nLab model structure for quasi-categories

Model structures

for ∞-groupoids

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

$(\infty,1)$-Category theory

(∞,1)-category theory

Contents

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

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

Remark

It is also a monoidal model category with respect to cartesian product 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

1. it is an inner fibration;

2. it is an “isofibration”: 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, cor. 2.4.6.5).

So every fibration in $sSet_{Joyal}$ is an inner fibration, but the converse is in general false. A notably 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.

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

$(grpdfy \dashv i \dashv Core) \;\; : \;\; (\infty,1)Cat \stackrel{\overset{grpdfy}{\to}}{\stackrel{\overset{i}{\leftarrow}}{\overset{Core}{\to}}} \infty Grpd \,,$

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

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

$(Id \dashv Id) \; :\; sSet_{Joyal} \stackrel{\leftarrow}{\to} sSet_{Quillen} \,.$

Notice that the left derived functor $\mathbb{L} Id : (sSet_{Joyal})^\circ \to (sSet_{Quillen})^\circ$ 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

$(k_! \dashv k^!) \;\; : sSet_{Quillen} \stackrel{\overset{k^!}{\leftarrow}}{\overset{k_!}{\to}} sSet_{Joyal}$

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 (JoTi, 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 (JoTi, prop 1.20)

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

The original construction of the Joyal model structure is in

Unfortunately, this is still not publicly available, but see the lecture notes

• Andre Joyal, The theory of quasi-categories and its applications, (pdf)

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 is given in detail following theorem 2.2.5.1 in

The relation to the model structure for complete Segal spaces is in

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