nLab
model structure on sSet-categories

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 $r$ shifted up by 1:

• there is a model structure on $SSet$-enriched categories whose fibrant objects are ∞-groupoid/Kan complex-enriched categories and which models (∞,1)-categories;

This we discuss below.

• there is another model structure on sSet-categories whose fibrant objects are (∞,1)-category/quasi-category-enriched categories, and which model (∞,2)-categories.

  For more on this see elsewhere

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

Model for $(\infty,1)$-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 $F : C \to D$ between sSet-categories is called a weak equivalence precisely if

• it is essentially surjective in that the induced functor of homotopy categories is an ordinary essentially surjective functor;

• it is an $\infty$-full and faithful functor in that for all objects $x,y \in C$ the morphism

  $$F_{x,y} : C(x,y) \to D(F(x), F(y))$$

  is a weak equivalence in the standard model structure on simplicial sets.

Such a morphism is also called a Dwyer-Kan weak equivalence after the work by Dwyer-Kan on simplicial localization.

Proposition

A Quillen equivalence $C \stackrel{\leftarrow}{\to} D$ between model categories induces a Dwyer-Kan-equivalence $L C \leftrightarrow L D$ between their simplicial localizations.

Proposition

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

• weak equivalences the Dwyer-Kan equivalences;

• fibrations those sSet-enriched functors $F : C \to D$ such that

  1. for all $x, y \in C$ the morphism $F_{x,y} : C(x,y) \to D(F(x), F(y))$ is a fibration in the standard model structure on simplicial sets;

  2. the induced functor $\pi_0(F) : Ho(C) \to Ho(D)$ on homotopy categories is an isofibration.

(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 right proper model category.

(Bergner 04, prop. 3.5)

Related concepts

• model structure for quasi-categories

• relation between quasi-categories and simplicial categories

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 $\mathcal{O}$-monoidal (∞,1)-categories (fourth table).

	general pattern
strict enrichment		(∞,1)-category/(∞,1)-operad
$\downarrow$		$\downarrow$
enriched (∞,1)-category	$\hookrightarrow$	internal (∞,1)-category
	(∞,1)Cat
SimplicialCategories	$-$homotopy coherent nerve$\to$	SimplicialSets/quasi-categories		RelativeSimplicialSets
$\downarrow$simplicial nerve		$\downarrow$
SegalCategories	$\hookrightarrow$	CompleteSegalSpaces
	(∞,1)Operad
SimplicialOperads	$-$homotopy coherent dendroidal nerve$\to$	DendroidalSets		RelativeDendroidalSets
$\downarrow$dendroidal nerve		$\downarrow$
SegalOperads	$\hookrightarrow$	DendroidalCompleteSegalSpaces
	$\mathcal{O}$Mon(∞,1)Cat
		DendroidalCartesianFibrations

References

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

• William Dwyer, Dan Kan, Simplicial localization of categories , J. Pure and Applied Algebra 17 (3) (1980),

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

• Julie Bergner, A model category structure on the category of simplicial categories , Trans. AMS (arXiv:0406507)

A survey is in section 3 of

• Julie Bergner, A survey of $(\infty,1)$-categories (arXiv:math/0610239)

See also section A.3.2 of

• Jacob Lurie, Higher Topos Theory

and

• Jacob Lurie, $(\infty,2)$-categories and the Goodwillie calculus, GoodwillieI.pdf

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).