A Cartan-Eilenberg approach to homotopical algebra

Abstract

In this paper we propose an approach to homotopical algebra w here the basic ingredient is a category with two classes of distinguished morphisms: s trong and weak equivalences. These data determine the cofibrant objects by an extension property ana logous to the classical lifting property of projective modules. We define a Cartan-Eilenberg categor y as a category with strong and weak equivalences such that there is an equivalence of categorie s between its localisation with respect to weak equivalences and the relative localisation of the subc ategory of cofibrant objets with respect to strong equivalences. This equivalence of categories allow s us to extend the classical theory of derived additive functors to this non additive setting. The main exa mples include Quillen model categories and categories of functors defined on a category endowed with a cotriple (comonad) and taking values on a category of complexes of an abelian category. In the latt er case there are examples in which the class of strong equivalences is not determined by a homotopy relation. Among other applications of our theory, we establish a very general acyclic models theor emPeer Reviewe