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 Astérisque 2007; 392 pp; softcover Number: 308 ISBN-10: 2-85629-225-9 ISBN-13: 978-2-85629-225-9 List Price: US$113 Individual Members: US$101.70 Order Code: AST/308 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. Table of Contents Partie I. Algèbre homotopique des préfaisceaux Constructions de catégories de modèles Yoga simplicial Densité homotopique Partie II. À la poursuite des modèles Correspondances fondamentales Factorisations de foncteurs Changements de base homotopiques Représentations et structures fibrées Partie III. Zoologies Exemples de catégories test locales Exemples de localisateurs fondamentaux Bibliographie Index Index des notations