nLab model structure for quasi-categories



Model category theory

model category, model \infty -category



Universal constructions


Producing new model structures

Presentation of (,1)(\infty,1)-categories

Model structures

for \infty-groupoids

for ∞-groupoids

for equivariant \infty-groupoids

for rational \infty-groupoids

for rational equivariant \infty-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general \infty-algebras

specific \infty-algebras

for stable/spectrum objects

for (,1)(\infty,1)-categories

for stable (,1)(\infty,1)-categories

for (,1)(\infty,1)-operads

for (n,r)(n,r)-categories

for (,1)(\infty,1)-sheaves / \infty-stacks

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



A quasi-category is a simplicial set satisfying weak Kan filler conditions that make it behave like the nerve of an (∞,1)-category.

There is a model category structure on the category SSet – the Joyal model structure or model structure for quasi-categories – such that the fibrant objects are precisely the quasi-categories and the weak equivalences precisely the correct categorical equivalences that generalize the notion of equivalence of categories.



The model structure for quasi-categories or Joyal model structure sSet JoyalsSet_{Joyal} on sSet has


As a Cisinski model structure

The model structure for quasi-categories is the Cisinski model structure on sSet whose class of weak equivalences is the localizer generated by the spine inclusions {Sp nΔ n}\{Sp^n \hookrightarrow \Delta^n\}. See (Ara).

General properties


The model structure for quasi-categories is


The image under the Cartesian product-functor of two weak categorical equivalences (Def. ) is again a weak categorical equivalence.

[Lurie (2009), Cor.]


The model structure for quasi-categories is a monoidal model category with respect to cartesian product (hence a cartesian closed model category) and thus is naturally an enriched model category over itself, hence is sSet JoyalsSet_{Joyal}-enriched (reflecting the fact that it tends to present an (infinity,2)-category). It is however not sSet QuillensSet_{Quillen}-enriched and thus not a “simplicial model category” with respect to this enrichment.


For p:𝒞𝒟p \colon \mathcal{C} \to \mathcal{D} a morphism of simplicial sets such that 𝒟\mathcal{D} is a quasi-category. Then pp is a fibration in sSet JoyalsSet_{Joyal} precisely if both of the following conditions hold:

  1. it is an inner fibration;

  2. it is an isofibration:

    in that for every equivalence in 𝒟\mathcal{D} and a lift of its domain through pp, there is also a lift of the whole equivalence through pp to an equivalence in 𝒞\mathcal{C}.

This is due to Joyal. [Lurie (2009), cor.]

So every fibration in sSet JoyalsSet_{Joyal} is an inner fibration, but the converse is in general false. A notable exception are the fibrations to the point:


The fibrant objects in sSet JoyalsSet_{Joyal} are precisely those that are inner fibrant over the point, hence those simplicial sets which are quasi-categories.

[Lurie (2009), theorem]

Relation to the model structure for \infty-groupoids

The inclusion of (∞,1)-categories ∞Grpd i\stackrel{i}{\hookrightarrow} (∞,1)Cat has a left and a right adjoint (∞,1)-functor

(grpdfyiCore):(,1)CatCoreigrpdfyGrpd, (grpdfy \dashv i \dashv Core) \;\; \colon \;\; (\infty,1)Cat \stackrel{\overset{grpdfy}{\longrightarrow}}{\stackrel{\overset{i}{\leftarrow}}{\underset{Core}{\longrightarrow}}} \infty Grpd \,,


  • CoreCore is the operation of taking the core, the maximal \infty-groupoid inside an (,1)(\infty,1)-category;

  • grpdfygrpdfy is the operation of groupoidification that freely generates an \infty-groupoid on a given (,1)(\infty,1)-category

(see HTT, around remark

The adjunction (grpdfyi)(grpdfy \dashv i) is modeled by the left Bousfield localization

(1)sSet QuillenIdIdsSet Joyal. sSet_{Quillen} \underoverset {\underset{Id}{\longrightarrow}} {\overset{Id}{\longleftarrow}} {\;\; \bot \;\;} sSet_{Joyal} \,.

Notice that the left derived functor 𝕃Id:(sSet Joyal) (sSet Quillen) \mathbb{L} Id \colon (sSet_{Joyal})^\circ \to (sSet_{Quillen})^\circ takes a fibrant object on the left – a quasi-category – then does nothing to it but regarding it now as an object in sSet QuillensSet_{Quillen} and then producing its fibrant replacement there, which is Kan fibrant replacement. This is indeed the operation of groupoidification .

The other adjunction is given by the following


There is a Quillen adjunction

(k !k !):sSet Quillenk !k !sSet Joyal (k_! \dashv k^!) \;\; \colon sSet_{Quillen} \stackrel {\overset{k^!}{\longleftarrow}} {\underset{k_!}{\longrightarrow}} sSet_{Joyal}

which arises as nerve and realization for the cosimplicial object

k:ΔsSet:[n]Δ[n], k : \Delta \to sSet : [n] \mapsto \Delta'[n] \,,

where Δ [n]=N({01n})\Delta^'[n] = N(\{0 \stackrel{\simeq}{\to} 1 \stackrel{\simeq}{\to} \cdots \stackrel{\simeq}{\to} n\}) is the nerve of the groupoid freely generated from the linear quiver [n][n].

This means that for XSSetX \in SSet we have

  • k !(X) n=Hom sSet(Δ[n],X)k^!(X)_n = Hom_{sSet}(\Delta'[n],X).

  • and k !(X) n= [k]X kΔ[k]k_!(X)_n = \int^{[k]} X_k \cdot \Delta'[k].

This is (Joyal & Tierney (2007), prop 1.19)

The following proposition shows that (k !k !)(k_! \dashv k^!) is indeed a model for (iCore)(i \dashv Core):

  • For any XsSetX \in sSet the canonical morphism Xk !(X)X \to k_!(X) is an acyclic cofibration in sSet QuillensSet_{Quillen};

  • for XsSetX \in sSet a quasi-category, the canonical morphism k !(X)Core(X)k^!(X) \to Core(X) is an acyclic fibration in sSet QuillensSet_{Quillen}.

This is (Joyal & Tierney (2007), prop 1.20)


André Joyal on the history of the Joyal model structure (also on MathOverflow):

I became interested in quasi-categories (without the name) around 1980 after attending a talk by Jon Beck on the work of Boardman and Vogt. I wondered if category theory could be extended to quasi-categories. In my mind, a crucial test was to show that a quasi-category is a Kan complex if its homotopy category is a groupoid. All my attempts at showing this have failed for about 15 years, until I stopped trying hard! I found a proof after extending to quasi-categories a few basic notions of category theory. This was around 1995. The model structure for quasi-categories was discovered soon after. I did not publish it immediately because I wanted to show that it could be used for proving something new in homotopy theory. I am a bit of a perfectionist (and overly ambitious?). I was hoping to develop a synthesis between category theory and homotopy theory (hence the name quasi-categories). I met Lurie at a conference organised by Carlos Simpson in Nice (in 2001?). I gave a talk on the model structure and Lurie asked for a copy of my notes afterward.

A similar model for (∞,n)-categories is discussed at

There are analogues of the Joyal model structure for cubical sets (with or without connection):


The original construction of the Joyal model structure is in

Unfortunately, this never became publicly available, but see the lecture notes:

or the construction of the model structure in Cisinski’s book

which closely follows Joyal’s original construction.

A proof that proceeds via homotopy coherent nerve and simplicially enriched categories:

On the relation to the model structure for complete Segal spaces:

Discussion with an eye towards Cisinski model structures and the model structure on cellular sets is in

See also

On a model structure for (infinity,2)-sheaves with values in quasicategories:

On transfer of the Joyal model structure to reduced simplicial sets, modelling quasi-categories with a single object:

Last revised on May 31, 2023 at 14:00:12. See the history of this page for a list of all contributions to it.