Dan Christensen and Mark Hovey,

Quillen model structures for relative homological algebra

Published as Math. Proc. Cambridge Philos. 
Soc. 133 Part 2 (2002), 261-293. See last page for
corrections made since publication.

Abstract. 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 gener- ated, 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.