The n-Category Café

So what is a “trace”, categorically speaking? In generality, it’s some kind of operation acting on endomorphisms in a category. And to be worthy of the name “trace”, it should at least be cyclic: $tr(f g) = tr(g f)$.

In a monoidal category, it turns out to be useful to generalize to “partial” or “twisted” traces: given a morphism $T\colon A\otimes C \to B\otimes C$, we can hope to “trace out” $C$ and end up with a morphism $A\to B$. In the category of vector spaces, $T$ can be written as a tensor with indices $T_{a c}^{b c'}$, and the partial trace is the contraction $\sum_{c} T_{a c}^{b c}$. Having partial traces in mind also makes it clear that the trace of an endomorphism $C\to C$ ought to result in an endomorphism of the unit object $I$.

At this point one can axiomatize the notion of “a trace” on a monoidal category (Joyal, Street, and Verity did this), or alternatively look for canonical ways to construct traces. The best example of the latter is that if the object $C$ is dualizable in a symmetric monoidal category, then we can define the trace of $T\colon A\otimes C \to B\otimes C$ to be the composite $$A \to A\otimes C\otimes C^\ast \xrightarrow{T} B\otimes C \otimes C^\ast \xrightarrow{\cong} B\otimes C^\ast \otimes C \to B.$$ If $C$ is a finite-dimensional vector space, with $C^\ast$ its dual space, then this does in fact compute the classical trace: the sum of the diagonal elements of a representing matrix. (More precisely, it computes the endomorphism of the ground field given by multiplying by the classical trace.) In string diagram notation, this trace looks like a loop: feeding one output strand back into the input.

The really interesting thing to me is what this all has to do with fixed points. Suppose $f\colon X\to X$ is an endomorphism of a finite set. Then the induced endomorphism of the free vector space on $X$ consists of 1s and 0s (like a permutation matrix, except we aren’t asking $f$ to be invertible), and its trace is exactly the number of fixed points of $X$.

By a massive categorification/homotopification of this, we can let $X$ be a closed smooth finite-dimensional manifold, or more generally a finite CW complex, and consider the suspension spectrum $\Sigma^\infty(X_+)$. Don’t worry; you don’t need to know much about spectra. All that matters is that they live in a symmetric monoidal category (the “stable homotopy category”), so we can take the trace therein of the map induced by $f\colon X\to X$. This gives precisely the Lefschetz number $L(f)$, or fixed-point index, of $f$: it counts the fixed points of $f$ “with multiplicity”. In particular, $L(id_X)$ is the Euler characteristic of $X$.

As usual, all sorts of useful things flow from identifying a numerical invariant with a categorical construction. For instance, we can prove by abstract manipulation that the Lefschetz number is additive on disjoint unions: $$L(f\sqcup g) = L(f) + L(g)$$ and multiplicative on cartesian products: $$L(f\times g) = L(f) \cdot L(g).$$ Also, since traces are preserved by strong monoidal functors, it follows that the Lefschetz number can be calculated after passage to (rational) homology. This yields what is probably the most common formula for the Lefschetz number: $$L(f) = \sum_{i\in\mathbb{Z}} (-1)^i H_i(f).$$

Now, to motivate Kate’s bicategorical context, we can go back to algebra and ask: what about traces for endomorphisms of (say finitely generated and free) modules over a noncommutative ring $R$? We can of course write such an endomorphism as a matrix and take the sum of its diagonal elements to get an element of $R$. But when $R$ is not commutative, this is no longer invariant under change of basis. (This is already a problem for $1\times 1$ matrices.)

The correct answer seems brutal at first sight: we just take the quotient of $R$ that forces its multiplication to become commutative: $$Tr(R) = R \Big/ (r \cdot s \sim s \cdot r )$$ and look at the image of the above sum in there. Note that $Tr(R)$ is not a ring any more, just an abelian group. Surprisingly (to me), this approach turns out to retain all the information that we want out of a trace; it’s called the Hattori-Stallings trace.

What Kate realized is that just as the trace for matrices over a commutative ring lives in the symmetric monoidal category of $R$-modules, the Hattori-Stallings trace lives in the bicategory of noncommutative rings and bimodules. A dualizable object in a monoidal category, of course, is the same as an adjoint 1-cell in the bicategory that we get from delooping. Thus, it’s natural to try to generalize the construction of the trace of an endomorphism of an object in a symmetric monoidal category to a trace for an endo-2-cell of an adjoint 1-cell in a bicategory. But it doesn’t quite work, because when we try to write down the composite $$I_X \to C\odot C^\ast \xrightarrow{T} C \odot C^\ast \xrightarrow{???} C^\ast \odot C \to I_Y.$$ we are reminded that in a bicategory there is no symmetry isomorphism. Indeed, not only are $C\odot C^\ast$ and $C^\ast\odot C$ not isomorphic, asking whether they are isomorphic is a type error: they live in different hom-categories.

What we need is to do something to these 1-cells so that $C\odot C^\ast$ and $C^\ast\odot C$ become isomorphic. Thus, we need some sort of operation on endo-1-cells in our bicategory which is (pseudo) cyclic. In other words, we need a vertically categorified trace for 1-cells of the bicategory, in order to define a horizontally categorified trace for endo-2-cells of adjoint 1-cells. A nice example of the microcosm principle! But to avoid confusion with so many different “traces” floating around, Kate invented a different name for the vertically categorified trace: a shadow.

How do we get a shadow on our bicategory? One good way is if our bicategory happens to be symmetric monoidal itself, and the objects (0-cells) in question are dualizable with respect to this monoidal structure: then we can simply categorify the construction above. In particular, the bicategory of rings and bimodules is symmetric monoidal, and every ring is dualizable: its dual is the same ring with the opposite multiplication.

In fact, all the examples of shadows that we know of can be constructed in this way. In practice, however, we usually just write down the shadow explicitly and check that it satisfies a few simple axioms; this is generally easier than constructing a symmetric monoidal structure on a bicategory. For rings and bimodules, the shadow is the quotient constructed above, generalized to $R$-$R$-bimodules other than $R$ itself. For the bicategory of categories and profunctors, the shadow is the trace of a category, generalized to profunctors other than the hom-profunctor.

Now once we have a shadow $Tr(-)$ on our bicategory, we can define the partial trace of any 2-cell $A\odot C \to C\odot B$, where $A$ and $B$ are endo-1-cells and $C$ has an adjoint. (I’m being purposely vague about whether our adjoints are right or left, to avoid getting into order-of-composition issues.) $$Tr(A) \to Tr(A\odot C\odot C^\ast) \xrightarrow{T} Tr(C\odot B \odot C^\ast) \xrightarrow{\cong} Tr(B\odot C^\ast \odot C) \to Tr(B).$$ Note that the bicategorical context has forced us to be more careful about the sort of morphisms we can take partial traces of: the 1-cell being traced out appears on one side of the domain and on the other side of the codomain.

For rings and bimodules, this construction produces precisely the Hattori-Stallings trace. It also has a meaning in fixed-point theory. Consider the bicategory obtained from the bicategory of spans of topological spaces by applying stabilization to all the hom-categories, in which a 1-cell from $X$ to $Y$ is a “parametrized spectrum” over $X\times Y$. (May and Sigurdsson wrote a nice long book about, among other things, the foundations and many applications of this bicategory.) If $X$ is a finite CW complex, then the span $1 \leftarrow X \to X$ becomes an adjoint 1-cell in this bicategory, written $\check{S_X}$. And an endomap $f\colon X\to X$ induces a 2-cell $$\check{S_X} \to \check{S_X} \odot X_f$$ where $X_f$ is a “twisting” or “base change” 1-cell from $X$ to $X$. The bicategorical trace of this 2-cell is precisely the Reidemeister trace of $f$, another classical fixed-point invariant, which refines the Lefschetz number and moreover supports a converse to the Lefschetz fixed-point theorem.

The paper that we’ve just posted makes a start at finding useful things that flow from identifying this numerical invariant with the bicategorical construction. By exploiting formal properties of bicategorical trace, we prove that the Lefschetz number and Reidemeister trace satisfy “multiplicativity” formulas not only for cartesian products, but for “twisted” cartesian products, i.e. fibrations. That is, given an endomorphism of a fibration sequence $$\array{F &\to& E& \to& B\\ ^f\downarrow && \downarrow^g && \downarrow ^h \\ F &\to& E& \to& B}$$ we would like to say something like “$L(g) = L(f) \cdot L(h)$”. Unfortunately, this is not true in general, because even when $B$ is connected, the Lefschetz number of the map $f$ induced on the fiber might depend on which fiber we look at! But the bicategorical context automatically gives us a formula which is true in generality, by bringing the Reidemeister trace into the act.

Now you have plenty of background to go read the paper and notice the bicategories. If you want more category theory, try this paper about symmetric monoidal traces, this paper about bicategorical traces, and this one about constructing such bicategories.