What structure of a monoidal simplicial model category is preserved by taking the opposite category?

Question

Suppose we have $(M,\otimes,1)$, a monoidal simplicial model category. Then we can consider the opposite model category $M^{op}$ with the opposite model structure (fibrations become cofibrations, etc.). This category is also still a monoidal category (the opposite of a monoidal category is canonically a monoidal category). How do all of these possible adjectives interact? Is it still closed? Does the push-out product axiom still hold? It seems like we must lose some things, for instance, if we wanted the axiom "for any cofibrant object $X$ there is an equivalence $1\otimes X\simeq X$" this can't in general be true anymore, can it, since the cofibrant objects are completely different now?

In general it seems clear that things break down, but I'm interested in precisely what breaks down. Do we still have a simplicial model category, but we're just missing some of the monoidal model category axioms? I'm also okay with dumping the simplicial requirement.

Answer 1

The general statement is that if $V$ is a monoidal model category (I will assume the unit object of $V$ is cofibrant, so that there are no funny extra axioms related to the unit) and $M$ is a $V$-model category, then $M^{\mathrm{op}}$ is also a $V$-model category. The enrichment comes from that of $M$, and the roles of the tensor and cotensor are interchanged.

To check that this really defines a $V$-model category structure, perhaps the clearest way is to use the cyclic structure of two-variable adjunctions. The well-known fact that a two-variable adjunction $F : C \times D \to E$ being a Quillen bifunctor can be checked using either cofibrations of $C$ and $D$, or cofibrations of $C$ and fibrations of $E$, or cofibrations of $D$ and fibrations of $E$ (Lemma 4.2.2 of Hovey's book) really says that $F$ is a Quillen bifunctor if and only if its cyclic shifts are. Specializing to the action $\otimes : M \times V \to M$, we get a Quillen bifunctor $V \times M^{\mathrm{op}} \to M^{\mathrm{op}}$ which gives the action of $V$ on $M^{\mathrm{op}}$.

In the situation of the question, this means that if $M$ is a monoidal simplicial model category, then $M^\mathrm{op}$ is a simplicial category, and also an $M$-model category. However, as Zhen Lin mentioned, there is no reason $M^{\mathrm{op}}$ should be a monoidal model category.