The n-Category Café

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.