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model structure for quasi-categories

Context

Model category theory

model category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

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

Model structures

for \infty-groupoids

for ∞-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general

specific

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

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 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

Definition

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

Properties

As a Cisinski model strucre

The model structure for quasi-categories is the Cisinski model structure on sSet induced by the localizer which consists of the spine inclusions {Sp nΔ n}\{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 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”.

Proposition

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

  1. it is an inner fibration;

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

So ever fibration in sSet JoyalsSet_{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 JoyalsSet_{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

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

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

where

  • 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 1.2.5.4)

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

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

Notice that the left derived functor 𝕃Id:(sSet Joyal) (sSet Quillen) \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 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

Proposition

There is a Quillen adjunction

(k !k !):sSet Quillenk !k !sSet Joyal (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: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 (JoTi, prop 1.19)

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

Proposition
  • 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 (JoTi, prop 1.20)

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

References

The original construction of the Joyal model structure is in

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

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

Revised on February 11, 2013 14:25:16 by Urs Schreiber (89.204.138.151)