nLab 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 VV 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 VHo_V is still itself closed monoidal.

A category enriched over such VV may be an enriched homotopical category.


A closed (symmetric) monoidal homotopical category V 0V_0 is a closed, symmetric monoidal homotopical category which is furthermore equipped with a closed monoidal deformation retract.


The most obvious example is:

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 VV is a closed symmetric monoidal homotopical category (such as a monoidal model category) and CC is any small category, then the functor category V CV^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 VV combined with the bar construction.



The homotopy category Ho VHo_V of a closed (symmetric) monoidal homotopical category VV 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.

Last revised on January 8, 2009 at 23:01:22. See the history of this page for a list of all contributions to it.