Cisinski model structure

Short description: Special kind of model structure

In higher category theory in mathematics, a Cisinski model structure is a special kind of model structure on 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:

$\mathrm{Cofib}={}^{\perp }(I^{\perp });$

$W\cap \mathrm{Cofib}={}^{\perp }(J^{\perp });$

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

$\mathrm{Mono}={}^{\perp }(I^{\perp }).$

Every topos admits a cellular model.

Examples

• Joyal model structure: Cofibrations (monomorphisms) are generated by the boundary inclusions $\partial {\mathrm{\Delta }}^n\hookrightarrow {\mathrm{\Delta }}^n$ and acyclic cofibrations (inner anodyne extensions) are generated by inner horn inclusions ${\mathrm{\Lambda }}_k^n\hookrightarrow {\mathrm{\Delta }}^n$ (with $n\ge 2$ and $0<k<n$).
• Kan–Quillen model structure: Cofibrations (monomorphisms) are generated by the boundary inclusions $\partial {\mathrm{\Delta }}^n\hookrightarrow {\mathrm{\Delta }}^n$ and acyclic cofibrations (anodyne extensions) are generated by horn inclusions ${\mathrm{\Lambda }}_k^n\hookrightarrow {\mathrm{\Delta }}^n$ (with $n\ge 2$ and $0\le k\le n$).

Literature

• Cisinski, Denis-Charles (September 2002). "Théories homotopiques dans les topos" (in fr). Journal of Pure and Applied Algebra 174 (1): 43–82. doi:10.1016/S0022-4049(01)00176-1. 
• Georges Maltsiniotis (2005), "La théorie de l'homotopie de Grothendieck", Astérisque 301, http://www.math.jussieu.fr/~maltsin/ps/prstnew.pdf 
• Denis-Charles Cisinski (2006), "Les préfaisceaux comme modèles des types d'homotopie", Astérisque 308, ISBN 978-2-85629-225-9, http://www.math.univ-toulouse.fr/~dcisinsk/ast.pdf 
• Cisinski, Denis-Charles (2019-06-30) (in en). Higher Categories and Homotopical Algebra. Cambridge University Press. ISBN 978-1108473200. https://cisinski.app.uni-regensburg.de/CatLR.pdf. 

External links

• Cisinksi model structure at the nLab