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Cohomological direct images in model categories. (Images directes cohomologiques dans les catégories de modèles.) (English) Zbl 1054.18005
The results of this paper are constructions of homotopy limits. In particular, a closed model category in the sense of Quillen with small projective limits admits arbitrary small homotopy limits. These methods give also a new kind of closed model categories stable under taking functor categories.
The main tool (and motivation) of the paper is a good presentation (with examples) of Grothendieck’s rich notion of derivator (very close to Heller’s homotopy theory) which appears particularly in every model category.
This is the first of three papers talking also about homotopy of small categories (studied with descent properties) and links with derivator structure: an assiociated derivator has a characterization by a universal property.

MSC:
18G55 Nonabelian homotopical algebra (MSC2010)
55U35 Abstract and axiomatic homotopy theory in algebraic topology
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