nLab model structure for quasi-categories (Rev #16, changes)

Showing changes from revision #15 to #16: Added | ~~Removed~~ | ~~Chan~~ged

Context

Model category theory

$(\infty,1)$ -Category theory

Idea
Definition
Properties
Related concepts
References

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 on 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

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 left adjoint of the homotopy coherent nerve to a weak equivalence in the model structure on simplicial categories.

Properties

As a Cisinski model structure

The model structure for quasi-categories is the Cisinski model structure on sSet induced by the localizer which consists of 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 and 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”.

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

it is an inner fibration;
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.

(Lurie, theorem 2.4.6.1)

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

\begin{proposition} The set of inner horn inclusions together with a set of representatives of weak equivalences $\{0\}\to B$ , where $B$ has two vertices and countably many nondegenerated simplices, is a set of generating acyclic cofibrations for the Joyal model structure. \end{proposition}

~~The~~ \begin{proof} ~~inclusion~~ See of \cite[Theorem B]{Stevenson}. \end{proof}~~(∞,1)-catgeories~~ ~~? Grpd?~~ ~~$\stackrel{i}{\hookrightarrow}$~~ ~~(∞,1)Cat~~ ~~has a left and a right~~ ~~adjoint (∞,1)-functor~~

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

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

The inclusion of (∞,1)-catgeories ? 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

(see HTT, around remark 1.2.5.4)

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

There is a Quillen adjunction

(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)

Related concepts

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

model structure on cellular sets

References

The original construction of the Joyal model structure is in

Andre Joyal, Theory of quasi-categories I

Unfortunately, this is still not publically available.

A proof that proceeds via homotopy coherent nerve and simplicially enriched categories is given in detail following theorem 2.2.5.1 in

Jacob Lurie, Higher Topos Theory

Another construction is given in

\bibitem{Stevenson} Danny Stevenson. Notes on the Joyal model structure.

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

Andre Joyal, Myles Tierney, Quasi-categories vs. Segal spaces (arXiv)

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)

nLab model structure for quasi-categories (Rev #16, changes)

Context

Model category theory

(∞,1)(\infty,1)-Category theory

Contents

Idea

Definition

Definition

Properties

As a Cisinski model structure

General properties

Proposition

Remark

Proposition

Proposition

Relation to the model structure for ∞\infty-groupoids

Relation to the model structure for ∞\infty-groupoids

Proposition

Proposition

Related concepts

References

$(\infty,1)$ -Category theory

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

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