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

(Dwyer-Kan equivalences)
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.

This notion is due to Dwyer & Kan (1980 FuncComp), §2.4 and now known as Dwyer-Kan equivalences.

Proposition

A Quillen equivalence $D \leftrightarrows C$ between model categories induces a Dwyer-Kan-equivalence $L C \leftrightarrow L D$ 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 (2009), Cor. A.3.1.12, under the additional assumptions that the model categories are simplicial, that every object of $C$ is cofibrant and that the right adjoint is an sSet enriched functor.

Proposition

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

• weak equivalences the Dwyer-Kan equivalences (Def. );

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

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

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

[Bergner (2004), Lurie (2009), thm. A.3.2.4]

Remark

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

Proposition

The cofibrations in the Dwyer-Kan-Bergner model structure (Prop. ) are in particular hom-object-wise cofibrations in the classical model structure on simplicial sets, hence hom-object wise monomorphism of simplicial sets:

$$\left. \begin{array}{l} \mathbf{D}, \, \mathbf{C} \,\in\, sSet\text{-}Cat, \\ F \colon \mathbf{D} \underset{\in Cof_{DKB}}{\longrightarrow} \mathbf{C}, \\ X,Y \,\in\, \mathbf{D} \end{array} \;\;\; \right\} \;\;\;\;\;\;\; \vdash \;\;\;\;\;\;\; F_{X,Y} \colon \mathbf{D}(X,Y) \underset{\in Cof_{KQ}}{\hookrightarrow} \mathbf{C}\big(F(X), F(Y)\big) \,.$$

[Lurie (2009), Rem. A.3.2.5]

Properties

Further model 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).

Remark

(failure of cartesian closed model structure)
While the underlying category $sSet Cat$ of sSet-enriched categories is cartesian closed (on general grounds, see here, also cf. Joyal (2008), §51.2), the construction of $sSet$-enriched product categories and $sSet$-enriched functor categories does not make the Bergner model structure of prop. into a monoidal model category hence not into a cartesian closed model category.

This is in contrast to the model structure on quasi-categories (see there) to which the Bergner structure is Quillen equivalent (see below).

Relation to simplicial groupoids

Proposition

The canonical inclusion functor

$$\iota \,\colon\, sSet\text{-}Grpd \xhookrightarrow{\phantom{--}} sSet\text{-}Cat$$

of the category of sSet-enriched groupoids (aka Dwyer-Kan simplicial groupoids) into that of sSet-enriched categories

1. has a left adjoint, given degreewise by the free groupoid-construction (localization at the class of all morphisms)

2. evidently preserves fibrations and weak equivalences between the above Bergner-model structure (Prop. ) and the Dwyer-Kan model structure on simplicial groupoids

hence we have a Quillen adjunction:

$$sSet\text{-}Grpd_{DK} \underoverset {\underset{\iota}{\hookrightarrow}} {\overset{F}{\longleftarrow}} {\;\;\; \bot_{\mathrm{Qu}} \;\;\;} sSet\text{-}Cat_{Berg} \,.$$

(see also Minichiello, Rivera & Zeinalian (2023), Prop. 2.8)

Relation to quasi-categories

The operations of forming homotopy coherent nerves, $N$, and of rigidification of quasi-categories, $\mathfrak{C}$, constitute a Quillen equivalence between the Bergner model structure on $sSet Cat$ (Prop. ) and the model structure for quasi-categories.

For more see at relation between sSet-enriched categories and quasi-categories.

Related concepts

• model structure on simplicial groupoids

• model structure for quasi-categories

• relation between quasi-categories and simplicial categories

• simplicial localization

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 collection of objects was first given in:

• William Dwyer, Daniel Kan, §7.1 in: Simplicial localizations of categories, J. Pure Appl. Algebra 17 3 (1980), 267-284 [doi:10.1016/0022-4049(80)90049-3]

The notion of what is now called Dwyer-Kan equivalences appears in

• William Dwyer, Daniel Kan, §2.4 in: Function complexes in homotopical algebra, Topology 19 (1980), 427-440 [doi:10.1016/0040-9383(80)90025-7, pdf]

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. Amer. Math. Soc. 359 (2007) 2043-2058 [arXiv:0406507, doi:10.1090/S0002-9947-06-03987-0]

Also:

• Jacob Lurie, Section A.3.2 of: Higher Topos Theory, Annals of Mathematics Studies 170, Princeton University Press (2009) [pup:8957, pdf]

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

Survey and review:

• Julie Bergner, Section 3 of: A survey of $(\infty,1)$-categories (arXiv:math/0610239)

• Emily Riehl, Section 16 of: Categorical Homotopy Theory, Cambridge University Press, 2014 (pdf, doi:10.1017/CBO9781107261457)

See also

• André Joyal, Notes on quasi-categories (2008) [pdf, pdf]

• Aaron Mazel-Gee, Quillen adjunctions induce adjunctions of quasicategories, New York Journal of Mathematics Volume 22 (2016) 57-93 (arXiv:1501.03146, publisher)

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).