higher category theory In mathematics, higher category theory is the part of category theory at a ''higher order'', which means that some equalities are replaced by explicit morphism, arrows in order to be able to explicitly study the structure behind those equalities. H ...

mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a Cisinski model structure is a special kind of

model structure A model is an informative representation of an object, person, or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin , . Models can be divided into ...

topoi In mathematics, a topos (, ; plural topoi or , or toposes) is a category that behaves like the category of sheaves of sets on a topological space (or more generally, on a site). Topoi behave much like the category of sets and possess a notion ...

. In homotopical algebra, the

category of simplicial sets Category, plural categories, may refer to: General uses *Classification, the general act of allocating things to classes/categories Philosophy *Category of being * ''Categories'' (Aristotle) *Category (Kant) *Categories (Peirce) *Category (Vais ...

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 ''Pursuing Stacks'' () is an influential 1983 mathematical manuscript by Alexander Grothendieck. It consists of a 12-page letter to Daniel Quillen followed by about 600 pages of research notes. The topic of the work is a generalized homotopy the ...

'' 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 In mathematics, in particular in category theory, the lifting property is a property of a pair of morphisms in a category. It is used in homotopy theory within algebraic topology to define properties of morphisms starting from an explicitly give ...

: :

\operatorname
=^\perp(I^\perp);

W\cap\operatorname
=^\perp(J^\perp);

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

\operatorname
=^\perp(I^\perp).

Every topos admits a cellular model.

Examples

Joyal model structure In higher category theory in mathematics, the Joyal model structure is a special model structure on the category of simplicial sets. It consists of three classes of morphisms between simplicial sets called ''fibrations'', ''cofibrations'' and ''wea ...

: 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 In higher category theory, the Kan–Quillen model structure is a special model structure on the category of simplicial sets. It consists of three classes of morphisms between simplicial sets called ''fibrations'', ''cofibrations'' and ''weak equi ...

: 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 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 Cambridge University Press was the university press of the University of Cambridge. Granted a letters patent by King Henry VIII in 1534, it was the oldest university press in the world. Cambridge University Press merged with Cambridge Assessme ...

, isbn=978-1108473200 , location= , language=en , authorlink=

References

External links

* Cisinksi model structure at the ''n''Lab Higher category theory