# 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
$\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
SimplicialOperads$-$homotopy coherent dendroidal nerve$\to$DendroidalSetsRelativeDendroidalSets
$\downarrow$dendroidal nerve$\downarrow$
SegalOperads$\hookrightarrow$DendroidalCompleteSegalSpaces
$\mathcal{O}$Mon(∞,1)Cat
DendroidalCartesianFibrations

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: