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December 14, 2012

The Additivity of Traces in Stable Monoidal Derivators

Posted by Mike Shulman

By a strange coincidence, three new papers that I’ve been working on (two of them coauthored) are going to appear this month (I hope). They are all very different papers, but each of them is about some kind of generalized category theory. The first one is out today:

This paper is about derivators. I blogged a bit about derivators a few years ago, but at the time I didn’t have anything concrete to point to in terms of their usefulness, only some thoughts about how they might be convenient for various things. This paper makes some progress on that front, by proving the additivity of traces in the context of derivators.

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

As a simple example, in the classical stable homotopy category, we might take XX to be the suspension spectrum of a manifold or a finite CW complex, with AA the suspension spectrum of a submanifold or subcomplex, and the ϕ\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 ϕ\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 (,1)(\infty,1)-categories — or, in our case, (,2)(\infty,2)-categories. But from what I’ve seen, (,1)(\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 (,1)(\infty,1)-categories. Any notion, ranging from (,1)(\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 (,1)(\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 (,1)(\infty,1)-categorical way of thinking has given us a lot of new insight. But to a certain extent, I feel as though (,1)(\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 (,1)(\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 AA-shaped diagram in a derivator 𝒟\mathcal{D}, it’s not enough to give an object of 𝒟\mathcal{D} for every object of AA and a morphism in 𝒟\mathcal{D} for every morphism in AA satisfying some commutativity relations. (Every AA-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 𝒟\mathcal{D} are, by definition, diagrams whose shape is the terminal category.

  2. Every morphism in 𝒟\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:ABf:A\to B is a functor, then every BB-shaped diagram gives rise to an AA-shaped diagram by restriction (“precomposition”).

  4. Similarly, if f:ABf:A\to B is a functor, then every AA-shaped diagram gives rise to BB-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 (,1)(\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.

Posted at December 14, 2012 3:34 AM UTC

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Re: The Additivity of Traces in Stable Monoidal Derivators

Let me take an opportunity to ask a basic question about derivators. As far as I understand we always assume that a derivator is a strict 2-functor (even though the associated Kan extensions won’t be strictly functorial). Is there any reason for that deeper than the fact that the examples constructed from model categories happen to be strict? Is there some potential loss of generality in restricting to strict 2-functors? If we didn’t insist on this restriction would some parts of the theory become considerably more difficult (or perhaps even impossible)?

Posted by: Karol Szumiło on December 14, 2012 10:16 AM | Permalink | Reply to this

Re: The Additivity of Traces in Stable Monoidal Derivators

I don’t think the theory would be any more difficult, but there’s no essential loss of generality in assuming strictness: any pseudofunctor with codomain Cat is equivalent to a strict one.

Posted by: Mike Shulman on December 14, 2012 2:27 PM | Permalink | Reply to this

Re: The Additivity of Traces in Stable Monoidal Derivators

any pseudofunctor with codomain Cat is equivalent to a strict one.

Could you tell me a reference to this fact (or perhaps briefly sketch the argument if it’s not too complicated)?

Posted by: Karol Szumiło on December 15, 2012 4:41 PM | Permalink | Reply to this

Re: The Additivity of Traces in Stable Monoidal Derivators

Power, J. A general coherence result, JPAA 57, 1989.

Posted by: Finn Lawler on December 15, 2012 6:09 PM | Permalink | Reply to this

Re: The Additivity of Traces in Stable Monoidal Derivators

Finn’s answer is a good reference for an abstract context. Here’s how it works a little more concretely in this case: given a pseudofunctor D:KCatD:K\to Cat, with KK a strict 2-category, define QD(k)Q D(k) to be the category whose objects are pairs (f,d)(f,d) with f:kf:\ell\to k in KK and dD()d\in D(\ell), and whose morphisms (f,d)(f,d)(f,d) \to (f',d') are morphisms D(f)(d)D(f)(d)D(f)(d) \to D(f')(d') in D(k)D(k). For g:k 1k 2g:k_1\to k_2 in KK, the functor QD(g):QD(k 1)QD(k 2)Q D(g):Q D(k_1)\to Q D(k_2) takes (f,d)(f,d) to (gf,d)(g f,d); this is strictly functorial since composition in KK is strictly associative. (If KK were merely a bicategory, we’d need to replace ff by a string of composable morphisms or something.) And it’s easy to show QDQ D is pseudonaturally equivalent to DD.

In fact, QDQ D has a universal property: for any 2-functor EE, strict 2-natural transformations QDEQ D \to E are in bijection with pseudonatural transformations DED \to E. Steve Lack’s paper Codescent objects and coherence shows that the strictifications obtained from Power’s coherence theorem always have an analogous universal property.

Posted by: Mike Shulman on December 15, 2012 6:49 PM | Permalink | Reply to this

Re: The Additivity of Traces in Stable Monoidal Derivators

Thanks for both your answers!

Posted by: Karol Szumiło on December 17, 2012 6:20 AM | Permalink | Reply to this

Re: The Additivity of Traces in Stable Monoidal Derivators

Here’s a question about derivators: is there a construction of a “slice derivator”?

Posted by: Charles Rezk on December 14, 2012 3:44 PM | Permalink | Reply to this

Re: The Additivity of Traces in Stable Monoidal Derivators

I think that’s one of the most important open questions! I don’t know of any way to define one; I suspect that some improved notion of derivator would be necessary.

Note, though, that some things we often state using slice categories can still be stated in a derivator. For instance, you can say what it means for colimits to be stable under pullback. I suspect that you can even say what it means to be locally cartesian closed.

Posted by: Mike Shulman on December 14, 2012 4:26 PM | Permalink | Reply to this

Re: The Additivity of Traces in Stable Monoidal Derivators

Maybe you already know this, but I think it is worth recalling the following (as it tells how important is this story of traces). Somewhere in section 69 of Pursuing stacks, Grothendieck explains that he started to think about a good generalisation of the notion of triangulated category (which led him to the theory of derivators), specifically because, in SGA5, they could not manage to express (whence compute) traces using only the theory of triangulated categories (in order to get nice formulas for L-functions).

Posted by: Denis-Charles Cisinski on December 16, 2012 2:10 AM | Permalink | Reply to this

Re: The Additivity of Traces in Stable Monoidal Derivators

Actually, no, I didn’t know that! Was he talking about the same kind of traces we are?

Posted by: Mike Shulman on December 16, 2012 2:16 AM | Permalink | Reply to this

Re: The Additivity of Traces in Stable Monoidal Derivators

I think so (in fact, maybe something a little more general, in the spirit of what you did with Ponto in the setting of bicategories). If you like French, you may have a look by yourself. See the second part of Exposé IIIB, by Illusie, starting on page 162 of

A. Grothendieck et al., Cohomologie l-adique et Fonctions L, Lecture Notes in Mathematics, vol. 589, Springer, 1977.

You will see that, instead of derivators, Illusie considers the derived categories of filtered complexes as well as the natural functors which relate it with the usual derived category (these data are in fact a derivator in disguise); see paragraph 5.5 and beyond in loc. cit.

Posted by: Denis-Charles Cisinski on December 16, 2012 10:39 PM | Permalink | Reply to this

Re: The Additivity of Traces in Stable Monoidal Derivators

Thanks! I’ll try to have a look.

Kate and I are currently working on generalizing additivity to bicategorical traces. It should be possible to adapt the derivator-ized version of May’s proof to work in the context of a derivator bicategory. However, that’s not what we’re doing! Instead, we have a totally new approach, also using derivators but taking advantage of the formal properties of bicategorical trace. This results in a much more general additivity theorem that has interesting special cases in non-homotopical contexts as well. Here are some slides from a talk I gave about the new approach.

Posted by: Mike Shulman on December 17, 2012 5:06 AM | Permalink | Reply to this

Re: The Additivity of Traces in Stable Monoidal Derivators

Is there any way to connect derivators with homotopy type theory? What would happen if you formed the homotopy prederivator of an (,1)(\infty, 1)-topos?

Posted by: David Corfield on December 17, 2012 10:06 AM | Permalink | Reply to this

Re: The Additivity of Traces in Stable Monoidal Derivators

Well, it should be possible to describe a notion of ‘derivator topos’ by expressing all the exactness properties of an (,1)(\infty,1)-topos in a derivator. I think someone ought to do that and then go through Higher Topos Theory writing everything with derivators. (For some things, it might be necessary to first solve the slice category problem.)

However, at present the only techniques we have for building models of type theory require the extra strictness of a model category. It would be lovely if you could make do with a derivator, but I’ve thought about it on and off for a while and not gotten anywhere yet.

The idea that’s seemed the most promising to me so far is to free ourselves from the ‘linear’ perspective on contexts and regard a context as indexed by any (finite) inverse category, describing its dependency structure. For instance, a context of the form (x:A),(y:B),(z:C(x,y))(x:A),(y:B),(z:C(x,y)), where A and B are independent but C depends on both of them, is structured by the inverse category (ACB)(A\leftarrow C \to B), while a context of the form (x:A),(y:B(x)),(z:C(x))(x:A),(y:B(x)),(z:C(x)) is structured by the inverse category BACB\to A\leftarrow C. I used this ‘diagrammatic’ view on contexts in this paper to build new models of homotopy type theory from old ones (special cases like (ACB)(A\leftarrow C\to B) have been used by type theorists before, but I think the categorical viewpoint is underappreciated).

Now of course derivators come with a notion of diagram on any such inverse category. It seems reasonable that you should be able to formulate all the basic type-forming operations as acting on contexts of this sort, and express them in a (locally cartesian closed) derivator. But that doesn’t say anything about how to deal with coherence. For this, it seems a little more promising to work with diagrams that encapsulate an entire derivation tree, rather that just a single context. But what happens to definitional equality? It seems like it must become isomorphism, and then we need some kind of coherence.

I would love to talk about this further with anyone who is interested. Here are a couple of concrete questions to begin with:

  • How do we express the action of type formers like dependent sum and dependent product as operations on generalized context diagrams in a derivator?

  • Is definitional equality coherent? For instance, consider the rewriting system of β\beta-reduction as giving generators for a category. Can we give some relations on these generators which result in a category, each component of which has a contractible nerve (such as by having a terminal object)? It seems like the answer to this might be known… does anyone know it?

Posted by: Mike Shulman on December 17, 2012 12:52 PM | Permalink | Reply to this

Re: The Additivity of Traces in Stable Monoidal Derivators

…to free ourselves from the ‘linear’ perspective on contexts

That reminds me of Hintikka’s work on independence-friendly logic, where you use branched quantifiers.

Posted by: David Corfield on December 17, 2012 9:05 PM | Permalink | Reply to this

Re: The Additivity of Traces in Stable Monoidal Derivators

Independence-friendly logic does look similar, thanks for the reference! The only other related reference I know of is a paper by Martin Hyland and Andy Pitts called The theory of constructions: categorical semantics and topos-theoretic models, which uses “contexts” that are partial orders on finite sets of variable declarations.

Posted by: Mike Shulman on December 18, 2012 3:12 PM | Permalink | Reply to this

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