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

##### 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
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