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 1983.¹

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 ${\displaystyle I}$ and ${\displaystyle J}$ of morphisms, on which the small object argument can be applied, so that they generate all cofibrations and trivial cofibrations using the lifting property:²

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

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

More generally, a small set generating the class of monomorphisms of a category of presheaves is called cellular model:³⁴

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

Every topos admits a cellular model.⁵

• Joyal model structure: Cofibrations (monomorphisms) are generated by the boundary inclusions ${\displaystyle \partial \Delta ^{n}\hookrightarrow \Delta ^{n}}$ and acyclic cofibrations (inner anodyne extensions) are generated by inner horn inclusions ${\displaystyle \Lambda _{k}^{n}\hookrightarrow \Delta ^{n}}$ (with ${\displaystyle n\geq 2}$ and ${\displaystyle 0<k<n}$).⁶⁷
• Kan–Quillen model structure: Cofibrations (monomorphisms) are generated by the boundary inclusions ${\displaystyle \partial \Delta ^{n}\hookrightarrow \Delta ^{n}}$ and acyclic cofibrations (anodyne extensions) are generated by horn inclusions ${\displaystyle \Lambda _{k}^{n}\hookrightarrow \Delta ^{n}}$ (with ${\displaystyle n\geq 2}$ and ${\displaystyle 0\leq k\leq n}$).⁶

• 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" [Presheaves as models for homotopy types] (PDF), Astérisque, 308, ISBN 978-2-85629-225-9, MR 2294028{{citation}}: CS1 maint: work parameter with ISBN (link)
• Cisinski, Denis-Charles (2019-06-30). Higher Categories and Homotopical Algebra (PDF). Cambridge University Press. ISBN 978-1108473200.