# nLab model structure on sSet-categories

Contents

### Context

#### Model category theory

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

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

Model structures

for $\infty$-groupoids

for ∞-groupoids

for equivariant $\infty$-groupoids

for rational $\infty$-groupoids

for rational equivariant $\infty$-groupoids

for $n$-groupoids

for $\infty$-groups

for $\infty$-algebras

general $\infty$-algebras

specific $\infty$-algebras

for stable/spectrum objects

for $(\infty,1)$-categories

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

for $(\infty,1)$-operads

for $(n,r)$-categories

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

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

(∞,1)-category theory

Background

Basic concepts

Universal constructions

Local presentation

Theorems

Extra stuff, structure, properties

Models

# 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 $r$ 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.

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

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

###### 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:

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

### 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.

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-categoriesRelativeSimplicialSets
$\downarrow$simplicial nerve$\downarrow$
SegalCategories$\hookrightarrow$CompleteSegalSpaces
(∞,1)Operad
SimplicialOperads$-$homotopy coherent dendroidal nerve$\to$DendroidalSetsRelativeDendroidalSets
$\downarrow$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:

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

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:

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 May 31, 2023 at 16:07:18. See the history of this page for a list of all contributions to it.