+-- {: .un_theorem} ###### 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 $p$ 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. =-- +-- {: .un_lemma #lemma2.2.5.2} ###### Lemma 2.2.5.2 Any inner anodyne map of simplicial sets is a categorial equivalence. =-- +-- {: .un_remark} ###### 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. =-- +-- {: .un_cor} ###### 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 (transclusion: +-- {: .un_theorem #theorem2.4.6.1} ###### 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. =-- ) +--{: .un_prop} ###### 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} ###### 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](#lemma2.2.5.2) 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 since $C$ is an $\infty$-category and consequently $*\to C$ is a fibration by [Theorem 2.4.6.1](#theorem2.4.6.1) and $A\times K\hookrightarrow B\times K$ 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 [[nLab:hom adjunction]] and passing to the homotopy category: 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: +-- {: .un_prop} ###### 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. +-- {: .un_theorem #theoremA.3.2.24} ###### Theorem 3.2.24 Let $S$ be an excellent model category. Then: 1. 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. 1. 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. =-- )