nLab
monoidal derivator

Definition

Monoidal prederivators

Let $\mathrm{Dia}$ be a suitable 2-category of diagram shapes, as used to define derivators. The 2-category $\mathrm{PDer}$ of prederivators is the functor 2-category $[{\mathrm{Dia}}^{\mathrm{op}},\mathrm{CAT}]$. As such, it is a naturally cartesian monoidal 2-category.

A monoidal prederivator is simply a pseudomonoid in $\mathrm{PDer}$. This is equivalent to saying that it is a pseudofunctor ${\mathrm{Dia}}^{\mathrm{op}}\to \mathrm{MonCat}$, where $\mathrm{MonCat}$ 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 $A\in \mathrm{Dia}$, the adjunction $r_{!}:D(A)\leftrightarrows D(*):r^{*}$ is a Hopf adjunction, where $r:A\to *$ 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

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

$${\mathrm{hocolim}}^A(r^{*}Y\otimes X)\to Y\otimes ({\mathrm{hocolim}}^{A}X)$$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 $p:B\to A$ is any opfibration, then we can conclude that $p_{!}⊣p^{*}$ 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

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

$$\begin{array}{ccc}{p}^{-1}(a)& \stackrel{g}{\to }& B\\ {}^q\downarrow & & {\downarrow }^p\\ *& \underset{a}{\to }& A\end{array}$$\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 $q_{!}⊣q^{*}$ is Hopf, we have

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