# nLab relation between quasi-categories and simplicial categories

### Context

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

(∞,1)-category theory

# Contents

## Idea

As a model for an (∞,1)-category a simplicially enriched category may be thought of as a semi-strictification of a quasi-category: composition along 0-cells is strictly associative and unital.

## The Quillen equivalence between the Joyal model structure and the Dwyer–Kan–Bergner model structure

Our notation follows (Bergner).

There is a Quillen equivalence (due to Joyal (unpublished) and Lurie)

$SC \stackrel{\overset{\mathfrak{C}}{\leftarrow}}{\underset{\tilde N}{\to}} sSet.$

Here $SC$ is the model category of simplicial categories equipped with the Dwyer–Kan–Bergner model structure and $sSet$ denotes the Joyal model structure on simplicial sets.

The functor $\tilde N$ is the homotopy coherent nerve functor. The functor $\mathfrak{C}$ is its left adjoint functor.

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

$\mathfrak{C}(\tilde N(C)) \to C$

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

$S \to \tilde N(R(\mathfrak{C}(S)))$

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

For more details, see, for example, \cite[§7.8]{Bergner} or Dugger–Spivak [DuggerSpivak.Rigidification], [DuggerSpivak.Mapping].

## 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 idea of a homotopy coherent nerve has been around for some time. It was first made explicit by Cordier in 1980, and the link with quasi-categories was then used in the joint work of him with Porter. That work owed a lot to earlier ideas of Boardman and Vogt about seven years earlier who had used a more topologically based approach. Precise references are given and the history discussed more fully at the entry, homotopy coherent nerve.

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

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

A detailed discussion of the map from quasi-categories to $SSet$-categories is in

More along these lines is in

• Emily Riehl, On the structure of simplicial categories associated to quasi-categories (pdf)

An expository account is in Section 7.8

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