nLab
relation between quasi-categories and simplicial categories

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 quasi-category corresponding to a simplicial category $C$ is its homotopy coherent nerve $N$

$$\mathrm{sSet}\mathrm{Cat}\stackrel{\stackrel{\mid -\mid }{\leftarrow }}{\underset{N}{\to }}\mathrm{sSet}.$$sSet Cat \stackrel{\overset{|-|}{\leftarrow}}{\underset{N}{\to}} sSet \,.

Relations

Via homotopy coherent nerve

For $C$ any SSet-enriched category, the canonical morphism

$$\mid N(C)\mid \to C$$|N(C)| \to C

is an equivalence in that it is essentially surjective on the underlying homotopyy categories and a weak eqivalence of simplicial sets hom-wise (…details/links…)

For $S$ any simplicial set, the canonical morphism

$$S\to N(\mid S\mid )$$S \to N(|S|)

is a categorical equivalence of simplicial sets.

Via $\overline{W}$-construction

We have an evident inclusion

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

of simplicially enriched categories into simplicial objects in Cat.

On the latter the $\overline{W}$-functor is defined as the composite

$$\overline{W}:{\mathrm{Cat}}^{\Delta }\stackrel{N^{\Delta }}{\to }{\mathrm{sSet}}^{\Delta }\stackrel{}{\to }\mathrm{sSet}$$\bar W : 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

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

from the homotopy coherent nerve

(Hinich)

Model category structures

The above relations constitute arrange into a Quillen equivalence between model category structures on quasicategories and simplicially enriched categories.

There is the

• model structure for quasi-categories

• model structure on sSet-categories.

The homotopy coherent nerve

$$\mathrm{sSet}\mathrm{Cat}\stackrel{N}{\to }{\mathrm{sSet}}_{\mathrm{Joyal}}$$sSet Cat \stackrel{N}{\to} sSet_{Joyal}

is the right adjoint part of a Quillen equivalence between these model structures.

Related concepts

• simplicial category

  • simplicially enriched category

  • simplicial object in Cat

• simplicial groupoid

References

The idea of a homotopy coherent nerve has been around for some time; see there for references.

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 $(\mathrm{\infty },1)$-categories (arXiv)

A detailed discussion of the map from quasi-categories to $\mathrm{SSet}$-categories is in

• Dan Dugger, David Spivak, Rigidification of quasi-categories (arXiv:0910.0814)

• Dan Dugger, David Spivak, Mapping spaces in quasi-categories (arXiv:0911.0469)

More along these lines is in

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

See also

• Vladimir Hinich, Simplicial nerve in Deformation theory (arXiv:0704.2503)

An introduction and overview of the relation between quasi-categories and simplicial categories is in section 1.1.5 of

• Jacob Lurie, Higher Topos Theory .

The details are in section 2.2