Derived Categories and Projective Classes

An important example of a model category is the category of unbounded chain complexes of R-modules, which has as its homotopy category the derived category of the ring R. This example shows that traditional homological algebra is encompassed by Quillen’s homotopical algebra. The goal of this paper is to show that more general forms of homological algebra also fit into Quillen’s framework. Specifically, any set of objects in a complete and cocomplete abelian category A generates a projective class on A, which is exactly the information needed to do homological algebra in A. The main result is that if the generating objects are small'' in an appropriate sense, then the category of chain complexes of objects of A has a model category structure which reflects the homological algebra of the projective class. The motivation for this work is the construction of the pure derived category’’ of a ring R. Finally, we explain how the category of simplicial objects in a possibly non-abelian category can be equipped with a model category structure reflecting a given projective class.