# Joyal's CatLab Les préfaisceaux comme type d'homotopie

Les préfaisceaux comme modèles des types d’homotopie.

Denis-Charles Cisinski

Astérisque 308

(pdf)

Abstract: Presheaves as models for homotopy types.

Grothendieck introduced in Pursuing Stacks the notion of test category . These are by definition small categories on which presheaves of sets are models for homotopy types of CW-complexes. A well known example is the category of simplices (the corresponding presheaves are then simplicial sets). Moreover, 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 some localizations of the homotopy category of CW-complexes. This text can be seen as a sequel of Grothendieck’s homotopy theory. We prove 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. Moreover, we show how a local version of the theory allows to consider in a unified setting 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.

Key words: homotopy, model category, presheaf, local test category, homotopy Kan extension, equivariant homotopy theory

Revised on April 11, 2013 at 20:53:43 by Anonymous Coward