Holmstrom Module over a monoidal category

A right module $D$ over a monoidal category $C$ is a a functor $\otimes: D \times C \to D$ together with two natural IMs expressing the obvious associativity and unit conditions. These must satisfy three coherence diagrams. Often we refer to these as just $C$-modules.

Example: Any category with all coproducts is a module over Set, if we take $A \otimes X$ to be the coproduct of $X$ with itself $|A|$ times.

Can define functor of $C$-modules, requiring it to commute with the tensor up to natural isomorphism, and satisfy two coherence diagrams. Get a 2-category of C-modules.

Example: A functor of Set-modules is a functor preserving coproducts.

See also Algebra over a monoidal category

Recall that $Ho \ Sset$ is a closed symmetric monoidal category. Hovey proves in chapter 5 that the homotopy category of any model category is naturally a closed $Ho \ Sset$-module. This implies that results about simplicial model cats often can be transferred to any model category. Also, the homotopy category of a monoidal MC is naturally a closed $Ho \ Sset$-algebra. The proof of these results uses simplicial and cosimplicial resolutions of objects in a model category, model structures on functor cats to a MC, Reedy categories, framings, latching space, matching space.

nLab page on Module over a monoidal category