nLab
enriched homotopical category

Idea

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 ${\mathrm{Ho}}_C$ of a category $C$ which is enriched over a suitable monoidal and homotopical category $V$ is itself a ${\mathrm{Ho}}_V$-enriched category.

Definition

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.

Examples

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

Consequences

Recall that for $V$ as above, ${\mathrm{Ho}}_V$ is closed monoidal.

Proposition

With $C$ a $V$-homotopical category, ${\mathrm{Ho}}_{C_0}$ is the underlying category of a ${\mathrm{Ho}}_V$-enriched category.

Write ${\mathrm{Ho}}_C$ for this ${\mathrm{Ho}}_V$-enriched category. This is the enriched analogue of the homotopy category of $C$.

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

$$(C\in V-\mathrm{Cat})\Rightarrow ({\mathrm{Ho}}_C\in {\mathrm{Ho}}_V-\mathrm{Cat})$$(C \in V-Cat) \Rightarrow (Ho_C \in Ho_V-Cat)

Construction of the enriched homotopy category

For $C$ an enriched homotopical $V$-category as above, the ${\mathrm{Ho}}_V$-category ${\mathrm{Ho}}_C$ is constructed from the homotopy category ${\mathrm{Ho}}_{C_0}$ of the ordinary category underlying $C$ by constructing a ${\mathrm{Ho}}_V$-module structure, essentially following section 4.3.2 of Hovey: Model categories.

References

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 ${\mathrm{Ho}}_C$ follows the proof of proposition 15.4, p. 45.