This is a subentry of [[HTT, A.3]] and of [[a reading guide to HTT]]. +-- {: .un_defn} ###### Definition A.3.1.1 (Quillen bifunctor) Let $A,B,C$ be model categories. A functor $F:A\times B\to C$ is called *Quillen bifunctor* if the following conditions are satisfied: (1) For cofibrations $i:a\to a^\prime$, and $j:b\to b^\prime$ in $A$ resp. in $B$, the induced map $$i\wedge j:F(a^\prime, b) \coprod_{F(a,b)}F(a,b^\prime)\to F(a^\prime,b^\prime)$$ is a cofibration in $C$. Moreover $i\wedge j$ is acyclic if either $i$ or $j$ is acyclic; where the pushout is $$\array{ F(a,b) &\stackrel{F(Id,j)}{\to}& F(a,b^\prime) \\ \;\;\downarrow^{F(i,Id)} && \downarrow \\ F(a^\prime,b) &\stackrel{}{\to}& F(a^\prime,b) \coprod_{F(a,b)} F(a,b^\prime) }$$ (2) $F$ preserves small colimits in each variable seperately. =-- +-- {: .num_remark} ###### Remark setting $i:0\hookrightarrow c^\prime$ shows that condition 1. in the previous definition reduces to the requirement on $F(c^\prime,-)$ to preserve cofibrations and acyclic cofibrations. =-- +-- {: .un_defn} ###### Definition A.3.1.2 (monoidal model category) A monoidal model category is a monoidal category $S$ equipped with a model structure satisfying the following: 1. The tensor product $\otimes:S\times S\to S$ is a left Quillen bifunctor. 1. The unit object $1\in S$ is cofibrant. 1. The monoidal structure is closed. =-- +-- {: .num_example} ###### Example A.3.1.4 The category $sSet$ is a monoidal model category with respect to the cartesian product and the Kan model structure. =-- +-- {: .un_defn} ###### Definition A.3.1.5 ($S$-enriched model category) Let $S$ be a monoidal model category. A $S$-enriched model category is defined to be a $S$-enriched category $A$ equiped with a model structure satisfying the following: 1. $A$ is tonsured and cotensored over $S$. 1. The tensor product $\otimes:A\times S\to A$ is a left Quillen bifunctor =-- +-- {: .un_remark} ###### Remark A.3.1.6 (alternative characterization of the Quillen bifunctor $\otimes:A\times S\to A$) =-- The following remark explicates the relation between simplicial homotopy theory and model-category-theoretic homotopy theory. +-- {: .un_remark} ###### Remark A.3.1.8 Let $C$ be a simplicial model category, let $X$ be a cofibrant object of $C$, let $Y$ be a fibrant object of $C$. Then we have (1) $Map_C (X,Y)$ is a Kan complex. (2) $hom_hC(X,Y)\simeq \pi_0 Map_C (X,Y)$ =-- +-- {: .un_prop} ###### Proposition A.3.1.10 Let $C$, $D$ be $S$-enriched model categories. Let $(F\dashv G):D\stackrel{G}{\to} D$ be a Quillen adjunction between the underlying model categories. Let every object of $C$ be cofibrant. Let $$\beta_{x,s}: s\otimes F(x)\to F(s\otimes x)$$ be a weak equivalence for every pair of cofibrant objects $x\in C$, $s\in S$. Then the following are equivalent: 1. $(F\dashv G)$ is a Quillen equivalence. 1. The restriction of $G$ determines a weak equivalence of $S$-enriched categories $D^\circ\to C^\circ$. =-- +-- {: .un_cor} ###### Corollary A.3.1.12 Let $(F\dashv G):D\stackrel{G}{\to} D$ be a Quillen equivalence between simplicial model categories where every object of $C$ is cofibrant. Let $G$ be a simplicial functor. Then $G$ induces an equivalence of $\infty$-categories $N(D^\circ)\to N(C^\circ)$. =--