nLab
model structure on sSet-categories

Contents

Context

Model category theory

model category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of (,1)(\infty,1)-categories

Model structures

for \infty-groupoids

for ∞-groupoids

for rational \infty-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general

specific

for stable/spectrum objects

for (,1)(\infty,1)-categories

for stable (,1)(\infty,1)-categories

for (,1)(\infty,1)-operads

for (n,r)(n,r)-categories

for (,1)(\infty,1)-sheaves / \infty-stacks

(,1)(\infty,1)-Category theory

Contents

Idea

Depending on the chosen model category structure, the category sSet of simplicial sets may model ∞-groupoids (for the standard model structure on simplicial sets) or (∞,1)-categories in the form of quasi-categories (for the Joyal model structure) on SSet.

Accordingly, there are model category structures on sSet-categories that similarly model (n,r)-categories with rr shifted up by 1:

This we discuss below.

Both are special cases of a model structure on enriched categories.

Model for (,1)(\infty,1)-categories

Here we describe the model category structure on SSet Cat that makes it a model for the (∞,1)-category of (∞,1)-categories.

Definition

An sSet-enriched functor F:CDF : C \to D between sSet-categories is called a weak equivalence precisely if

This notion is due to Dwyer & Kan 80 FuncComp, Sec. 2.4 and known as Dwyer-Kan equivalences.

Proposition

A Quillen equivalence DCD \leftrightarrows C between model categories induces a Dwyer-Kan-equivalence LCLDL C \leftrightarrow L D between their simplicial localizations.

This is made explicit in Mazel-Gee 15, p. 17 to follow from Dwyer & Kan 80, Prop. 4.4 with 5.4.

The analogous statement under further passage to Joyal equivalences of quasi-categories is Lurie 09, Cor. A.3.1.12, under the additional assumptions that the model categories are simplicial, that every object of CC is cofibrant and that the right adjoint is an sSet enriched functor.

Proposition

The category SSet Cat of small simplicially enriched categories carries the structure of a model category with

(Bergner 04)

Remark

In particular, the fibrant objects in this structure are the Kan complex-enriched categories, i.e. the strictly ∞-groupoid-enriched ones (see (n,r)-category).

Properties

Proposition

The Bergner model structure of prop. is a proper model category.

A reference for right properness is (Bergner 04, prop. 3.5). A reference for left properness is (Lurie, A.3.2.4) and (Lurie, A.3.2.25).

The entries of the following table display model categories and Quillen equivalences between these that present the (∞,1)-category of (∞,1)-categories (second table), of (∞,1)-operads (third table) and of 𝒪\mathcal{O}-monoidal (∞,1)-categories (fourth table).

general pattern
strict enrichment(∞,1)-category/(∞,1)-operad
\downarrow\downarrow
enriched (∞,1)-category\hookrightarrowinternal (∞,1)-category
(∞,1)Cat
SimplicialCategories-homotopy coherent nerve\toSimplicialSets/quasi-categoriesRelativeSimplicialSets
\downarrowsimplicial nerve\downarrow
SegalCategories\hookrightarrowCompleteSegalSpaces
(∞,1)Operad
SimplicialOperads-homotopy coherent dendroidal nerve\toDendroidalSetsRelativeDendroidalSets
\downarrowdendroidal nerve\downarrow
SegalOperads\hookrightarrowDendroidalCompleteSegalSpaces
𝒪\mathcal{O}Mon(∞,1)Cat
DendroidalCartesianFibrations

References

A model category structure on the category of sSetsSet-categories with a fixed set of objects was first given in

Dywer, Spalinski and later Rezk then pointed out that there ought to exist a model category structure on the collection of all sSetsSet-categories that models the (∞,1)-category of (∞,1)-categories. This was then constructed in

Also:

Survey and review:

See also

Recall the slight but crucial difference between the two notions of “simplicial categories”, the other being an internal category in sSet. But also for this latter concept there is a model category structure which presents (infinity,1)-categories, see

  • Geoffroy Horel, A model structure on internal categories, Theory and Applications of Categories, Vol. 30, 2015, No. 20, pp 704-750 (arXiv:1403.6873).

Last revised on June 12, 2021 at 09:34:52. See the history of this page for a list of all contributions to it.