**enriched category theory
* category theory
* monoidal category
, closed monoidal category
, double category
, virtual double category
## Basic concepts
* enriched category
* enriched functor
* enriched functor category
## Universal constructions
* weighted limit
## Extra stuff, structure, property
ing (tensoring), power
### 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 of a category which is enriched over a suitable monoidal and homotopical category is itself a -enriched category.
For a closed monoidal homotopical category, a -enriched category with powers and copowers and with the structure of a homotopical category on its underlying category is a -homotopical category when equipped with a deformation retract for the enrichment.
If is a monoidal model category, then any -enriched model category is automatically a -homotopical category.
Recall that for as above, is closed monoidal.
With a -homotopical category, is the underlying category of a -enriched category.
Write for this -enriched category. This is the enriched analogue of the homotopy category of .
So schematically we have (with all of the above qualifiers suppressed):
Construction of the enriched homotopy category
For an enriched homotopical -category as above, the -category is constructed from the homotopy category of the ordinary category underlying by constructing a -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 follows the proof of proposition 15.4, p. 45.