category with duals (list of them)
dualizable object (what they have)
A monoidal prederivator is simply a pseudomonoid in . This is equivalent to saying that it is a pseudofunctor , where consists of monoidal categories, strong monoidal functors, and monoidal natural transformations. We may similarly define braided and symmetric monoidal prederivators.
A monoidal semiderivator is a monoidal prederivator which is a semiderivator.
We could define a monoidal derivator to be simply a monoidal prederivator which is a derivator. This is what Groth does. However, we might also want to ask that the tensor product be well-behaved with respect to homotopy Kan extensions, and the natural requirement is that it preserve homotopy colimits in each variable.
Precisely, let be a monoidal prederivator which is a derivator; we say that its tensor product preserves homotopy colimits in each variable separately if for any , the adjunction is a Hopf adjunction, where is the unique functor to the terminal category. (Note that this is automatically a comonoidal adjunction, since is strong monoidal.
Explicitly, this means that for any diagram and any object , the canonical transformations
are isomorphisms. In this case we say that is a monoidal derivator.
The above preservation condition, though stated only for colimits, also applies to certain homotopy Kan extensions. If is any opfibration, then we can conclude that is also a Hopf adjunction as follows. It suffices to show that for any , , and , the induced transformation
is an isomorphism (and dually). But since is an opfibration, the square
is homotopy exact. Therefore, , and (since the square commutes) , so using the fact that is Hopf, we have
We leave it to the reader to verify that this composite isomorphism is, in fact, the transformation in question.
Any representable prederivator represented by a monoidal category is a monoidal prederivator. If the monoidal category is complete and cocomplete, and its tensor product preserves colimits on each side, then this is a monoidal derivator.
The homotopy derivator of any monoidal model category is a monoidal derivator.