Holmstrom Module over a monoidal MC

Let $C$ be a monoidal model category. A $C$-model category is a $C$-module $D$ with a model structure such that the following conditions hold:

1. The action map $D \times C \to D$ is a Quillen bifunctor.
2. The map $1 \otimes q$ is a WE for all cofibrant $X$. Here $q$ is cofibrant replacement of unit, and $1$ is the identity on $X$. This is automatic when the unit is cofibrant.

A $Sset$-model category is called a simplicial model category. This def is different from Quillen’s. For Quillen, Top is a simplicial MC, but not with this def. The cateogries $Sset$, k-spaces, CG spaces and their pointed versions are all simplicial MCs, but categories of chain complexes are not. However, $Ch(R)$ is a $Ch(\mathbb{Z})$-model category.

If $C$ is pointed, then so is every $C$-model category as well. Under some conditions, if $D$ is a $C$-model caat, then $D_*$ is a $C_*$-model category. Get equivalence between pointed $C$-model cats and $C_*$-model cats.

Can also define algebras: Let $C$ be as above. A monoidal $C$-model category is a monoidal model category $D$ together with a monoidal Quillen functor $C \to D$. Get a 2-category again.

Example: k-spaces is a symmetric monoidal $Sset$-model category.

nLab page on Module over a monoidal MC