Spahn Joyal model structure in HTT (changes)

Skip the Navigation Links | Home Page | All Pages | Latest Revisions | Authors |

Showing changes from revision #3 to #4: Added | Removed | Changed

Theorem 2.2.5.1

(existence of the Joyal model structure, Quillen equivalence to the model structure on simplicial categories)

The exists a left proper, combinatorial model structure on the category of simplicial sets such that

(C) Cofibrations are precisely monomorphisms

(W) A map pp is a categorical equivalence iff S(p)S(p) is an equivalence of simplicial categories. Where S:sSet→sCatS:sSet\to sCat denotes the functor induced via Kan extension by the cosimplicial object ℭ:Δ→sCat\mathfrak{C}:\Delta\to sCat, Definition 1.1.5.1, HTT.

This is a subentry of a reading guide to HTT.

Contents

Existence and proof of the equivalence to the model structure on simplicial categories
Theorem 2.2.5.1

(existence of the Joyal model structure, Quillen equivalence to the model structure on simplicial categories)

The exists a left proper, combinatorial model structure on the category of simplicial sets such that

(C) Cofibrations are precisely monomorphisms

(W) A map pp is a categorical equivalence iff S(p)S(p) is an equivalence of simplicial categories. Where S:sSet→sCatS:sSet\to sCat denotes the functor induced via Kan extension by the cosimplicial object ℭ:Δ→sCat\mathfrak{C}:\Delta\to sCat, Definition 1.1.5.1, HTT.
Lemma 2.2.5.2

Any inner anodyne map of simplicial sets is a categorial equivalence.

Remark 2.2.5.3

The Joyal model structure is not right proper.

The horn inclusion Λ 1 2⊂Δ 2\Lambda_1^2\subset \Delta^2 is a categorical equivalence but its pullback along the fibration Δ {0,2}↪Δ 2\Delta^{\{0,2\}}\hookrightarrow \Delta^2 is not.

Corollary 2.2.5.4

Let f:A→Bf:A\to B be a categorical equivalence of simplicial sets. Let KK be a simplicial set.

Then the induced map A×K→B×KA\times K\to B\times K is a categorical equivalence.

Proof of the fibrancy of functor categories

Application: proof of the fibrancy of functor categories

(transclusion:

Theorem 2.4.6.1

Let XX be a simplicial set.

Then XX is fibrant in the Joyal model structure iff XX is an ∞\infty-category.

)

Proposition 1.2.7.3

(stated again and proved in 2.2.5)

Let KK be a simplicial set. Then

(1) For every ∞\infty-category CC, the simplicial set Fun(K,C)Fun(K,C) is an ∞\infty-category.

(2) Let C→DC\to D be a categorial equivalence of ∞\infty-categories. Then the induced map Fun(K,C)→Fun(K,D)Fun(K,C)\to Fun(K,D) is a categorial equivalence.

(3) Let CC be an ∞\infty-category. Let K→K ′K\to K^\prime be a categorial equivalence of simplicial sets. Then the induced map Fun(K ′,C)→Fun(K,C)Fun(K^\prime,C)\to Fun(K,C) is a categorial equivalence.

Proof

(1) Fun(K,C)Fun(K,C) is an ∞\infty-category if it is fibrant in the Joyal model structure on sSetsSet. This is the case if Fun(K,C)→*Fun (K,C)\to * has the right lifting property wrt. all acyclic cofibrations.

By Lemma 2.2.5.2 it suffices to show that it has the extension property with respect to every inner anodyne monomorphism A↪BA\hookrightarrow B.

A → * ↓ ↓ B → Fun(K,C)\array{ A&\to& * \\ \downarrow&&\downarrow \\ B&\to&Fun(K,C) }

This lifting problem is equivalent to the assertion that CC has the right lifting property wrt. the monomorphism A×K↪B×KA\times K\hookrightarrow B\times K.

A×K → * ↓ ↓ B×K → C\array{ A\times K&\to& * \\ \downarrow&&\downarrow \\ B\times K&\to&C }

But since CC is an ∞\infty-category and consequently *→C*\to C is a fibration by Theorem 2.4.6.1 and A×K↪B×KA\times K\hookrightarrow B\times K is inner anodyne (Corollary 2.3.2.4) this lifting problem is solvable what shows (1)

The proof of (2) and (3) consists of translating the statement via the hom adjunction? and passing to the homotopy category: Let hsSeth sSet denote the homotopy category of sSetsSet wrt. the Joyal model structure. Let [X]∈hsSet[X]\in h sSet denote the homotopy class of XX.

For X,Y∈sSetX,Y\in sSet, we have that [X×Y][X\times Y] is a product for [X][X] and [Y][Y]. (If XX and YY are fibrant this is a general fact. If not, we take fibrant replacements and apply Proposition 2.2.5.7.)

If CC is an ∞\infty-category, CC is a fibrant in sSets Set by Theorem 2.4.6.1.

By Proposition 2.2.5.7 we identify hom hsSet([X],[C])hom_{h sSet}([X],[C]) with the set of equivalence classes of objects in the ∞\infty-category Fun(X,C)Fun (X,C) and there are canonical bijections

hom hsSet([X]×[K],[C])≃hom hsSet([X×K],[C])≃hom hsSet([X],[Fun(K,C])hom_{h sSet}([X]\times [K],[C])\simeq hom_{h sSet}([X\times K],[C])\simeq hom_{h sSet}([X],[Fun(K,C])

It follows that [Fun(K,C)][Fun(K,C)] is determined up to canonical isomorphism by [K][K] and [C][C] in that it is an exponential [C] [K][C]^{[K]} in hsSeth sSet. This proves (2) and (3).

(transclusion:

Proposition A.3.2.4

Let SS be a combinatorial monoidal model category. Let every object of SS be cofibrant. Let the collection of all weak equivalences in SS be stable under filtered colimits.

Then there exists a left proper, combinatorial model structure on SCatS Cat such that:

(C) The class of cofibrations in SCatS Cat is the smallest weakly saturated class of morphisms containing the set of morphisms C 0C_0 defined in A.3.2.3. (C 0C_0 is some class of ‘’indicating morphisms’’).

(W) The weak equivalences in SCatS Cat are those functors F:C→DF:C\to D which are essentially surjective on the level of homotopy categories and such that Map C(X,Y)→Map C ′(F(X),F(Y))Map_C(X,Y)\to Map_{C^\prime}(F(X),F(Y)) for every X,Y∈CX,Y\in C.

Recall that sSetsSet equipped with the Kan model structure is an excellent model category.

Theorem 3.2.24

Let SS be an excellent model category. Then:

  1. An SS-enriched category CC is a fibrant object of SCatS Cat iff it is locally fibrant: i.e. for all X,Y∈CX,Y\in C the hom object Map C(X,Y)∈SMap_C (X,Y)\in S is fibrant.

  2. Let F:C→DF:C\to D be a SS-enriched functor where DD is a fibrant object of SCatS Cat. Then FF is a fibration iff FF is a local fibration.

)

Relation to the standard (Kan) model structure

In higher category theory, for the inclusion ii, we have an adjoint triple
(gpdfy⊣i⊣core):∞Cat→core∞Gprd(gpdfy\dashv i\dashv core):\infty Cat\stackrel{core}{\to}\infty Gprd

which we expect to be presented by a Quillen adjunction between suitable model categories on sSetsSet. Suitable are here: the Kan model structure (hence also called model structure for ∞\infty-groupoids) and the Joyal model structure (hence also called model structure for ∞\infty-categories).

(gpdfy⊣i)(gpdfy\dashv i) is presented by
(id⊣id):sSet Quillen→sSet Joyal(id\dashv id):sSet_Quillen\to sSet_Joyal

and (i⊣Core)(i\dashv Core) is presented by
(k !⊣k !):sSet Joyal→k !sSet Quillen(k_!\dashv k^!):sSet_Joyal\stackrel{k^!}{\to} sSet_Quillen

where Δ ′[n]=N({0→≃1→≃⋯→≃n})\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 X∈SSetX \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 k⋅Δ′[k]k_!(X)_n = \int^{[k]} X_k \cdot \Delta'[k].

Proof This is (JoTi, prop 1.19)

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

Last revised on June 29, 2012 at 14:27:34. See the history of this page for a list of all contributions to it.

EditDiscussPrevious revisionHide changesHistory (3 revisions) Print Source