closed monoidal homotopical category

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.

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.

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.

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

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.