Background
Basic concepts
equivalences in/of $(\infty,1)$-categories
Universal constructions
Local presentation
Theorems
Extra stuff, structure, properties
Models
As a model for an (∞,1)-categories, sSet-enriched categories (“simplicial categories”) may be thought of as a semi-strictification of a quasi-category, where composition along objects is strictly associative and unital.
The equivalence between quasi-categories and sSet-enriched categories as models for (∞,1)-categories is exhibited by the operation of sending an $sSet$-category $\mathbf{C}$ to the simplicial set $N(\mathcal{C})$ called its homotopy-coherent nerve (in generalization of the simplicial nerve of an ordinary category). Together with its left adjoint operation $\mathfrak{C}$ (“rigidification of quasi-categories”) this constitutes a Quillen equivalence between the two models.
Our notation partly follows Bergner (2018).
There is a Quillen equivalence:
where:
$SC$ denotes the category of sSet-enriched categories equipped with the Dwyer–Kan–Bergner model structure,
$sSet$ denotes the Joyal model structure on simplicial sets,
$\tilde N$ denotes the homotopy coherent nerve functor,
$\mathfrak{C}$ denotes rigidification of quasi-categories.
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 a fibrant replacement functor in the Dwyer–Kan–Bergner model structure.
For more details on the rigidification of quasi-categories see also Dugger & Spivak (2009a), (2009b).
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 notion of the homotopy coherent nerve (see there for more) goes back to Cordier (1982); Cordier & Porter(1986).
The Quillen equivalence between the model structure for quasi-categories and the model structure on sSet-categories is described in
after an unpublished proof by André Joyal was announced in:
Further discussion of rigidification of quasi-categories $\mathfrak{C}$:
Dan Dugger, David Spivak, Rigidification of quasi-categories, Algebr. Geom. Topol. 11 (2011) 225-261 [arXiv:0910.0814, doi:10.2140/agt.2011.11.225]
Dan Dugger, David Spivak, Mapping spaces in quasi-categories, Algebr. Geom. Topol. 11 (2011) 263-325 [arXiv:0911.0469, doi:10.2140/agt.2011.11.263]
More along these lines is in
An expository account is in Section 7.8
See also
Last revised on May 31, 2023 at 14:05:15. See the history of this page for a list of all contributions to it.