A COHERENT HOMOTOPY CATEGORY OF 2-TRACK COMMUTATIVE CUBES. K.A. HARDIE, K.H. KAMPS, AND P.J. WITBOOI Abstract. We consider a category H ⊗ (the homotopy category of homotopy squares) whose objects are homotopy commutative squares of spaces and whose morphisms are cubical diagrams subject to a coherent homotopy relation. The main result characterises the isomorphisms of H ⊗ to be the cube morphisms whose forward arrows are homotopy equivalences. As a first application of the new category we give a direct 2-track theoretic definition of the quaternary Toda bracket operation.

Subject classifications : [2000] 18D05, 18B30, 55P10. Keywords : track, semitrack, homotopy 2-groupoid, triple category, homotopy pair, interchange 2-track, Toda bracket. Department of Mathematics and Applied Mathematics, University of Cape Town, 7701 Rondebosch, RSA E-mail address: [email protected] ¨t, Postfach 940, D-58084 Hagen, Germany Fachbereich Mathematik, Fernuniversita E-mail address: [email protected] Department of Mathematics, University of the Western Cape, 7535 Bellville, RSA E-mail address: [email protected]

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