equivalences in/of $(\infty,1)$-categories
category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
monoidal dagger-category?
Let $Dia$ be a suitable 2-category of diagram shapes, as used to define derivators. The 2-category $PDer$ of prederivators is the functor 2-category $[Dia^{op},CAT]$. As such, it is a naturally cartesian monoidal 2-category.
A monoidal prederivator is simply a pseudomonoid in $PDer$. This is equivalent to saying that it is a pseudofunctor $Dia^{op} \to MonCat$, where $MonCat$ 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 $D$ be a monoidal prederivator which is a derivator; we say that its tensor product preserves homotopy colimits in each variable separately if for any $A\in Dia$, the adjunction $r_! \colon D(A) \leftrightarrows D(*) : r^*$ is a Hopf adjunction, where $r\colon A\to *$ is the unique functor to the terminal category. (Note that this is automatically a comonoidal adjunction, since $r^*$ is strong monoidal.
Explicitly, this means that for any diagram $X\in D(A)$ and any object $Y\in D(*)$, the canonical transformations
are isomorphisms. In this case we say that $D$ is a monoidal derivator.
The above preservation condition, though stated only for colimits, also applies to certain homotopy Kan extensions. If $p\colon B\to A$ is any opfibration?, then we can conclude that $p_! \dashv p^*$ is also a Hopf adjunction as follows. It suffices to show that for any $a\in A$, $X\in D(B)$, and $Y\in D(A)$, the induced transformation
is an isomorphism (and dually). But since $p$ is an opfibration, the square
is homotopy exact. Therefore, $a^* p_! \cong q_! g^*$, and (since the square commutes) $g^* p^* \cong q^* a^*$, so using the fact that $q_! \dashv q^*$ 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.