[[!redirects HTT, A.3]] This is a subentry of [[a reading guide to HTT]]. # Contents * Automatic table of contents {: toc} +-- {: .num_remark} ###### Remark (morphisms in enriched categories) In a model category $A$ there are stricly speaking no morphism defined but only hom objects. So if we wish to define the notion of an *enriched model category* where we expect to have distinguished classes of morphisms we need to refer to an associated (ordinary - i.e. $Set$-enriched) model category where we have morphisms and to qualify our morphisms there as cofibrations, fibrations and weak equivalences, respectively. This is explicated in the following way: =-- +-- {: .num_defn} ###### Definition ([Street, Chapter 1.3](#Street)) Let $V$ be a monoidal category. Let $V_0$ denote the set of objects of $V$. Let $\mathcal{I}$ denote the terminal $V$-category $\mathcal{I}:=\{0,I\}$; i.e. $\mathcal{I}$ has precisely one object $0$ and the monoidal unit is defined to be the hom object $I:=hom(0,0)$. Let $*$ denote the terminal category. Let $V:=V_0(I,-):V_0\to Set$. Let $V Cat$ denote the [[nLab:2-category]] of $V$-categories. Then there is a functor $$(-)_0:=V Cat(\mathcal{I},-):\begin{cases} V Cat\to Cat \\ A\stackrel{id}{\mapsto}A \\ (f:I\to A(a,b))\mapsto (*\to V A(a,b)) \end{cases}$$ called the *underlying set functor*. =-- So if we speak of a *cofibration, fibration* or *weak equivalences* $f:a\to b$ in an enriched category $A$ we mean in fact $(-)_0(f:I\to A(a,b))$. ## A.3.1 Enriched monoidal model categories +-- {: .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 $\otimes:A\times S\to A$) =-- +-- {: .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 $N(D^\circ)\to N(C^\circ)$. =-- ## A.3.2 The model structure on $S$-enriched categories +--{: .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. =-- ## A.3.3 Model structures on diagram categories ## A.3.4 Path spaces in $S$-enriched categories ## A.3.5 Homotopy colimits in $S$-enriched categories ## A.3.6 Exponentiation in model categories ## A.3.7 Localizations of simplicial model categories ## References * Ross Street, basic concepts of enriched category theory, [pdf](http://www.tac.mta.ca/tac/reprints/articles/10/tr10.pdf){#Street} * [[nLab:enriched model category]]