### Traces in Bicategories

#### Posted by Urs Schreiber

*guest post by Mike Shulman*

Regular readers of the cafe may be familiar with the microcosm
principle: certain
algebraic structures can be defined in any category equipped with a
categorified version of that same structure. Kate Ponto and I are
working on a paper about a perhaps less familiar instance of this
principle, regarding the horizontal
categorification
of traces; and we’d appreciate
any feedback you have to offer. Here’s the link:

- Kate Ponto and Mike Shulman, Shadows and traces in bicategories

Normally (at least, normally for a certain type of person) traces live
in a symmetric monoidal category. Specifically, if $M$ is an object
of a symmetric monoidal category which is
dualizable, and $f:M\to M$ is an endomorphism, then we define the **trace** of $f$ to
be the composite

$I \overset{\eta}{\to} M M^{\ast} \overset{f.id}{\to} M M^{\ast} \overset{\cong}{\to} M^{\ast} M \overset{\epsilon}{\to} I.$

It’s a nice exercise to show that in the category $Vect$, the dualizable objects are the finite-dimensional vector spaces, and the trace of a linear endomorphism is the same as the trace of any matrix representing it (in the more explicit sense, namely $tr(A) = \sum_i a_{i i}$). The same thing works in $R Mod$ for any commutative ring $R$.

Another important examples is the stable homotopy category, in which traces can be identified with “fixed point indices,” which count the number of fixed points of a function (with multiplicities). Now homology is a symmetric monoidal functor, hence it preserves traces—so it follows automatically that if $f$ has a fixed point, then $tr(f)$ is nonzero, and hence $tr(H_{\ast}(f))$ is also nonzero. This is the Lefschetz fixed point theorem!

This is wonderful and fairly well-known, but sometimes one needs a
more general notion of trace. One generalization is a “traced
monoidal category,” which is just abstractly equipped with an
operation called “trace,” not necessarily arising in this way.
However, we’re interested in a different generalization, namely a
trace for modules over a *noncommutative* ring, and these aren’t even
a monoidal category; instead we have a *bicategory* of noncommutative
rings, bimodules, and bimodule maps. But while we can define
dualizable objects (aka adjoints) in any bicategory, we can’t define
traces, since we don’t have any symmetry isomorphisms. What Kate
realized (generalizing an old construction of Hattori and Stallings)
is that we *can* define traces as long as the bicategory comes
equipped with a (categorified) trace of its own.

What does this mean? Well, since a trace on a category is a function on endomorphisms which is cyclic, so a categorified trace on a bicategory is a functor $\langle\langle-\rangle\rangle$ defined on endo-1-cells equipped with cyclicity isomorphisms $\langle\langle m n\rangle\rangle \cong \langle\langle n m\rangle\rangle$, satisfying some natural axioms. Then if $m:a\to b$ is a dualizable 1-cell and $f:m\to m$ is an endo-2-cell, we can define the trace of $f$ to be the composite

$\langle\langle 1_b \rangle\rangle \overset{\eta}{\to} \langle\langle m m^{\ast} \rangle\rangle \overset{f.id}{\to} \langle\langle m m^{\ast} \rangle\rangle \overset{\cong}{\to} \langle\langle m^{\ast} m \rangle\rangle \overset{\epsilon}{\to} \langle\langle 1_a \rangle\rangle$

Kate christened the “categorified trace”
$\langle\langle-\rangle\rangle$ on a bicategory a **shadow**, to avoid
confusion when discussing traces of 2-cells in the same breath. If
our bicategory is a symmetric monoidal category (regarded as a
bicategory with one object), we can take the shadow
$\langle\langle-\rangle\rangle$ to be the identity functor, and we
recover the previous notion of trace in a symmetric monoidal category.

In the motivating case of the bicategory of rings and bimodules, the shadow maps an $R$-$R$-bimodule $M$ to the abelian group $\langle\langle M \rangle\rangle$ obtained by quotienting $M$ by the relations $r m = m r$ for all $m\in M$, $r\in R$. In this case the resulting notion of trace for endomorphisms of $R$-modules is the same one defined by Hattori and Stallings, but now the same idea can be applied in bicategories of chain complexes, parametrized topological spaces, spectra, equivariant things, categories and profunctors, etc. etc.

Kate’s original goal in all this was to find a good context in which to talk about generalizations and converses of the Lefschetz fixed point theorem (e.g. see here and here). However, we thought it would be nice to publish the definitions of shadows and traces in bicategories somewhere more widely visible than papers on fixed point theory; hence the present paper. It starts out with a review of traces in symmetric monoidal categories, since not everyone learns about those in the cradle, and then goes on to bicategories, including lots of examples. Of course, as often seems to be the case with the papers I write, double categories snuck in towards the end. Not much background is required; we’ve tried to make it at least vaguely accessible even to fixed point theorists who’ve never seen a bicategory before.

Any comments or suggestions are very welcome. In particular, we’d especially appreciate any feedback about the examples (are there too many? too few? any good ones we left out?) and the organization (we had a hard time getting things in a sensible order, especially in the later sections). And if you know of any other work in this direction that we should refer to, please send that along as well.

## Re: Traces in Bicategories

Okay, so first I should say I haven’t read the paper yet, although I’m looking forward to doing so! However, I just wanted to say I was collecting lots of examples of monoidal bicategories and traces at one point. Some of them made it into the slides for my talk Two $2$-traces, but others didn’t. I have various half-written attempts at writing this stuff into a paper but could never quite decide on an audience: I was either getting bogged down in details of many, many examples or else too stuck in technicalities. Writing this up properly is next on my list once I get some more of the magnitude of metric spaces material out of the way. Aaron has been hassling me to write this up, in particular the 3d string diagram bits; in the slides for the talk that I mentioned above there’s no 3d string diagrams, partly because I didn’t get round to LaTeXing them for the talk and partly because in earlier versions of the talk it was a bit much for people to absorb.

I’m glad you got the trace for a map of metric spaces in – one of my favourite examples – and also that you have parametrized spectra in there, I look forward to seeing what you say about that as it was on my list of examples to work out in detail.

Todd: I see Trimble and McIntyre is cited as “to appear in Adv. Math.” is this on the cards? Hint, hint.