Quillen model structures for relative homological algebra. (English) Zbl 1016.18008

Christensen, J. Daniel; Hovey, Mark

Math. Proc. Camb. Philos. Soc. 133, No. 2, 261-293 (2002).

The paper deals with Quillen’s homotopical algebra. There are many examples of concepts for classical homological algebra to be encompassed by Quillen’s homotopical algebra. The aim of this paper is to give more general concepts of homological algebra that can be “the right thing” for Quillen’s framework.
The main result of the paper is that, under weak hypotheses, the category of chain complexes of objects of a complete and cocomplete abelian category \({\mathcal R}\) has a model category structure that reflects the homological algebra of the projective class. There are also some successful attempts for the possibility of equipping, with a model category structure reflecting a given projective class, the category of simplicial objects in a possible non-abelian category.

Reviewer: Viorel Mihai Gontineac (Iasi)

MSC:

18G55	Nonabelian homotopical algebra (MSC2010)
55P99	Homotopy theory
18G35	Chain complexes (category-theoretic aspects), dg categories
18G25	Relative homological algebra, projective classes (category-theoretic aspects)
18E10	Abelian categories, Grothendieck categories

Keywords:

Quillen’s homotopical algebra; model category structure; projective class; simplicial objects