  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, a projective class on a complete and cocomplete
abelian category A is exactly the information needed to do homological algebra
in A. The main result is that, under weak hypotheses, the category of chain
complexes of objects of A has a model category structure that reflects the
homological algebra of the projective class in the sense that it encodes the
Ext groups and more general derived functors. Examples include the "pure
derived category" of a ring R, and derived categories capturing relative
situations, including the projective class for Hochschild homology and
cohomology. We characterize the model structures that are cofibrantly
generated, and show that this fails for many interesting examples. 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, and give examples that include equivariant homotopy theory
and bounded below derived categories.
