nLab relation between quasi-categories and simplicial categories


(,1)(\infty,1)-Category theory

Enriched category theory



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 sSetsSet-category C\mathbf{C} to the simplicial set N(𝒞)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.

The Quillen equivalence

Our notation partly follows Bergner (2018).


There is a Quillen equivalence:

sSet-CatN˜sSet. sSet\text{-}Cat \underoverset {\underset{\tilde N}{\longrightarrow}} {\overset{\mathfrak{C}}{\longleftarrow}} {\;\;\; \bot \;\;\;} sSet.


In particular, for CC a fibrant SSet-enriched category, the canonical morphism

(N˜(C))C \mathfrak{C}\big(\tilde N(C)\big) \longrightarrow C

given by the counit of the above adjunction is derived, hence a Dwyer–Kan weak equivalence of simplicial categories.

For SS any simplicial set, the canonical morphism

SN˜(R((S))) S \longrightarrow \tilde N\Big( R\big( \mathfrak{C}(S) \big) \Big)

is a categorical equivalence of simplicial sets, where RR denotes a fibrant replacement functor in the Dwyer–Kan–Bergner model structure.

This was claimed and credited to Joyal in Bergner (2007), Thm. 7.8; detailed proof was produced in Lurie (2009), Thm. & p. 91.

For more details on the rigidification of quasi-categories see also Dugger & Spivak (2009a), (2009b).


Via W¯\bar W-construction

We have an evident inclusion

sSetCatCat ΔsSet Cat \hookrightarrow Cat^{\Delta}

of simplicially enriched categories into simplicial objects in Cat.

On the latter the W¯\bar W-functor is defined as the composite

W¯:Cat ΔN ΔsSet ΔsSet \bar W \colon Cat^\Delta \stackrel{N^\Delta}{\to} sSet^\Delta \stackrel{}{\to} sSet

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 Δ nΔ n×Δ n\Delta^n \to \Delta^n \times \Delta^n).


For CC a simplicial groupoid there is a weak homotopy equivalence

𝒩(C)W¯(C)\mathcal{N}(C) \to \bar W(C)

from the homotopy coherent nerve


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}:

More along these lines is in

An expository account is in Section 7.8

  • Julie Bergner, The homotopy theory of (∞,1)-categories, London Mathematical Society Student Texts 90, 2018.

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.