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 homotopy 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

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

References

The idea of a homotopy coherent nerve has been around for some time. It seems first to have been made explicit by Cordier in 1980, and the link with quasi-categories was first made explicit in the joint work of him with Porter, although that work owed a lot to earlier ideas of Boardman and Vogt about seven years earlier. 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 $(\mathrm{\infty },1)$-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