Les préfaisceaux comme modèles des types d'homotopie
By Denis-Charles Cisinski. Astérisque, 2018
162 citations recorded by the source corpus.
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.
Subfields: Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/b0757b26e7401f6a64a0fa1a1771525226316049
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In Geometry and Topology
In Mathematical Physics
Citations in the corpus
- The Frobenius condition, right properness, and uniform fibrations (2015, methodology): “A naive Cisinski fibration would be defined as a map with the right lifting property with respect to the class I ′⊗ =def {(m ⊗̂ (−)) n(δk ⊗̂ i) | n ∈ N , k ∈ {0 , 1} , i ∈ I} = {m ⊗̂ . . . ⊗̂m ⊗̂ δk ⊗̂ i | k ∈ {0 , 1} , i ∈ I}.”
- La catégorie cubique avec connexions est une catégorie test stricte (2009, unknown): “En réalité, la catégorie des cubes est bien une catégorie test [8], mais elle n’est pas une catégorie test stricte.”