Spahn Joyal model structure in HTT (Rev #1, changes)

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Theorem 2.2.5.1

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.

(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.

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(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).

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Revision on June 28, 2012 at 15:29:17 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.