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

• 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.

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.

References

The homotopy coherent nerve is an old idea. 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 $(\mathrm{\infty }\mathrm{,1})$-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)

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