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

External links

• Cisinksi model structure at the nLab