nLab simplicial monoidal model category

Idea

A simplicial monoidal model category is a $sSet_{Qu}$-enriched monoidal model category, namely a model category equipped with

1. the structure of a monoidal model category,

2. the structure of a simplicial model category,

3. the compatibility structure that makes the underlying monoidal and sSet-enriched structure an

   sSet-enriched monoidal category.

Examples

Example

It ought to be true that the Bousfield localization at the “realization equivalences” (see this Prop.) of the Reedy model structure on simplicial objects in chain complexes is simplicial monoidal under the objectwise tensor product of chain complexes.

References

The notion may be implicit in many discussions of monoidal model category such as used, notably, to build symmetric monoidal smash products of spectra. One reference making more explicit the need to specify compatibility structure is:

• Jacob Lurie, Def. 4.1.7.8 in Higher Algebra [pdf]

with a precursor in

• Jacob Lurie, Def. 1.6.4 in: Derived Algebraic Geometry II: Noncommutative Algebra [arXiv:math/0702299]

but the actual compatibility is maybe not made very explicit there either, for this see the references at enriched monoidal category.