The n-Category Café

The additivity of traces is a theorem which says that in any “symmetric monoidal stable homotopy theory”, if you have a map $A\to X$, where $X$ and $A$ are dualizable objects, and compatible endomorphisms $\phi_X:X\to X$ and $\phi_A:A\to A$, then $$\tr(\phi_X) = \tr(\phi_A) + \tr(\phi_{X/A}).$$ Here $\phi_{X/A}$ is the induced map on the quotient $X/A$, and $\tr(-)$ denotes the symmetric monoidal trace.

As a simple example, in the classical stable homotopy category, we might take $X$ to be the suspension spectrum of a manifold or a finite CW complex, with $A$ the suspension spectrum of a submanifold or subcomplex, and the $\phi$’s to be identity maps. In this case, the traces reduce to the classical Euler characteristic, and the additivity theorem describes the additivity of the Euler characteristic on subcomplexes. For more general $\phi$, we obtain Lefschetz numbers, and in other stable homotopy theories we obtain generalizations of these invariants.

The additivity theorem was first proven in generality by Peter May, in his paper “The additivity of traces in triangulated categories”. He began by writing down several axioms, some of them rather complicated-looking, which ought to be satisfied by any triangulated category having a compatible symmetric monoidal structure. Then he proved that:

• These axioms suffice to prove the additivity of traces of identity morphisms (i.e., abstract Euler characteristics).
• The homotopy category of a well-behaved, stable, enriched, symmetric monoidal model category satisfies the axioms.
• Moreover, such a homotopy category also satisfies the more general additivity theorem for traces.

Kate and I first got interested in this because we wanted to prove an analogous additivity theorem for bicategorical traces. For instance, such a theorem would imply additivity formulas for the Reidemeister trace, which is a refinement of the Lefschetz number, and its various generalizations. However, generalizing May’s approach turned out to be quite sticky, and the reason is that triangulated categories are annoying.

Triangulated categories are basically homotopy categories of stable homotopy theories, which remember some of the homotopical structure, namely suspensions and (co)fiber sequences. But they don’t remember enough structure in order to characterize homotopy limits and colimits in any way, which means that various things are asserted to exist, but not to be unique. In any reasonable example, there are particular well-behaved choices of such things, arising from homotopy limits and colimits, but the triangulated category doesn’t have enough structure to characterize them.

This means that when you start adding more axioms, such as May’s monoidal compatibility properties, you start to say things like “there exist objects and morphisms as asserted to exist by axioms 1, 2, and 3, which are moreover compatible in ways 4, 5, and 6, and some additional data satisfying properties 7 and 8 with respect to them all”. You can see a bit of this in May’s axioms already, but his are significantly simplified because he was in a symmetric monoidal situation. The corresponding bicategorical axioms are much worse. Kate and I tried for a while, but eventually we threw up our hands and said “there’s got to be a better way”.

One alternative would be to use model categories, as May did for the proof of more general traces. However, this would also have been tricky in the bicategorical case, because while “locally-model bicategories” do give rise to “locally-triangulated bicategories” in the same way that monoidal model categories give rise to monoidal triangulated categories, not all the bicategories we care about arise in this way. In fact, one of the ones we care about most, the bicategory of parametrized spectra, doesn’t. So if we used model categories, we would have to consider at least two different ways in which a model-categorical structure gives rise to a locally-triangulated bicategory.

Moreover, in my humble opinion, model categories are a pain to work with, because you have to keep fibrantly and cofibrantly replacing things. For concrete calculation, one may want explicit presentations, but for proving formal properties like the additivity of traces, it gets tedious. This is especially true when you start mixing left and right adjoints — such as tensor products and homs — which is necessary when talking about duality. In fact, due to complications of this sort, May didn’t even give a complete proof of the additivity of traces except when all objects of the model category are fibrant — which is true in some classical model categories of spectra built from topological spaces, but not of course for spectra built out of simplicial sets, or sheaves of pretty much anything, or even parametrized topological spectra.

In some sense, of course, the “right” thing to do is to use $(\infty,1)$-categories — or, in our case, $(\infty,2)$-categories. But from what I’ve seen, $(\infty,1)$-categories aren’t much easier to work with than model categories. There’s still a huge amount of technology; it’s just different technology. Instead of fibrations and cofibrations in model categories, you end up talking about various kinds of fibrations and cofibrations between $(\infty,1)$-categories. Any notion, ranging from $(\infty,1)$-categories themselves, to functors, profunctors, limits, colimits, presheaves, etc., tends to have half a dozen different definitions, related by a web of Quillen equivalences constructed by a dozen different authors. And every time you define something new, you have to think about how to express all $\infty$ levels of homotopy coherence in some clever combinatorial way.

Don’t get me wrong: I think $(\infty,1)$-categories are great. They’re a big improvement on model categories in some ways, largely because of the things they can say that model categories aren’t so good at. They give us a language in which to talk directly about the “invariant notion” that model categories have been trying to approximate. The $(\infty,1)$-categorical way of thinking has given us a lot of new insight. But to a certain extent, I feel as though $(\infty,1)$-categories have moved the problem one level up, rather than getting rid of it entirely.

Obviously, tastes differ. I expect some people will always like model categories; others may find $(\infty,1)$-categories easy. But personally, I find derivators to be easier, more intuitive, and less technically demanding than either one. Amazingly, derivators are a purely 1-categorical notion (although we use some 2-categorical technology when working with them, just as we do when working with other sorts of structured categories). In a derivator, you can talk about “limits” and “colimits”, all of which behave very much like they do in plain old comfortable 1-category theory. Everything you do has an honest 1-categorical universal property and is determined up to unique isomorphism. But magically, everything you do is also homotopically meaningful.

(Derivators were invented essentially independently by Grothendieck, Heller, and Franke, and studied further by Cisinski, Maltsiniotis, and others including us. See the paper for references.)

Essentially, the only new thing you have to get used to is treating diagrams as an irreducible notion, rather than something that you can build up by hand out of pieces. This requires a bit of an adjustment, but it’s not really that bad. To give an $A$-shaped diagram in a derivator $\mathcal{D}$, it’s not enough to give an object of $\mathcal{D}$ for every object of $A$ and a morphism in $\mathcal{D}$ for every morphism in $A$ satisfying some commutativity relations. (Every $A$-shaped diagram contains such objects and morphisms, but in general it also contains more data. It can be useful — but is not necessary — to think of that extra data as consisting of specified homotopies exhibiting it as a homotopy coherent diagram.)

Instead, a derivator requires you to limit yourself to a few particular ways to build diagrams, such as:

1. Objects of $\mathcal{D}$ are, by definition, diagrams whose shape is the terminal category.

2. Every morphism in $\mathcal{D}$ gives rise to a diagram whose shape is the interval category. This diagram is determined up to isomorphism, but not up to unique isomorphism (because a homotopy-commutative square can contain more than one homotopy witnessing its commutativity).

3. If $f:A\to B$ is a functor, then every $B$-shaped diagram gives rise to an $A$-shaped diagram by restriction (“precomposition”).

4. Similarly, if $f:A\to B$ is a functor, then every $A$-shaped diagram gives rise to $B$-shaped diagrams by left and right Kan extension.

This may seem very limiting at first, but with a bit of practice, it becomes second nature to manipulate diagrams in this way. It turns out that almost any diagram you want can be built out of these basic operations. And in exchange for this minor limitation, a derivator allows you to do $(\infty,1)$-category theory without worrying about homotopy coherence.

(I’ve posted once or twice before about how the “calculus of Kan extensions” works in a derivator. But I’m a bit hesitant to link to those posts, which were made while I was still struggling to get the hang of derivators, because now I think that the proofs as I gave them there may look overly combinatorial and give a poor flavor of the subject. Better to read the new paper!)

After Kate and I decided to go the derivator route, we teamed up with Moritz Groth, who had written a nice article on the basic theory of derivators. In particular, he showed that every stable derivator gives rise to a triangulated category. And he had separately done a lot of work on monoidal structures on derivators, and come up with good notions of coend and closed structure for derivators.

In the paper we’ve just put out, we develop all the basic theory of monoidal derivators, proving some new facts about stable derivators along the way. We also recall the basic definitions, so that you can read the paper without knowing anything about derivators (although we refer to Moritz’ paper for proofs of basic facts.) Then we apply all that theory to prove that May’s axioms hold for the triangulated category underlying a stable monoidal derivator, and conclude the additivity of traces by copying May’s proof.

From the point of view of concrete applications, this is not really such a significant improvement on May’s theorem; it’s just the use of a different underlying technology for essentially the same proof. But we think that from the point of view of further generalizations, derivators will be significantly easier to work with. In particular, Kate and I are already working on proving the additivity theorem for bicategorical traces (and, in fact, a significantly more general version of it) using derivators.