nLab model structure for quasi-categories

Idea

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.

Definition

Definition

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

• cofibrations are the monomorphisms,

• weak equivalences are those maps that are taken by the rigidification functor $\mathfrak{C}$ (the left adjoint to the homotopy coherent nerve) to a Dwyer-Kan equivalence (a weak equivalence in the Dwyer-Kan-Bergner model structure on sSet-enriched categories).

Properties

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 $\{Sp^n \hookrightarrow \Delta^n\}$. See (Ara).

General properties

Proposition

The model structure for quasi-categories is

• a left proper model category,

• a combinatorial model category.

Lemma

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

[Lurie (2009), Cor. 2.2.5.4]

Remark

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 $sSet_{Joyal}$-enriched (reflecting the fact that it tends to present an (infinity,2)-category). It is however not $sSet_{Quillen}$-enriched and thus not a “simplicial model category” with respect to this enrichment.

Proposition

For $p \colon \mathcal{C} \to \mathcal{D}$ a morphism of simplicial sets such that $\mathcal{D}$ is a quasi-category. Then $p$ is a fibration in $sSet_{Joyal}$ 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 $\mathcal{D}$ and a lift of its domain through $p$, there is also a lift of the whole equivalence through $p$ to an equivalence in $\mathcal{C}$.

This is due to Joyal. [Lurie (2009), cor. 2.4.6.5]

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

Proposition

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

[Lurie (2009), theorem 2.4.6.1]

Relation to the model structure for $\infty$-groupoids

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

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

where

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

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

(see HTT, around remark 1.2.5.4)

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

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

Notice that the left derived functor $\mathbb{L} Id \colon (sSet_{Joyal})^\circ \to (sSet_{Quillen})^\circ$ takes a fibrant object on the left – a quasi-category – then does nothing to it but regarding it now as an object in $sSet_{Quillen}$ 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

Proposition

There is a Quillen adjunction

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

which arises as nerve and realization for the cosimplicial object

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

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

This means that for $X \in SSet$ we have

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

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

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

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

Proposition

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

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

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

History

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.

Related concepts

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

• model structure on cellular sets

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

• model structure for cubical quasicategories

References

The original construction of the Joyal model structure is in

• Andre Joyal, Theory of quasi-categories I

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

• André Joyal, The Theory of Quasi-Categories and its Applications, lectures at Advanced Course on Simplicial Methods in Higher Categories, CRM 2008 (pdf, pdf)

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

• Denis-Charles Cisinski, Higher categories and homotopical algebra, (pdf)

which closely follows Joyal’s original construction.

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

• Jacob Lurie, following Thm. 2.2.5.1 in: Higher Topos Theory, Annals of Mathematics Studies 170, Princeton University Press (2009) [pup:8957, pdf]

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

• Andre Joyal, Myles Tierney, Quasi-categories vs. Segal spaces, in Categories in Algebra, Geometry and Mathematical Physics, Contemporary Mathematics 431 (2007) [arXiv:math/0607820, doi:10.1090/conm/431]

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

• Dimitri Ara, Higher quasi-categories vs higher Rezk spaces (arXiv)

See also

• Denis-Charles Cisinski, Algèbre homotopique et catégories supérieures, course notes, 2009, pdf.

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

• Nicholas Meadows, The Local Joyal Model Structure (arXiv:1507.08723)

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

• Burke (2021)