### 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 $a \otimes_x b = \Delta_x^\ast (a\boxtimes b)$ where $\Delta_x \colon x\to x\times x$ is the diagonal in $B$. Conversely, given a $B$-indexed monoidal category, we make it into a monoidal fibration by defining, for $a\in E_x$ and $b\in E_y$, $a\boxtimes b = \pi_y^\ast(a) \otimes_{x\times y} \pi_x^\ast(b)$ where $\pi_y$ and $\pi_x$ are the two projections $x\times y \to x$ and $x\times y \to y$, respectively.

I’d be ecstatic with a string diagram calculus including the case when $B$ is not cartesian, but I’d be pretty happy with one that only works when $B$ is cartesian. There are a bunch of monoidal categories around, and also (in the second case) a bunch of monoidal functors; we know how string diagrams for monoidal categories work, and Micah McCurdy recently showed us one nice way to do string diagrams for monoidal functors. Can we put those together somehow and incorporate the monoidal structure of $B$?

## Re: The Geometry of Monoidal Fibrations?

Nice, thanks. I’ve used similar technique:

(sorry for Russian subscripts, but I hope the principle is clear).

It was originally placed in my blog record.

Btw I made all calculations in Appendix A of that paper using this technique.