### The Geometry of Monoidal Fibrations?

#### Posted by Mike Shulman

In my paper framed bicategories and monoidal fibrations I wrote down a definition of a “monoidal fibration,” which you can think of as a family of monoidal categories indexed by a (usually cartesian) monoidal category. (I certainly wasn’t the first person to write down such a thing.) A canonical example is that if $V$ is any monoidal category, then the categories $V^X$, for $X$ a set, are each monoidal and are indexed by the cartesian monoidal category $\mathrm{Set}$.

What I want to know now is – is there a string diagram calculus for these things? Never mind proving anything about it; I can’t even visualize what it ought to look like!

That’s all I want to ask, but in case you don’t want to search through the long paper looking for the definition, let me give it to you, in two different ways.

If $B$ is a *cartesian* monoidal category, then a **$B$-indexed monoidal category**, is just a pseudofunctor
$B^{op} \to MonCat$
where $MonCat$ is the 2-category of monoidal categories, strong monoidal functors, and monoidal transformations.

On the other hand, if $B$ is any monoidal category, then a **monoidal fibration** over $B$ is a fibration $p:E\to B$ such that $E$ is also a monoidal category, $p$ is a strict monoidal functor, and the tensor product of $E$ preserves cartesian arrows. If you convert this fibration to a pseudofunctor $B^{op}\to Cat$ sending $x\mapsto E_x$, then what the extra structure gives you is a collection of functors
$\boxtimes: E_x \times E_y \to E_{x\otimes y}$
with suitably coherent associativity and unit isomorphisms. Note that in general, the individual fiber categories $E_x$ are *not* monoidal categories in their own right.

However, when $B$ is cartesian monoidal, we can pass back and forth between the two definitions as follows. Given a monoidal fibration $E\to B$, we define a monoidal structure on the fiber $E_x$ by $a \otimes_x b = \Delta_x^* (a\boxtimes b)$