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.{{Cite web |last=Grothendieck |date= |title=Pursuing Stacks |url=https://thescrivener.github.io/PursuingStacks/ |url-status=live |archive-url=https://web.archive.org/web/20200730015735/https://thescrivener.github.io/PursuingStacks/ps-online.pdf |archive-date=30 Jul 2020 |access-date=2020-09-17 |website=thescrivener.github.io}}

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:Cisinski 2019, 2.4.1.

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

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

More generally, a small set generating the class of monomorphisms of a category of presheaves is called cellular model:Cisinski 2002, Définition 1.28.Cisinski 2019, Definition 2.4.4.

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

Every topos admits a cellular model.Cisinski 2002, Proposition 1.29.

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 $n\geq 2$ and $0). Cisinski 2019, Example 2.4.5. Cisinski 2019, Definition 3.2.1.$
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 $n\geq 2$ and $0\leq k\leq n$ ).

Literature

{{cite journal |last1=Cisinski |first1=Denis-Charles |author-link=Denis-Charles Cisinski |date=September 2002 |title=Théories homotopiques dans les topos |journal=Journal of Pure and Applied Algebra |language=fr |volume=174 |issue=1 |pages=43–82 |doi=10.1016/S0022-4049(01)00176-1}}
{{citation |author=Georges Maltsiniotis |title=La théorie de l'homotopie de Grothendieck |journal=Astérisque |volume=301 |year=2005 |trans-title=Grothendieck's homotopy theory |url=http://www.math.jussieu.fr/~maltsin/ps/prstnew.pdf |mr=2200690}}
{{citation |author=Denis-Charles Cisinski |title=Les préfaisceaux comme modèles des types d'homotopie |journal=Astérisque |volume=308 |year=2006 |trans-title=Presheaves as models for homotopy types |url=http://www.math.univ-toulouse.fr/~dcisinsk/ast.pdf |isbn=978-2-85629-225-9 |mr=2294028}}
{{cite book |last=Cisinski |first=Denis-Charles |author-link=Denis-Charles Cisinski |url=https://cisinski.app.uni-regensburg.de/CatLR.pdf |title=Higher Categories and Homotopical Algebra |date=2019-06-30 |publisher=Cambridge University Press |isbn=978-1108473200 |location= |language=en |authorlink=}}

References

External links

Cisinksi model structure at the nLab

Category:Higher category theory