Presheaves as models for homotopy types. (Les préfaisceaux comme modèles des types d’homotopie.) (French) Zbl 1111.18008

Cisinski, Denis-Charles

Astérisque 308. Paris: Société Mathématique de France (ISBN 978-2-85629-225-9/pbk). xxiv, 392 p. (2006).

A. Grothendieck introduced in “À la poursuite des champs” 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, A. 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 can be seen as a sequel of Grothendieck’s homotopy theory. The author proves in particular two conjectures made by A. 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, he shows how a local version of the theory allows to consider in a unified setting the equivariant homotopy as well. The realization of this program goes through the construction and the study of a model category structure 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.
Chapter 1 is devoted to the study of the structure of Quillen model categories on categories of presheaves whose cofibrations are the monomorphisms. In chapter 2 the author illustrated the techniques introduced in chapter 1, rediscovering the homotopy theory of simplicial sets. In chapter 3, he studies the relation between homotopy limit and fundamental localizer. Chapter 4 is devoted to the proofs of the two Grothendieck’s conjectures mentioned above. In chapter 5 different structures of a Quillen model category on the category \({\mathcal C}at\) of small categories are constructed. In chapter 6 homotopy pullback diagrams in \({\mathcal C}at\) relative to a fundamental accessible localizer are studied. In chapter 7 the equivariant case is developed. In chapter 8, the author produces non trivial examples of test categories having interesting combinatorial properties. In chapter 9 particular fundamental localizers are constructed.

Reviewer: Jean Claude Thomas (Angers)

MSC:

18F20	Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects)
55-02	Research exposition (monographs, survey articles) pertaining to algebraic topology
18G50	Nonabelian homological algebra (category-theoretic aspects)
18G55	Nonabelian homotopical algebra (MSC2010)
18G30	Simplicial sets; simplicial objects in a category (MSC2010)

Keywords:

test category; localizers