called sober, if it is homeomorphic to Pt (X) = Pt (Loc (X)). Any Hausdorff space is sober. [30] 65
Remark B.2.10. The notion of spatiality leads to development of pointless (or pointfree) topology, since there are locales that have no (global) points. Apart from that it is very interesting idea in itself, the application of pointless structures in physics is shown in Sec. 5.1. It is interesting to notice following connection between locales and models of theories. Remark B.2.11. It is elaborated in Sec. 4.4 that propositions of a theory can be identified with subsets (possibly open) of a given set, while logical operations correspond to appropriate operations on these subsets. The same goes for locales, which generalize this situation. Recall that sentences of a theory give rise to the Lindenbaum–Tarski algebra, for example for intuitionistic theory T a Lindenbaum–Tarski algebra is a Heyting algebra H (T ) of sentences in T modulo entailment (see Sec. A.2). In case of a propositional theory T , which gives rise to complete Heyting algebra (i.e. a frame) H (T ) ≡ O (T ), points of a locale X correspond to standard models of T by definition: any arrow of the form O (T ) → {0, 1} assigns truth values to propositions from T . As seen above, locales give rise to a true generalization of the concept of space. Furthermore, there are famous connections between algebraic and topological structures. However, classical topology (geometry) reduces to commutative algebra. The idea of noncommutative geometry began with the so-called Gelfand duality [12]. Theorem B.2.1. (Gelfand–Naimark) If A is unital, commutative C ∗ -algebra, then there exists a compact Hausdorff space X such that C (X) is isomorphic to A. Moreover, if A is unital C ∗ -algebra, then there exists a Hilbert space H such that A is isomorphic to the C ∗ -algebra B (H). Firstly, In the commutative case, the isomorphism is given by f : X → C (X) and f −1 : A → Σ (A), where Σ (A) is a (Gelfand) spectrum of A, i.e. a space of nonzero, multiplicative linear functionals φ : A → C (characters of A). As a result, the commutative algebra can be translated to geometry, and vice versa. Then a natural question ”What is the geometry that corresponds to noncommutative algebras?” gives rise to noncommutative geometry. It is not a purpose of this thesis to give a standard description of noncommutative geometry. Instead, in Sec. 5.1 there is given an operation that might be viewed as optional approach to such geometry. Whole construction relies on the following categorical version of Gelfand duality [28]. Theorem B.2.2. Let KRegLoc be a category of compact, regular locales.39 Then categories KRegLoc and cCStar are dually equivalent. Hence one obtains Gelfand duality inside any topos that is able to internalize constructions from above theorem.
39
Without going into details, it is enough to say that the category KRegLoc is equivalent to category KHaus of compact Hausdorff spaces.
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