## September 4, 2009

### 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:

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.

Posted at September 4, 2009 8:16 PM UTC

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### 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.

Posted by: Simon Willerton on September 4, 2009 10:04 PM | Permalink | Reply to this

### Re: Traces in Bicategories

Looks very interesting! I hadn’t realized in particular that the usual picture of traces requires symmetry.. sounds like a cool generalization.

For the case of modules over a noncommutative ring though aren’t you living in a symmetric monoidal context? i.e. the “Morita” bicategory you describe (which is a full subthingy of all categories, by sending an algebra to its category of modules) has a symmetric monoidal structure, given by tensor product of rings (which corresponds to tensor product of module categories) (sorry for getting some of my terminologies wrong..) I realize your story is more general, so this might not be at all relevant to your motivations, but in this particular case, for a module over a noncommutative ring (or ring spectrum etc), there’s a canonical trace (or Chern character) living in Hochschild homology of the ring (which is defined exactly the way you defined the trace of a matrix, i.e., using the dualizability of the category of modules over any ring). More generally there’s a trace map (Dennis trace?) from K-theory of a category (or $(\infty,1)$-category if you prefer) to the Hochschild homology (or topological Hochschild homology) of the category. Anyway, probably irrelevant comment for your project, sorry.

Posted by: David Ben-Zvi on September 4, 2009 10:57 PM | Permalink | Reply to this

### Re: Traces in Bicategories

For the case of modules over a noncommutative ring though aren’t you living in a symmetric monoidal context? i.e. the “Morita” bicategory you describe… has a symmetric monoidal structure,

Yes, the bicategory itself is symmetric monoidal, but the tensor product of bimodules, which is what’s used to define traces in the classical situation, is not symmetric.

The symmetric monoidal structure of the bicategory does play a role in defining traces, though, see below.

for a module over a noncommutative ring (or ring spectrum etc), there’s a canonical trace (or Chern character) living in Hochschild homology of the ring

Is the following what you mean? An $R$-module $M$ is a 1-cell $Z \to R$ in the bicategory of rings and modules. This bicategory can be extended to the bicategory of small $Ab$-enriched categories and profunctors, and in that bicategory $R$ is (Morita) equivalent to the category $R Mod_f$ of dualizable $R$-modules (its Cauchy completion). Hence $M$ is equivalently a profunctor $Z\to R Mod_f$. If $M$ is itself dualizable, then this profunctor is representable by some functor $Z\to R Mod_f$, which just picks out $M$ in $R Mod_f$ (since $Z$ is the unit $Ab$-category). Now this functor induces an element of $\langle\langle R Mod_f \rangle\rangle = \int^{N\in R Mod_f} R Mod_f(N,N)$ (the 2-trace/shadow of $R Mod_f$ in $Ab Prof$) which I think is what you are calling the trace of $M$? And since $R$ is equivalent to $R Mod_f$ in $Ab Prof$, they have the same 2-trace and so this is also an element of the 2-trace/shadow of $R$, which is an “underived” version of $H H$ (the homotopy version would land in actual $H H$).

If that’s right, then this trace is the same as what we would call the trace of the identity map of $M$ (aka the dimension or Euler characteristic of $M$). The fact that these are the same can be extracted from Thm 14.5 (and I took a slightly circuitous route in the previous paragraph to make this clearer), but we didn’t include it explicitly because we thought profunctors were already fairly esoteric for the audience. But if this really is the Dennis trace, which I’ve heard about but never figured out, perhaps we should.

Posted by: Mike Shulman on September 5, 2009 1:57 AM | Permalink | Reply to this

### Re: Traces in Bicategories

A similar argument also works for nonidentity endomorphisms, of course; the trace of $f$ is just given by the image of $f$ itself inside $\langle\langle R Mod_f \rangle\rangle$.

Posted by: Mike Shulman on September 5, 2009 8:27 PM | Permalink | Reply to this

### Re: Traces in Bicategories

Something I may not have made entirely clear in the post: there are 3 different kinds of trace to be distinguished.

• The trace of an endo 1-morphism in a (symmetric monoidal) 1-category. This is the classical case.
• The trace of an endo 1-morphism in a (symmetric monoidal) 2-category. This looks to be what Simon’s talk was about. Also the span trace that’s gotten attention around here is a special case of it. This is a vertical categorification of the ordinary trace. In our paper we call this the shadow, to avoid confusion with…
• The trace of an endo 2-morphism in a 2-category, which uses the trace of endo 1-morphisms to deal with the fact that composition of 1-morphisms is not symmetric in a bicategory. This is a horizontal categorification of the ordinary trace.

Now we don’t say hardly anything about monoidal bicategories in the paper. Most examples of shadows (= 2-traces) do in fact arise because the bicategory in question is autonomous symmetric monoidal—but writing for an audience who may be more than a little skeptical of autonomous symmetric monoidal bicategories, we instead axiomatize the necessary properties of a shadow.

Posted by: Mike Shulman on September 5, 2009 1:15 AM | Permalink | Reply to this

### Re: Traces in Bicategories

Gavin Wraith writes:

Dear John

Maybe this is a superfluous remark about Traces in Bicategories. A student of Anders Kock called Maan Justersen wrote a thesis on traces back in the 1960s. It might be interesting to ask Anders if he still has a copy. The microcosm principle had not then been formulated, but the fact that Trace was a trace on categories and profunctors was noted.

Posted by: John Baez on September 7, 2009 7:07 PM | Permalink | Reply to this

### Re: Traces in Bicategories

Hello everyone,

I really enjoyed this, especially the thorough examples. I was especially interested in the example of parametrized spaces (just after 2.12). If I have a particular fibration (or even finite cover) of compact manifolds in mind, is it possible to compute its Euler characteristic? I imagine this involves the Euler characteristics of the total space and fiber (and the base?), but it will be an endomorphism of the unit, so maybe it will be more complicated than just a pair of Euler characteristics?

I also am interested in the comparisons/differences between fiberwise and total duality for indexed symmetric monoidal categories (section 12), but I have to think some more before I’m ready to talk about that :)

Posted by: Niles Johnson on September 11, 2009 3:30 PM | Permalink | Reply to this

### Re: Traces in Bicategories

Good question. I’m not really your go-to man for actually computing particular examples (although Peter keeps threatening to get me to compute something one of these days), but I can give you a general sort of answer. Hopefully Kate will correct me if I get something wrong.

The important thing is that duality and traces in the symmetric monoidal category $Sp_B$ are fiberwise, just like the smash product is. Notice we said that a fibration is dualizable just when each of its fibers is dualizable—no condition on the base space $B$ is required. That means, in particular, that traces and Euler characteristics can carry little to no information about the base space; it might not even be dualizable, and so not even have a well-defined Euler characteristic. This is one of the reasons for the introduction of Costenoble-Waner duality, which lives in the bicategory of parametrized spectra; see 6.24 and §12.

Continuing in $Sp_B$, however, any trace (such as an Euler characteristic) will be an endomorphism of the unit, which is the parametrized sphere spectrum $S_B$. This can essentially be thought of as the projection $B\times S \to B$, i.e. it is one copy of the usual sphere spectrum over each point of $B$. Since an endomorphism of $S$ is an integer, an endomorphism of $S_B$ consists essentially of an integer for each point of $B$. In particular, the trace of an endomorphism $f:M\to M$ of a dualizable fibration over $B$ essentially tells you, for each point $b\in B$, what the trace of $f\big|_{M_b}$ is. Same with the Euler characteristic; it all happens fiberwise in $Sp_B$.

Would it be useful to include something about this in the example 2.12?

Posted by: Mike Shulman on September 12, 2009 7:42 PM | Permalink | Reply to this

### Re: Traces in Bicategories

Thanks Mike–this makes a lot of sense. Since it’s not too hard to explain, I’d vote for including the comment that trace and Euler characteristic are computed fiberwise in this case. But, of course, if you had included it in the draft I read, it would not have piqued my curiosity so much!

As for Constenoble-Waner duality, I still feel like I’m not quite connecting something all the way. I like the result that total duality implies finiteness for the “base” (12.9), but I wasn’t sure where to go from there. I think that a fibration between closed smooth manifolds is totally dualizable as a parametrized space–is that true? If so, is 12.9 saying that it’s Euler characteristic is still an integer? ($I_* = S$ in this case?) Either way, does this factorization of the trace have some interesting meaning? Or a simple explanation in the case I keep thinking about? In the proof of 12.9, you explain that the second map is the trace of the identity on a certain unit; is that (roughly) the Euler characteristic of $A$ (at least in the fibration of closed manifolds example)? Example 12.15 starts to address this family of questions, but I guess I wish there was a little more there.

Or maybe I don’t–this has been fun to think about ;)

Posted by: Niles Johnson on September 13, 2009 10:44 PM | Permalink | Reply to this

### Re: Traces in Bicategories

First, these kinds of traces and Euler characteristics are not so easy to compute. There is a really nice example computed by Dold in his original paper on fiberwise fixed point indices. (Dold, The fixed point index of fibre preserving maps.)

Think of $S^1$ as the complex numbers of unit length. Define a map $f: S^1\times S^1\rightarrow S^1\times S^1$ by $f(x,y)=(x,xy)$. This is a fiberwise map over $S^1$ using the first coordinate projection. The trace of this map is an element of the first stable homotopy group of $S^0$. This group is isomorphic to $Z/2$, and the trace of $f$ is the nontrivial element in this group (the Hopf fibration).

On each fiber this map is a rotation, so the trace restricted to each fiber is trivial.

This example generalizes to $S^3$ and $S^7$. In those cases the traces are the corresponding Hopf fibrations. (The explicit computation is in: Jeremy Scofield, Nielsen Fixed Point Theory for fiber-preserving maps, 1985.)

For your question about Costenoble-Waner duality- one familiar example comes from the stable homotopy category. Dual pairs for spaces in the (symmetric monoidal) stable homotopy category are composites of a Costenoble-Waner dual pair and a base change dual pair. You can see this in the original descriptions of the coevaluation and evaluation for the dual pairs, they really are composites. In this case, the factoring in 12.9 is an interesting factoring of the classical fixed point index.

Posted by: Kate Ponto on September 14, 2009 3:38 PM | Permalink | Reply to this

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