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 is its homotopy coherent nerve
Via homotopy coherent nerve
For any SSet-enriched category, the canonical morphism
is an equivalence in that it is essentially surjective on the underlying homotopy categories and a weak equivalence of simplicial sets hom-wise (…details/links…)
For any simplicial set, the canonical morphism
is a categorical equivalence of simplicial sets.
We have an evident inclusion
of simplicially enriched categories into simplicial objects in Cat.
On the latter the -functor is defined as the composite
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 ).
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
The homotopy coherent nerve
is the right adjoint part of a Quillen equivalence between these model structures.
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 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
A detailed discussion of the map from quasi-categories to -categories is in
More along these lines is in
- Emily Riehl, On the structure of simplicial categories associated to quasi-categories (pdf)
An introduction and overview of the relation between quasi-categories and simplicial categories is in section 1.1.5 of
The details are in section 2.2