Spahn HTT, A.3 simplicial categories (Rev #2)

This is a subentry of a reading guide to HTT.

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

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:

Definition

(Street, Chapter 1.3)

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 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
enriched model category

Revision on June 25, 2012 at 21:52:07 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.