equivalences in/of $(\infty,1)$-categories
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 $N$
For $C$ 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 $S$ 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 $\bar W$-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 $\Delta^n \to \Delta^n \times \Delta^n$).
For $C$ a simplicial groupoid there is a weak homotopy equivalence
from the homotopy coherent nerve
(Hinich)
The above relations constitute arrange into a Quillen equivalence between model category structures on quasicategories and simplicially enriched categories.
There is the
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 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
A detailed discussion of the map from quasi-categories to $SSet$-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
See also
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