nLab closed monoidal homotopical category

Idea

In the context of homotopy coherent category theory one wishes to nicely pair enriched category theory with homotopy theory for the case that the category $V$ being enriched over is a model category or at least a homotopical category.

The idea is that a closed (symmetric) monoidal homotopical category is a closed monoidal category which is also homotopical such that its homotopy category $Ho_V$ is still itself closed monoidal.

A category enriched over such $V$ may be an enriched homotopical category.

Definition

A closed (symmetric) monoidal homotopical category $V_0$ is a closed, symmetric monoidal homotopical category which is furthermore equipped with a closed monoidal deformation retract.

Examples

The most obvious example is:

• Any (symmetric) monoidal model category is a closed (symmetric) monoidal homotopical category.

In this case, the cofibrant and fibrant replacements serve as a closed monoidal deformation retract. Closed monoidal homotopical categories that are not monoidal model categories seem surprisingly hard to come by, given how much stronger the axioms of a monoidal model category appear. One other example that can probably be extracted from the theory of homotopy tensor products in Shulman (below) is:

• If $V$ is a closed symmetric monoidal homotopical category (such as a monoidal model category) and $C$ is any small category, then the functor category $V^C$, with its Day tensor product and levelwise homotopical structure, is a closed symmetric monoidal homotopical category. The closed monoidal deformation retract is provided by the deformations in $V$ combined with the bar construction.

Properties

Proposition

The homotopy category $Ho_V$ of a closed (symmetric) monoidal homotopical category $V$ is itself closed (symmetric) monoidal.

References

The definition appears as definition 15.3, p. 44 in Shulman: Homotopy limits and colimits in enriched homotopy theory, the proposition as proposition 15.4, p. 45.