Cisinski model structure

In higher category theory in mathematics, a Cisinski model structure is a special kind of model structure on Grothendieck topoi. In homotopical algebra, the category of simplicial sets is of particular interest. Cisinski model structures are named after Denis-Charles Cisinski, who introduced them in 2001. His work is based on unfinished ideas presented by Alexander Grothendieck in his script Pursuing Stacks from 1983.

Definition

A cofibrantly generated model structure on a topos, for the cofibrations are exactly the monomorphisms, is called a Cisinski model structure. Cofibrantly generated means that there are small sets $I$ and $J$ of morphisms, on which the small object argument can be applied, so that they generate all cofibrations and trivial cofibrations using the lifting property:

Cofib =^{⊥} (I^{⊥});

W \cap Cofib =^{⊥} (J^{⊥});

More generally, a small set generating the class of monomorphisms of a category of presheaves is called cellular model:

Mono =^{⊥} (I^{⊥}) .

Every topos admits a cellular model.

Examples

Joyal model structure: Cofibrations (monomorphisms) are generated by the boundary inclusions $\partial Δ^{n} ↪ Δ^{n}$ and acyclic cofibrations (inner anodyne extensions) are generated by inner horn inclusions $Λ_{k}^{n} ↪ Δ^{n}$ (with $n \geq 2$ and $0 < k < n$ ).
Kan–Quillen model structure: Cofibrations (monomorphisms) are generated by the boundary inclusions $\partial Δ^{n} ↪ Δ^{n}$ and acyclic cofibrations (anodyne extensions) are generated by horn inclusions $Λ_{k}^{n} ↪ Δ^{n}$ (with $n \geq 2$ and $0 \leq k \leq n$ ).

Literature

Cisinski, Denis-Charles (September 2002). "Théories homotopiques dans les topos". Journal of Pure and Applied Algebra (in French). 174 (1): 43–82. doi:10.1016/S0022-4049(01)00176-1.
Georges Maltsiniotis (2005), "La théorie de l'homotopie de Grothendieck" [Grothendieck's homotopy theory] (PDF), Astérisque, 301, MR 2200690
Denis-Charles Cisinski (2006), "Les préfaisceaux comme modèles des types d'homotopie" (PDF), Astérisque [Presheaves as models for homotopy types], vol. 308, ISBN 978-2-85629-225-9, MR 2294028
Cisinski, Denis-Charles (2019-06-30). Higher Categories and Homotopical Algebra (PDF). Cambridge University Press. ISBN 978-1108473200.