nLab monoidal derivator

Definition

Monoidal prederivators

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.

Monoidal derivators

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

$$hocolim^A (X \otimes r^* Y) \to (hocolim^A X) \otimes Y$$

$$hocolim^A (r^* Y\otimes X) \to Y\otimes (hocolim^A X)$$

are isomorphisms. In this case we say that $D$ is a monoidal derivator.

Properties

Preservation of certain homotopy Kan extensions

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

$$a^* p_! (X\otimes p^* Y) \to a^* (p_! X \otimes Y)$$

is an isomorphism (and dually). But since $p$ is an opfibration, the square

$$\array{ p^{-1}(a) & \overset{g}{\to} & B \\ ^q\downarrow & & \downarrow^p \\ * & \underset{a}{\to} & A }$$

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

$$a^* p_! (X\otimes p^* Y) \cong q_! g^* (X \otimes p^* Y) \cong q_! (g^* X \otimes g^* p^* Y) \cong q_! (g^* X \otimes q^* a^* Y) \cong q_! g^* X \otimes a^* Y \cong a^* p_! X \otimes a^* Y \cong a^* (p_! X \otimes Y).$$

We leave it to the reader to verify that this composite isomorphism is, in fact, the transformation in question.

Examples

• 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.

References

• Moritz Groth, Derivators, pointed derivators, and stable derivators (pdf)