enriched homotopical category

**enriched category theory**
## Background
* category theory
* monoidal category, closed monoidal category
* cosmos, multicategory, bicategory, double category, virtual double category
## Basic concepts
* enriched category
* enriched functor, profunctor
* enriched functor category
## Universal constructions
* weighted limit
* end, coend
## Extra stuff, structure, property
* copowering (tensoring), powering (cotensoring)
### Homotopical enrichment
* enriched homotopical category
* enriched model category
* model structure on homotopical presheaves

The concept of *enriched homotopical category* is the generalization of the concept of homotopical category to the context of enriched category theory and hence to homotopy coherent category theory.

The idea is that the homotopy category $Ho_C$ of a category $C$ which is enriched over a suitable monoidal and homotopical category $V$ is itself a $Ho_V$-enriched category.

For $V$ a closed monoidal homotopical category, a $V$-enriched category $C$ with powers and copowers and with the structure of a homotopical category on its underlying category $C_0$ is a **$V$-homotopical category** when equipped with a deformation retract for the enrichment.

If $V$ is a monoidal model category, then any $V$-enriched model category is automatically a $V$-homotopical category.

Recall that for $V$ as above, $Ho_V$ is closed monoidal.

With $C$ a $V$-homotopical category, $Ho_{C_0}$ is the underlying category of a $Ho_V$-enriched category.

Write $Ho_C$ for this $Ho_V$-enriched category. This is the enriched analogue of the homotopy category of $C$.

So schematically we have (with all of the above qualifiers suppressed):

$(C \in V-Cat) \Rightarrow (Ho_C \in Ho_V-Cat)$

For $C$ an enriched homotopical $V$-category as above, the $Ho_V$-category $Ho_C$ is constructed from the homotopy category $Ho_{C_0}$ of the ordinary category underlying $C$ by constructing a $Ho_V$-module structure, essentially following section 4.3.2 of Hovey: *Model categories*.

The definition appears as definition 16.1, p. 46 in Shulman: Homotopy limits and colimits in enriched homotopy theory, the proposition is proposition 16.2, p. 46. The construction of $Ho_C$ follows the proof of proposition 15.4, p. 45.

Revised on June 18, 2010 08:23:14
by Urs Schreiber
(87.212.203.135)