Recall that the *[[HTT, A.2.2|homotopy category]]* $h S$ of a model category $S$ was defined to have the same objects as $S$ and the $hom_hS$ objects consist of the equivalence classes of the morphisms in the $hom_S$ objects wrt. the *homotopy equivalence relation*. Here two morphisms of $S$ were called to be *homotopic* if their product map factors through some path object of their codomain. +-- {: .num_remark} ###### Remark Let $S$ be a monoidal model category. Let $S Cat$ denote the category of small $S$-enriched categories. Given a monoidal structure on $S$ also its [[HTT, A.2.2|homotopy category]] (this was defined to have the same objects and the $hom_hS$ objects consist of the equivalence classes of the morphisms in the $hom_S$ objects wrt. homotopy are those of $S$$h S$ carries a monoidal structure which is determined up to a unique isomorphism by the requirement that there exists a monoidal functor $$S\to hS$$ from $S$ to its homotopy category. =-- +--{: .un_defn} ###### Definition A.3.2.1 Let $S$ be a monoidal model category. A functor $F:C\to C^\prime$ in $sSet Cat$ is a weak equivaleence if the induced functor $hC\to h C^\prime$ is an equivalence of $h S$-enriched categories. In other words: F is a weak equivalence iff (1) For every pair $X,Y\in C$, the induced map $$Map_C (X,Y)\to Map_{C^\prime} (F(X), F(Y))$$ is a weak equivalence in $S$. (2) $F$ is essentially surjective on the level of homotopy categories. =-- The following definition says a functor between categories is called a quasi fibrations if every isomorphism has a lift with respect to $F$. +-- {: .un_defn} ###### Definition A.3.2.7 Let $F$:C\to D$ be a functor between classical categories. $F$ is called a *quasi-fibration* if, for every object $x\in C$ and every isomorphism $f:F(x)\to y$ in $D$, there exists an isomorphism $\overline f:x\to \overline y$ in $C$ such that $F(f)=f$. =-- +-- {: .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 $sSet Cat$ 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 $sSet Cat$. Then $F$ is a fibration iff $F$ is a local fibration. =-- +-- {: .un_defn} ###### Definition Let $S$ be a monoidal category. Let $C$ be an $S$-enriched category. (1) A morphism $f$ in $C$ is called an *equivalence* if the homotopy class $[f]$ of $f$ is an isomorphism in $h C$. (2) $C$ is called *locally fibrant object* if for every pair of objects $X,Y\in C$, the mapping space $Map_C(X,Y)$ is a fibrant object of $S$. (3) An $S$-enriched functor $F:C\to C^\prime$ is called a *local fibration* if the following conditions are satisfied: (3.i) $Map_C (X,Y)\to Map_{C^\prime} (FX,FY)$ is a fibration in $S$ for every $X,Y\in C$. (3.ii) The induced map $h C\to h C^\prime$ is a quasi-fibration of categories. =-- +-- {: .un_defn} ###### Definition A.3.2.16 (excellent model category) A model category $S$ is called *excellent model category* if it is equipped with a symmetric monoidal structure and satisfies the following conditions (A1) $S$ is combinatorial. (A2) Every monomorphism in $S$ is a cofibration and the collection of cofibrations in $S$ is stable under products. (A3) The collection of weak equivalencies in $S$ is stable under filtered colimits. (A4) $\otimes:S\times S\to S$ is a Quillen bifunctor. (A5) The monoidal model category $S$ satisfies the invertibility hypothesis. =--