### Traces in Indexed Monoidal Categories

#### Posted by Mike Shulman

After a gestation period of many years, the following paper is finally ready for the light of day:

- Kate Ponto and Michael Shulman,
*Duality and traces in indexed monoidal categories*. We’re currently having issues with the arXiv compiler, but you can get it from my web page. There are three versions:- If you’re going to read it on a screen, or print it in color, get this version
- If you’re going to print it in black and white, get this version
- If you’re going to do both, but you only want to download one file, get this version.

There are also (at least) two ways to read this paper. If you’re mainly interested in duality and trace (especially their applications in fixed-point theory), then you can read sections 1–8 and stop there. On the other hand, if you’re mainly interested in indexed monoidal categories and their string diagrams, you can read sections 1–3, then 9–10. (But you may still want to go back and read sections 4–8 and 11–12, to see the applications of the string diagrams.)

I’m not going to say anything about traces today—you can read the paper for that. But I’ll say a bit about indexed monoidal categories and string diagrams, which I expect will be of interest to several people here; especially since we got the basic idea of these string diagrams from another discussion here.

An **indexed monoidal category** is one of my favorite categorical structures. In its simplest form it consists of a category $S$ and a pseudofunctor $S^{op} \to MonCat$, which we write as $A\mapsto (C^A, \otimes_A, I_A)$. By the usual Grothendieck construction, this pseudofunctor can of course be regarded as a fibration. And if $S$ has finite products, then then the “fiberwise” monoidal structures $\otimes_A$ can also be “Grothendieckified” into an “external product”
$\boxtimes\colon C^A \times C^B \to C^{A\times B}$
defined by $M\boxtimes N = \pi_2^\ast M \otimes_{A\times B} \pi_1^\ast N$. This makes the total category of the fibration a monoidal category and the fibration itself a strict monoidal functor (when $S$ has its cartesian monoidal structure); I call this a **monoidal fibration**. Moreover, we can recover $\otimes_A$ from $\boxtimes$ via $M\otimes_A N = \Delta_A^\ast (M\boxtimes N)$, so the two structures have the same information.

Here are some nice examples:

- $S=Sets$, $C^A=$ $A$-indexed families of objects of $C$, for any monoidal category $C$.
- $S=Gpd$, $C^A=$ $A$-diagrams of objects of $C$
- $S=$ any category with pullbacks, $C^A = S/A$
- $S=Top$, $C^A=$ spectra parametrized over $A$
- $S=Grp$ or $TopGrp$, $C^A=$ sets or spaces with an action by $A$
- The homotopy category of any of the above equipped with a homotopy theory

In many cases, the reindexing functors $f^\ast\colon C^B \to C^A$ induced by a morphism $f\colon A\to B$ in $S$ all have left adjoints $f_!$. If these left adjoints satisfy the Beck-Chevalley condition for all pullback squares in $S$, then the indexed category is traditionally said to have **indexed coproducts**. For many applications, though, we only need condition for a few pullback squares, which coincidentally (?) happen to be those that are pullbacks in any category with finite products (whether or not it even has all pullbacks).

What’s especially interesting, though, is that the cases we care about for fixed-point theory are mostly homotopy categories, and in this case the Beck-Chevalley condition is usually *not* satisfied for all these pullback squares; only for those that are *homotopy* pullbacks. Fortunately, this includes most of the pullback squares built out of products; see section 3 of the paper for a list.

Back in August of last year, I asked whether there was a string diagram calculus for these things, and was pointed to this paper by Todd Trimble and Geraldine Brady, about representing predicate calculus categorically using C.S. Peirce’s “system beta”. This turned out to be just what we needed, except that we needed a three-dimensional “surface diagram” sort of calculus too in order to represent morphisms. (Predicate calculus is all about indexed posets, so there’s no need to keep track of morphisms, only $\le$ relationships.) Then Daniel Schäppi suggested a “schematic” sort of surface diagram picture as being easier to draw and understand, and that led to the pictures we ended up with in the paper. (Daniel’s diagrams are actually even more schematic than ours; hopefully we’ll get a guest post from him soon when his thesis is finished.)

I’d like to show some pictures of the string diagrams, but it would be a lot of work to extract them into a form that I can post on the blog. So to get a taste of the string diagrams, I suggest you have a look at the first half of these slides from my talk last week at the Category Theory Octoberfest. (The second half of the slides gives an overview of the part of the paper that’s about duality, traces, and fixed-point theory. I’m still looking for a good answer to the question on the final slide, which is more or less the same question I asked here back in August.) Then, if you’re hooked, you can read the paper.

## Re: Traces in Indexed Monoidal Categories

Wowee! Congratulations on getting this finished — it looks really nice!