### The Linearity of Traces

#### Posted by Mike Shulman

At long last, the following two papers are up:

- Kate Ponto and Mike Shulman, The linearity of traces in monoidal categories and bicategories
- Kate Ponto and Mike Shulman, The linearity of fixed-point invariants

I’m super excited about these, and not just because I like the results. Firstly, these papers are sort of a culmination of a project that began around 2006 and formed a large part of my thesis. Secondly, this project is an excellent “success story” for a methodology of “applied category theory”: taking seriously the structure that we see in another branch of mathematics, but studying it using honest category-theoretic tools and principles.

For these reasons, I want to tell you about these papers by way of their history. (I’ve mentioned some of their ingredients before when I blogged about previous papers in this series, but I won’t assume here you know any of it.)

To begin with, recall that an object $X$ of a symmetric monoidal category is dualizable if, when regarded as a 1-cell in the associated one-object bicategory, it has an adjoint $D X$. Then any endomorphism $f:X\to X$ has a trace defined by

$I \xrightarrow{\eta} X \otimes D X \xrightarrow{f\otimes 1} X\otimes D X \xrightarrow{\cong} D X \otimes X \xrightarrow{\epsilon} I.$

In $Vect$, the dualizable objects are the finite-dimensional ones, traces reproduce the usual trace of a matrix (incarnated as a $1\times 1$ matrix), and in particular $tr(1_X) = dim(X)$. In the stable homotopy category, this is Spanier-Whitehead duality, traces produce the *Lefschetz number* $L(f)$ (incarnated as the degree of a self-map of a sphere), and we have $L(1_X) = \chi(X)$, the *Euler characteristic*. The Lefschetz fixed point theorem follows by abstract nonsense.

The recent part of the story began in 2001, when Peter May wrote “The additivity of traces in triangulated categories”. The Euler characteristic (and Lefschetz number) are *additive*: if $X$ is a cell complex and $A\subseteq X$ a subcomplex, then $\chi(X) = \chi(A) + \chi(X/A)$ and $L(f) = L(f|_A) + L(f/A)$. Peter showed an abstract version of this: if a symmetric monoidal category is compatibly triangulated, then for any distinguished triangle $X\to Y\to Z\to \Sigma X$, we have $\chi(Y) = \chi(X) + \chi(Z)$.

A few years later, Peter and Johann Sigurdsson realized that “Costenoble-Waner duality” for parametrized spaces was naturally about adjunctions in a *bicategory* whose objects were topological spaces and whose 1-cells from $A$ to $B$ are spectra “parametrized over $A\times B$”. (The 2-cells are fiberwise stable maps; note the conspicuous absence of continuous maps of base spaces.) Peter thus wondered whether additivity generalized to bicategories. In the book that he and Johann wrote, they generalized *some* of his axioms for triangulated monoidal categories to “locally triangulated” bicategories, but the final axiom (TC5) used the symmetry, which doesn’t make sense in a bicategory. It was also not clear how to generalize “traces”, since the definition of trace also uses symmetry.

Enter Kate Ponto, who was studying topological fixed-point theory. This subject “begins” with the Lefschetz fixed point theorem, but continues with more refined invariants such as the **Reidemeister trace**, which supports a converse to this theorem (under suitable hypotheses). One definition of the Reidemeister trace uses the **Hattori-Stallings trace**, which is a sort of trace for matrices over a noncommutative ring: you’d like to sum along the diagonal, but the result is basis-dependent until you map it from $R$ into the quotient abelian group $\langle\langle R \rangle\rangle = R / (r s \sim s r)$. Kate realized that the Hattori-Stallings trace, and hence also the Reidemeister trace, was a sort of “bicategorical trace” that she was able to define for endo-2-cells of dualizable 1-cells in any bicategory equipped with some extra structure that she named a **shadow**.

Pleasingly to fans of the microcosm principle, a shadow on a bicategory $\mathbf{B}$ is a “categorified trace”, consisting of functors $\langle\langle-\rangle\rangle:\mathbf{B}(A,A) \to \mathbf{T}$ that are “cyclic up to isomorphism”:
$\langle\langle X \odot Y \rangle\rangle \cong \langle\langle Y \odot X \rangle\rangle$
plus some coherence axioms. Given this, if $X:A\to B$ has an adjoint $D X$ and $f:X\to X$, Kate defined its **trace** $\tr(f)$ to be
$\langle\langle U_A \rangle\rangle \xrightarrow{\eta} \langle\langle X \odot D X\rangle \xrightarrow{f\odot 1} \langle\langle X\odot D X \rangle\rangle \xrightarrow{\cong} \langle\langle D X \odot X \rangle\rangle \xrightarrow{\epsilon} \langle\langle U_B \rangle\rangle$
where $U_A$, $U_B$ are the unit 1-cells. I blogged about this here. So Kate had solved half of the problem of generalizing additivity to bicategories.

At about the same time, I was intrigued by a different aspect of Peter and Johann’s bicategory. Parametrized spaces and spectra can be “pulled back” and “pushed forward” along maps of base spaces. Moreover, pushforward and “copushforward” generalize homology and cohomology, hence should preserve duality. But how can we show this abstractly, since the maps between base spaces are missing from the bicategory of parametrized spectra? Peter and Johann solved this with “base change objects”: for any continuous map $f:A\to B$ they defined spectra $S_f$ and ${}_f S$ over $B\times A$ and $A\times B$ such that composing with them was equivalent to pulling back and pushing forward. Moreover, $S_f$ and ${}_f S$ are dual; thus, since adjunctions compose, if $X$ is Costenoble-Waner dualizable, so is its pushforward to a point $(\pi_A)_! X$. This clean and easy argument, when they noticed it, replaced a long and messy calculation.

I, however, was unsatisfied with the fact that the maps of base spaces were not actually present in the bicategory, leading me to invent framed bicategories, which are actually double categories with extra properties. The horizontal arrows give it an underlying bicategory, while the vertical arrows supply the missing morphisms, and the additional 2-cells let us characterize the base change objects with a universal property. Soon, I realized that a “framing” on a bicategory was equivalent to giving pseudofunctorial “base change objects” with adjoints, a structure which had been defined by Richard Wood under the name proarrow equipment. However, the double-categorical viewpoint has certain advantages: e.g. it looks a little less *ad hoc*, it makes it easier to define functors and transformations between such structures (this had already been observed by Dominic Verity), and it generalizes to situations where the horizontal 1-cells can’t be composed.

Another thing that bothered me about Peter and Johann’s bicategory was that, to be honest, they hadn’t finished constructing it. They defined the composition and units and constructed associativity and unit isomorphisms, but didn’t prove the coherence axioms. In order to remedy this cleanly and abstractly, I isolated the properties of parametrized spectra that were necessary for the construction, leading to the notion of **monoidal fibration** or **indexed monoidal category**: a pseudofunctor $\mathbf{C}:S^{op} \to MonCat$. The only assumptions needed beyond this are that $S$ is cartesian monoidal and that the “pullback” functors $f^\ast:\mathbf{C}(B) \to \mathbf{C}(A)$ have “pushforward” Hopf left adjoints $f_!$ satisfying the Beck-Chevalley condition for pullback squares (or homotopy pullback squares). Thus, from any such $\mathbf{C}$ we can construct a (framed) bicategory $Fr(\mathbf{C})$, whose objects are those of $S$ and with $Fr(\mathbf{C})(A,B)=\mathbf{C}(A\times B)$. This was the main result of Framed bicategories and monoidal fibrations.

Now since Kate and I were both graduate students of Peter’s at the time, it was natural to put our work together. The mass of material that we produced eventually got sorted into three papers:

Traces in symmetric monoidal categories, an expository paper containing the background we wanted to assume in the other papers, plus some fun unusual examples.

Shadows and traces in bicategories. Kate originally defined shadows and traces in her thesis, but here we took a more systematic category-theoretic perspective. We described a string diagram calculus for shadows, and generalized the basic properties of symmetric monoidal trace that had been axiomatized by Joyal, Street, and Verity in

*Traced monoidal categories*. For example, we showed that if $X$ and $Y$ are right dualizable and $f:X\to X$ and $g:Y\to Y$, then $\tr(f\odot g) = \tr(g) \circ \tr(f)$. You might say the intent was to make bicategorical traces “category-theoretically respectable”.Duality and traces for indexed monoidal categories, in which we finally combined our theses. Using another string diagram calculus, we showed that $Fr(\mathbf{C})$ has a shadow and related its bicategorical traces to symmetric monoidal traces in the $\mathbf{C}(A)$s.

To elaborate on this last one, any $X\in \mathbf{C}(A)$ can be regarded as a 1-cell in $Fr(\mathbf{C})$ in two ways: from $A$ to $1$ or from $1$ to $A$. We denote these by $\hat{X}$ and $\check{X}$ respectively. Then $\hat{X}$ has a (right) adjoint just when $X$ is dualizable in $\mathbf{C}(A)$. If we think of $X$ as an “$A$-indexed family” $(X_a)_{a\in A}$, or as a map $X\to A$ with fiber $X_a$ over $a\in A$, then this generally means just that each $X_a$ is dualizable. However, the trace of $tr(\hat{f})$ contains more information than $tr(f)$, and sometimes strictly more. The former has domain $\langle\langle U_A \rangle\rangle$, which is generally like the free loop space of $A$, and $tr(\hat{f})$ maps a loop $\alpha$ to the trace of $f_a\circ X_\alpha$, where $X_\alpha$ is the monodromy around $\alpha$ and $f_a$ is the action of $f$ over some point $a\in \alpha$. By contrast, the trace of $f$ in $\mathbf{C}(A)$ only knows about these traces for constant loops.

(Right) dualizability of $\check{X}$ is a stronger condition; in parametrized spectra, for $\check{I_A}$ (with $I_A$ the unit of $\mathbf{C}(A)$) it is Costenoble-Waner duality. The composing-adjunctions argument mentioned above shows that if $\check{X}$ is right dualizable, then $(\pi_A)_! X$ is dualizable in $\mathbf{C}(1)$. In particular, a Costenoble-Waner dualizable space is also Spanier-Whitehead dualizable. Now Kate and I showed that $tr(\check{f})$ also contains more information than $tr((\pi_A)_!(f))$: the latter is the composite $I_1 \xrightarrow{tr(\check{f})} \langle\langle A \rangle\rangle \to I_1$. This *also* follows completely formally, from the basic property of bicategorical traces that I mentioned above: if you compose two dualizable 1-cells, then the trace of an induced endomorphism is the composite of the original two traces. (The map $\langle\langle A \rangle\rangle \to I_1$ is the trace of the identity map of the base change object for $\pi_A$.) In particular, this explains how the Reidemeister trace refines the Lefschetz number.

As we worked on these papers, Kate and I were also trying to generalize additivity to bicategories. This was harder than we expected, mainly because triangulated categories are no good. Since their axioms are about nonunique existence, when you add more axioms like Peter’s, you get “there exists an X as in axiom A, and also a Y as in axiom B, together satisfying axiom C, and also …”. Peter’s axioms were manageable, but the bicategorical generalization was too much for us. If we had believed triangulated categories were a “correct thing”, we might have pushed through; but clearly the “correct thing” is a stable (∞,1)-category. However, we weren’t really enthusiastic about using those either. This led us to derivators; which may not really be a “correct thing” either, but their structure is categorically sensible and characterizes objects by universal properties, so they are *much* nicer to work with than triangulated categories.

The obvious place to start was to prove that symmetric monoidal derivators satisfy Peter’s axioms. In May 2011 I visited Kate in Kentucky, and we spent an intense week filling blackboards with string diagrams and checking that squares were homotopy exact. I even wrote a little computer program to do the latter for us. Eventually we joined forces with Moritz Groth, who contributed (among other things) the right definition of “closed monoidal derivator”. But we stayed stuck on things like Peter’s axiom (TC3).

Then in November 2011 we discovered a totally different approach to additivity. Consider the bicategory $Prof(\mathbf{V})$ of categories and profunctors enriched in a symmetric monoidal $\mathbf{V}$. We have embeddings like $\hat{(-)}$ and $\check{(-)}$, but with variance: a functor $X:A\to \mathbf{V}$ becomes a profunctor $\hat{X}:A ⇸ 1$, while a functor $\Phi:A^{op}\to \mathbf{V}$ becomes a profunctor $\check{\Phi}:1 ⇸ A$. As before, $\hat{X}$ is right dualizable when each $X_a$ is dualizable, and $tr(\hat{f})$ records the traces of $f_a\circ X_\alpha$ as $\alpha$ ranges over endomorphisms in $A$. And right dualizability of $\check{\Phi}$ says that $\Phi$ is a weight for absolute colimits in $\mathbf{V}$; thus the composing-adjoints argument implies

**Theorem:** *If $X:A\to \mathbf{V}$ is such that each $X_a$ is dualizable, while $\Phi$ is a weight for absolute colimits, then the weighted colimit $\colim^{\Phi}(X)$ is dualizable.*

I would be surprised if no one had noticed this before, but I don’t recall seeing it written down. Even more interestingly, however, the “composition of traces” property now implies:

**Theorem:** *In the above situation, given $f:X\to X$, the trace of $\colim^\Phi(f)$ is the composite $I \xrightarrow{\tr(1_\Phi)} \langle\langle U_A \rangle\rangle \xrightarrow{\tr(\hat{f})} I$.*

If $\mathbf{V}$ is additive and $A$ is finite, $\langle\langle U_A \rangle\rangle$ is a direct sum of copies of $I$ over “conjugacy classes” of endomorphisms in $A$. Thus, $\tr(\colim^\Phi(f))$ is a *linear combination* of the traces of $f_a\circ X_\alpha$, with coefficients determined by $\Phi$. So for completely formal reasons, we have a very general “linearity formula” (hence the paper titles) for traces of absolute colimits. We obtain Peter’s original additivity theorem by generalizing $\mathbf{V}$ to be a symmetric monoidal *derivator*, with $\Phi$ the weight for cofibers. Absoluteness of this weight is equivalent to stability of $\mathbf{V}$, and its coefficients are $1$ and $-1$, yielding the original formula in a rewritten form:

$L(f/A) = L(f) - L(f|_A).$

Finally, *this* argument can be entirely straightforwardly generalized to bicategories, since we know how to define “categories and profunctors enriched in a bicategory”.

Before going on, I want to emphasize why I consider this a success story for applied category theory. We started out by looking at something that arose naturally in another branch of mathematics; in this case, the Reidemeister trace in topological fixed-point theory. Its definition looked somewhat *ad hoc*, but it was a generalization of something that did have a nice category-theoretic description (the Lefschetz number), so we (and here I mean Kate) trusted that it probably had one too. So we (i.e. Kate) wrote down a categorical description of the structure being used, and then abstracted away the particulars to arrive at a general definition: shadows and bicategorical traces.

This general definition might have looked a bit peculiar to a category theorist, but we took it seriously and went on to study it using category-theoretic tools. We proved a coherence theorem (the string diagram calculus), ensuring that the definition was not missing any axioms. We investigated its abstract properties, not because we had any particular reason to need them at the moment, but because past experience suggested that they would eventually be necessary to know, and useful to have collected in one place.

It then turned out that one of these abstract properties — the composition theorem for traces — enabled a clean and essentially completely formal proof, and generalization, of a result (additivity) that used to require long calculations and lots of commutative diagrams. It took us a while to notice this. But I dare say it would have taken much longer if we hadn’t previously written down the composition theorem. That’s why I say it was a success story for applying category theory seriously.

In fact, there are a couple more similar success stories hiding inside this larger story. The first involves shadows on *framed* bicategories, which were slated for inclusion in *Shadows and traces in bicategories* but got omitted out of consideration for the intended readership. Such a shadow is easiest to define using the double-categorical perspective: it’s a single functor whose domain is the category whose objects are all the endo-horizontal-1-cells and whose morphisms are the squares with equal horizontal sources and targets:

$\array{ A & \xrightarrow{X} & A \\ ^f\downarrow & \Downarrow & \downarrow^f \\ B & \xrightarrow{Y} & B. }$

Such a shadow can be defined on any double category, but in the framed case, a shadow on the horizontal bicategory *extends uniquely* to one on the framed bicategory — by the construction of twisted traces! When we first noticed it, this seemed like just a cute bit of trivia. But in the linearity paper, it turned out to be crucial in identifying the components of $\tr(\hat{f})$, which we did by applying the composition theorem *again* using a base change object, whose trace we identified using this characterization of framed shadows. I’ll omit the details; you can find them in the paper. The point is that just as before, having previously found and studied abstractly the correct categorical structure gave us the tools we needed later on for a concrete result.

The second additional success story has to do with *derivator bicategories*: bicategories enriched over the monoidal bicategory of derivators. We needed these to get linearity for the Reidemeister trace, which is a bicategorical trace and also requries “stable” additivity. In particular, we needed to extend Peter and Johann’s bicategory to a derivator bicategory. This might have been a lot of work, except that in *Framed bicategories and monoidal fibrations* I had already shown that $Fr$ was *2-functorial*. My motivation for this was pure category-theoretic principle: every construction should be a functor. But now, since a monoidal derivator is a 2-functor $Cat^{op}\to MONCAT$ (with extra properties), we can essentially just apply the 2-functor $Fr$ to an “indexed monoidal derivator” to obtain a derivator bicategory. And the indexed monoidal derivator is essentially right there in Peter and Johann’s book. (When we shared these papers with Peter, he remarkede “so that is what we were doing way back then!”)

I’ll finish this long post by mentioning a story that has yet to be told, relating to the construction of $Prof(\mathbf{V})$ for a derivator $\mathbf{V}$. Kate and I needed this bicategory for the linearity story, so we joined forces with Moritz Groth (who had the first idea of how to construct it) to do it in a separate paper. However, the three of us then discovered that $Prof(\mathbf{V})$ would also solve the original problem of proving that Peter’s axioms hold in a stable monoidal derivator. This seemed a good way to make the bicategory paper stand on its own, so we retitled it The additivity of traces in monoidal derivators (and eventually split it in two as well).

(We still don’t know whether Peter’s proof generalizes *directly* to bicategorical trace. Even using derivators, there seems to be another roadblock or two. I’d be happy to elaborate if anyone is interested; it’s possible they could be circumvented with a little thought.)

Now unfortunately, the objects of $Prof(\mathbf{V})$ are not actually categories enriched in $\mathbf{V}$, but ordinary unenriched categories. (No one knows how to define “categories (coherently) enriched in a monoidal derivator”; it may be impossible with the current definition of derivator.) Now given a monoidal derivator $\mathbf{V}:Cat^{op}\to MONCAT$, the hom-category $Prof(\mathbf{V})(A,B)$ should be $\mathbf{V}(A\times B^{op})$. This should look familiar! Indeed, a monoidal derivator *is* a $Cat$-indexed monoidal category, and the construction of $Prof(\mathbf{V})$ is very similar to that of $Fr(\mathbf{C})$ (recall $Fr(\mathbf{C})(A,B) = \mathbf{C}(A\times B)$). However, the pushforward functors in a derivator don’t satisfy the Beck-Chevalley condition for (homotopy) pullback squares, which we required for $Fr$; instead, they satisfy it for comma squares, or more generally homotopy exact squares.

The unit and composition in $Fr(\mathbf{C})$ and $Prof(\mathbf{V})$ also look very similar. For instance, in $Fr(\mathbf{C})$ we compose $X\in \mathbf{C}(A\times B)$ and $Y\in \mathbf{C}(B\times C)$ by pulling them both back to $A\times B\times B\times C$ and tensoring them there, pulling back again along the diagonal to $A\times B\times C$, then pushing forward to $A\times C$. In $Prof(\mathbf{V})$, we compose $X\in \mathbf{V}(A\times B^{op})$ and $Y\in \mathbf{V}(B\times C^{op})$ by pulling them both back to $A\times B^{op}\times B\times C^{op}$ and tensoring them there, pulling back again along the projection of the twisted arrow category to $A\times tw(B)^{op}\times C^{op}$, then pushing forward to $A\times C^{op}$. Note that if $A$, $B$, and $C$ are *groupoids*, then $B^{op} \cong B$ and $tw(B)\simeq B$, and the two constructions do agree. This leads to a natural

**Question:** *Is there an abstract construction producing a (framed) bicategory from some input data, which reduces in one case to $Fr$ and in another case to $Prof$?*

If such a thing existed, maybe we could apply it to “derivators” with $Cat$ replaced by something else, such as the 2-category of internal categories in a topos, or a 2-category of (∞,1)-categories. The latter would include in particular the (∞,0)-categories, i.e. spaces; thus when $\mathbf{V}=Spectra$ it should reproduce Peter and Johann’s bicategory (c.f. also Ando-Blumberg-Gepner).

In fact, Kate and I had already used a version of the linearity story with Peter and Johann’s bicategory replacing $Prof(\mathbf{V})$ to prove the multiplicativity of the Lefschetz number and Reidemeister trace. Roughly, multiplicativity means that given a fibration $E\to B$ with fiber $F$, and compatible endomorphisms $f:E\to E$ and $\bar{f}:B\to B$, we have $L(f) = L(f|_F) \cdot L(\bar{f})$. However, if $B$ is not simply-connected, then $L(f|_F)$ can differ between fibers; thus instead of a simple product we need a sum of fiberwise traces over loops in $B$ — whose coefficients turn out to be none other than the *Reidemeister trace* of $\bar{f}$. In other words, it is another *linearity formula*, with $B$ acting like the weight $\Phi$ and the Reidemeister trace acting like its coefficient vector. And we proved it in the same way: composing the Costenoble-Waner dualizable $\check{I_B}:1⇸ B$ with the fiberwise dualizable $\hat{E}:B⇸1$ yields the ordinary space $E:1⇸1$, and we can apply the composition-of-traces theorem.

Now a fibration $E\to B$ is equivalently an (∞,1)-functor $B \to \infty Gpd$, and the total space $E$ is its (homotopy) *colimit*. Thus, additivity and multiplicativity are really two special cases of a single theorem about absolute colimits of (∞,1)-diagrams; the only thing missing is a construction of the appropriate bicategory.

## Re: The Linearity of Traces

Bravo! And thanks for explaining the story behind how all this came to be; it’s not something that goes in a paper, but certainly helps others to understand the flow of ideas.