**Astérisque**

Volume: 308;
2007;
392 pp;
Softcover

MSC: Primary 55; 18; 54;
**Print ISBN: 978-2-85629-225-9
Product Code: AST/308**

List Price: $113.00

Individual Member Price: $101.70

# Les Préfaisceaux Comme Modèles des Types d’Homotopie

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*Denis-Charles Cisinski*

A publication of the Société Mathématique de France

Grothendieck introduced in *Pursuing Stacks* the notion of
*test categories*. These, by definition, are small categories on which
presheaves of sets are models for homotopy types of CW-complexes. A well-known
example of this is the category of simplices. (The corresponding presheaves are
then simplicial sets.) Furthermore, Grothendieck defined the notion of
*basic localizer*, which gives an axiomatic approach to the homotopy
theory of small categories and gives a natural setting to extend the notion of
test category with respect to some localizations of the homotopy category of
CW-complexes. This text is the sequel to Grothendieck's homotopy theory. The
author proves in particular two conjectures made by Grothendieck: any category
of presheaves on a test category is canonically endowed with a Quillen closed
model category structure, and the smallest basic localizer defines the homotopy
theory of CW-complexes.

The author shows how a local version of the theory allows consideration in a unified setting of the equivariant homotopy theory as well. The realization of this program goes through the construction and the study of model category structures on any category of presheaves on an abstract small category, as well as the study of the homotopy theory of small categories following and completing the contributions of Quillen, Thomason and Grothendieck.

A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list.

#### Readership

Graduate students and research mathematicians interested in homotopy, model category, presheaf, local test category, homotopy Kan extension, or equivariant homotopy theory.