Cubical homotopy theory: a beginning

This paper displays a closed model structure for the category of cubical sets and shows that the resulting homotopy category is equivalent to the ordinary homotopy category for topological spaces. The main results are Theorem 19, which gives the model structure, and Theorem 29 and Corollary 30 which together imply the equivalence of homotopy categories. The cofibrations and weak equivalences for the theory are what one might expect, namely levelwise inclusions and maps which induce weak equivalences of topological spaces respectively. The closed model structure is relatively easy to derive, once one gets away from the preconception that fibrations should be defined by analogy with Kan fibrations. A fibration is defined to be a map which has the right lifting property with respect to all trivial cofibrations. The verification of the closed model axioms is essentially formal, and is displayed here (see also [4]) as a consequence of standard tricks from localization theory having to do with a bounded cofibration condition for countable complexes. The equivalence of the homotopy category of cubical complexes with the ordinary homotopy category is much more interesting, and follows from the assertion that the cubical singular functor satisfies excision in a non-abelian sense. There is an underlying category of models, namely the box category , which is used to define cubical sets in the same way that the category of ordinal numbers defines simplicial sets. This means that a cubical set X is defined as a contravariant functor X : op → Set on the box category, taking values in the category of sets. The box category and its basic properties are the subject of the first section of this paper, while the first properties of cubical sets are described in the second section. The closed model structure is derived in Section 3, and appears as Theorem 19. The assertion that the homotopy categories of cubical sets and topological spaces (or simplicial sets) are equivalent involves the final three sections of this paper. One needs a good subdivision operator. There is certainly an obvious subdivision of an n-cube, which is just a product of barycentric subdivisions of