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 ## A.3.2 The model structure on $S$-enriched categories ## 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]]