Function complexes in homotopical algebra

Abstract

1 .l Summary IN [l] QUILLEN introduced the notion of a model category (a category together with three classes of maps: weak equivalences, fibrations and cofibrations, satisfying certain axioms (1.4 (iv))) as a general framework for “doing homotopy theory”. To each model category M there is associated a homotopy category. If W C M denotes the subcategory of the weak equivalences, then this homotopy category is just the localization M[W-‘I, i.e. the category obtained from M by formally inverting the maps of W, and it thus depends only on the weak equivalences and not on the fibrations and the cofibrations. Moreover, if two model categories are connected by a pair of adjoint functors satisfying certain conditions, then their homotopy categories are equivalent. The homotopy category of a model category M does not capture the “higher order information” implicit in M. In the pointed case, however, Quillen was able to recover some of this information by adding some further structure (a loop functor, a suspension functor and fibration and cofibration sequences) to the homotopy category. His fundamental comparison theorem then stated that, if two pointed model categories are connected by a pair of adjoint functors satisfying certain conditions, then their homotopy categories are equivalent in a manner which respects this additional structure. The aim of the present paper is to go back to an arbitrary model category M and construct a simplicial homotopy category which does capture the “higher order information” implicit in M. This simplicial homotopy category is defined as the hummock localization L”(M, W) (for short LHM) of [2]. It is a simplicial category (1.4) with the following basic properties: (i) The simplicial homotopy category LHM depends (by definition) only on the weak equivalences and not on the fibrations and cofibrations. (ii) If two model categories are connected by a pair of adjoint functors satisfying Quillen’s conditions, then their simplicial homotopy categories are weakly equivalent (1.4). (iii) The “category of components” of the simplicial homotopy category of M is just the homotopy category of M. (iv) If M, is a closed simplicial model category [I], then, as one would expect, the full simplicial subcategory M$ C M* generated by the objects which are both cofibrant and jibrant is weakly equivalent (1.4) to LHM. (v) “LHM provides M with function complexes”, i.e. for every two objects X, YE M, the simplicial set LHM(X, Y) has the correct homotopy type for a function complex, in the sense that, for every cosimplicial resolution X* of X and every simplicial resolution Y, of Y (4.31, it has the same homotopy type as diag M(X*, Y*).