# When does the canonical model structure on $\mathcal V$-$\mathbf{Cat}$ give a structure of monoidal model category?

Let $\mathcal V$ be a closed symmetric monoidal model category. It is well known that the category $\mathcal V$-$\mathbf{Cat}$ of $\mathcal V$-enriched categories is itself a closed symmetric monoidal category. The paper by Berger-Moerdijk shows that in some cases $\mathcal V$-$\mathbf{Cat}$ admits the so-called canonical model structure (weak equivalences are Dwyer-Kan equivalences). Typical examples are obtained taking $\mathcal V = \mathbf{Set}$, $\mathcal V = \mathbf{sSet}$ or $\mathcal V = \mathbf{Ch}(k)$ (the category of complexes of $k$-modules). My question is the following:

Are there reasonable hypothesis on $\mathcal V$ which ensure that $\mathcal V$-$\mathbf{Cat}$ is a closed symmetric monoidal model category?

The case $\mathcal V = \mathbf{Ch}(k)$ gives the known model structure on the category $\mathbf{dgCat}$ of dg-categories, and the homotopy category $\mathrm{Ho}(\mathbf{dgCat})$ is indeed a closed symmetric monoidal category - even if I actually don't know if $\mathbf{dgCat}$ is a monoidal model category (with the right compatibility conditions). Is a similar result reasonable also for $\mathcal V = \mathbf{sSet}$? I couldn't find any reference about internal homs in the homotopy category of simplicially enriched categories, even if I have in mind some possible candidates...

PS: this question is possibly related to this one.

• For the case V = simplicial sets this model category has a name, it is the Bergner model structure on simplicially enriched categories. It is well-known that this model structure is not Cartesian (and hence is not a closed symmetric monoidal model category in the way you ask). A nice counter example is given in this MO answer mathoverflow.net/a/198310/184. However it is Quillen equivalent to Cartesian model categories, e.g. Quasicategories, and so the homotopy category still inherits a nice monoidal structure. – Chris Schommer-Pries Mar 3 '15 at 14:42
• dgCat is also not a monoidal model category (see mathoverflow.net/questions/195178/…). – AAK Mar 3 '15 at 14:54
• I can't think of any nontrivial monoidal model categories $\mathcal{V}$ for which the standard monoidal structure on $\mathcal{V}\text{-}\mathbf{Cat}$ makes it a monoidal model category. (Of course, it is true in trivial cases like $\mathcal{V}=\mathbf{Set}$.) The closest thing I know of is that for $\mathcal{V}=\mathbf{Cat}$, there is a different monoidal structure on $2\text{-}\mathbf{Cat}$ called the Gray tensor product that makes it a monoidal model category (web.science.mq.edu.au/~slack/papers/qmc2cat.html). – Mike Shulman Dec 24 '18 at 11:48