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.
Our notation follows (Bergner).
There is a Quillen equivalence (due to Joyal (unpublished) and Lurie)
Here $SC$ is the model category of simplicial categories equipped with the Dwyer–Kan–Bergner model structure and $sSet$ denotes the Joyal model structure on simplicial sets.
The functor $\tilde N$ is the homotopy coherent nerve functor. The functor $\mathfrak{C}$ is its left adjoint functor.
In particular, for $C$ a fibrant SSet-enriched category, the canonical morphism
given by the counit of the above adjunction is derived, hence a Dwyer–Kan weak equivalence of simplicial categories.
For $S$ any simplicial set, the canonical morphism
is a categorical equivalence of simplicial sets, where $R$ denotes the fibrant replacement functor in the Joyal model structure.
For more details, see, for example, \cite[§7.8]{Bergner} or Dugger–Spivak [DuggerSpivak.Rigidification], [DuggerSpivak.Mapping].
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)
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
More along these lines is in
An expository account is in Section 7.8
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
Last revised on November 26, 2020 at 04:30:35. See the history of this page for a list of all contributions to it.