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) =-- +-- {: .un_defn} ###### Definition A.3.1.2 (monoidal model category) =-- +-- {: .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) =-- +-- {: .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.3.1.5 ($S$-enriched model category) Let $S$ be a monoidal model category. A functor $F:C\to C^\prime$ in $sSet Cat$ is called a *weak equivalence* if the induced functor $h C\to hC^\prime$ is an equivalence of $hS$-enriched categories. In other words 1. For every $X,Y\in C$, the induced map $Map_C(X,Y)\to Map_{C^\prime}(F(X),F(Y))$$ is a weak equivalence in $S$. 1. $F$ is essentially surjective on the homotopy categories. =-- A.3.2.1 A.3.2.24 A.3.2.7 A.3.2.9 A.3.2.16 ## 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]]