Spahn Joyal model structure in HTT (Rev #2, changes)

Showing changes from revision #1 to #2: Added | ~~Removed~~ | ~~Chan~~ged

Theorem 2.2.5.1

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

(C) Cofibrations are precisely monomorphisms

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

Joyal model structure in HTT

Compare the following theorem with Theorem 2.4.6.1: Let $X$ be a simplicial set. Then $X$ is fibrant in the Joyal model structure iff $X$ is an $\infty$ -category.

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

(W) A map $p$ is a categorical equivalence iff $S(p)$ is an equivalence of simplicial categories. Where $S:sSet\to sCat$ denotes the functor induced via Kan extension by the cosimplicial object $\mathfrak{C}:\Delta\to sCat$ , 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 $\Lambda_1^2\subset \Delta^2$ is a categorical equivalence but its pullback along the fibration $\Delta^{\{0,2\}}\hookrightarrow \Delta^2$ is not.

Corollary 2.2.5.4

Let $f:A\to B$ be a categorical equivalence of simplicial sets. Let $K$ be a simplicial set.

Then the induced map $A\times K\to B\times K$ is a categorical equivalence.

Proof of the fibrancy of functor categories

Proposition 1.2.7.3

(stated again and proved in 2.2.5)

Let $K$ be a simplicial set. Then

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

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

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

Proof

(1) $Fun(K,C)$ is an $\infty$ -category if it is fibrant in the Joyal model structure on $sSet$ . This is the case if $Fun (K,C)\to *$ 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 $A\hookrightarrow B$ .

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

This lifting problem is equivalent to the assertion that $C$ has the right lifting property wrt. the monomorphism $A\times K\hookrightarrow B\times K$ .

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

But $C$ is an $\infty$ -category and consequently $*\to C$ is a fibration by Theorem 2.4.6.1 and $A\times K\hookrightarrow B\times K$ is inner anodyne (Corollary 2.3.2.4).

Let $h sSet$ denote the homotopy category of $sSet$ wrt. the Joyal model structure. Let $[X]\in h sSet$ denote the homotopy class of $X$ .

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

If $C$ is an $\infty$ -category, $C$ is a fibrant in $s Set$ by Theorem 2.4.6.1.

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

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

(transclusion:

Proposition A.3.2.4

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

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

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

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

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

Theorem 3.2.24

Let $S$ be an excellent model category. Then:

An $S$ -enriched category $C$ is a fibrant object of $S Cat$ iff it is locally fibrant: i.e. for all $X,Y\in C$ the hom object $Map_C (X,Y)\in S$ is fibrant.
Let $F:C\to D$ be a $S$ -enriched functor where $D$ is a fibrant object of $S Cat$ . Then $F$ is a fibration iff $F$ is a local fibration.

)
~~(transclusion: here: give proof of Proposition 1.2.7.3~~

Proposition 1.2.7.3

(stated again and proved in 2.2.5)

Let $K$ be a simplicial set. Then

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

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

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

Proof

(1) $Fun(K,C)$ is an $\infty$ -category if it is fibrant in the Joyal model structure.

It suffices to show that it has the extension property with respect to every inner anodyne inclusion $A\subseteq B$ .

This is equivalent to the assertion that $C$ has the right lifting property wrt. the inclusion $A\times K\subseteq B\times K$ .

But $C$ is an $\infty$ -category and $A\times K\subseteq B\times K$ is inner anodyne (Corollary 2.3.2.4).

Let $h sSet$ denote the homotopy category of $sSet$ wrt. the Joyal model structure. Let $[X]\in h sSet$ denote the homotopy class of $X$ .

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

If $C$ is an $\infty$ -category, $C$ is a fibrant in $s Set$ by Theorem 2.4.6.1.

By Proposition 2.2.5.7 we identify $hom_{h sSet}([X]\times [X],[C])\simeq Fun (X,C)$ and there are canonical bijections

$hom_{h sSet}([X]\times [K],[C])\simeq hom_{h sSet}([X\times K],[C])\simeq hom_{h sSet}([X],[Fun(K,C])$

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

)

Revision on June 28, 2012 at 21:44:13 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.