# nLab relation between quasi-categories and simplicial categories

### Context

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

(∞,1)-category theory

Background

Basic concepts

Universal constructions

Local presentation

Theorems

Extra stuff, structure, properties

Models

#### Enriched category theory

enriched category theory

# Contents

## Idea

As a model for an (∞,1)-categories, sSet-enriched categories (“simplicial categories”) may be thought of as a semi-strictification of a quasi-category, where composition along objects is strictly associative and unital.

The equivalence between quasi-categories and sSet-enriched categories as models for (∞,1)-categories is exhibited by the operation of sending an $sSet$-category $\mathbf{C}$ to the simplicial set $N(\mathcal{C})$ called its homotopy-coherent nerve (in generalization of the simplicial nerve of an ordinary category). Together with its left adjoint operation $\mathfrak{C}$ (“rigidification of quasi-categories”) this constitutes a Quillen equivalence between the two models.

## The Quillen equivalence

Our notation partly follows Bergner (2018).

###### Proposition

There is a Quillen equivalence:

$sSet\text{-}Cat \underoverset {\underset{\tilde N}{\longrightarrow}} {\overset{\mathfrak{C}}{\longleftarrow}} {\;\;\; \bot \;\;\;} sSet.$

where:

In particular, for $C$ a fibrant SSet-enriched category, the canonical morphism

$\mathfrak{C}\big(\tilde N(C)\big) \longrightarrow C$

given by the counit of the above adjunction is derived, hence a Dwyer–Kan weak equivalence of simplicial categories.

For $S$ any simplicial set, the canonical morphism

$S \longrightarrow \tilde N\Big( R\big( \mathfrak{C}(S) \big) \Big)$

is a categorical equivalence of simplicial sets, where $R$ denotes a fibrant replacement functor in the Dwyer–Kan–Bergner model structure.

This was claimed and credited to Joyal in Bergner (2007), Thm. 7.8; detailed proof was produced in Lurie (2009), Thm. 2.2.5.1 & p. 91.

For more details on the rigidification of quasi-categories see also Dugger & Spivak (2009a), (2009b).

## Relations

### Via $\bar W$-construction

We have an evident inclusion

$sSet Cat \hookrightarrow Cat^{\Delta}$

On the latter the $\bar W$-functor is defined as the composite

$\bar W \colon Cat^\Delta \stackrel{N^\Delta}{\to} sSet^\Delta \stackrel{}{\to} sSet$

where first we degreewise form the ordinary nerve of categories and then take the total simplicial set of bisimplicial sets (the right adjoint of pullback along the diagonal $\Delta^n \to \Delta^n \times \Delta^n$).

###### Proposition

For $C$ a simplicial groupoid there is a weak homotopy equivalence

$\mathcal{N}(C) \to \bar W(C)$

from the homotopy coherent nerve

(Hinich)

There is an operadic analog of the relation between quasi-categories and simplicial categories, involving, correspondingly, dendroidal sets and simplicial operads.

The notion of the homotopy coherent nerve (see there for more) goes back to Cordier (1982); Cordier & Porter(1986).

The Quillen equivalence between the model structure for quasi-categories and the model structure on sSet-categories is described in

after an unpublished proof by André Joyal was announced in:

Further discussion of rigidification of quasi-categories $\mathfrak{C}$:

More along these lines is in

An expository account is in Section 7.8

• Julie Bergner, The homotopy theory of (∞,1)-categories, London Mathematical Society Student Texts 90, 2018.