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November 4, 2011

Traces in Indexed Monoidal Categories

Posted by Mike Shulman

After a gestation period of many years, the following paper is finally ready for the light of day:

  • Kate Ponto and Michael Shulman, Duality and traces in indexed monoidal categories. We’re currently having issues with the arXiv compiler, but you can get it from my web page. There are three versions:
    • If you’re going to read it on a screen, or print it in color, get this version
    • If you’re going to print it in black and white, get this version
    • If you’re going to do both, but you only want to download one file, get this version.

There are also (at least) two ways to read this paper. If you’re mainly interested in duality and trace (especially their applications in fixed-point theory), then you can read sections 1–8 and stop there. On the other hand, if you’re mainly interested in indexed monoidal categories and their string diagrams, you can read sections 1–3, then 9–10. (But you may still want to go back and read sections 4–8 and 11–12, to see the applications of the string diagrams.)

I’m not going to say anything about traces today—you can read the paper for that. But I’ll say a bit about indexed monoidal categories and string diagrams, which I expect will be of interest to several people here; especially since we got the basic idea of these string diagrams from another discussion here.

An indexed monoidal category is one of my favorite categorical structures. In its simplest form it consists of a category SS and a pseudofunctor S opMonCatS^{op} \to MonCat, which we write as A(C A, A,I A)A\mapsto (C^A, \otimes_A, I_A). By the usual Grothendieck construction, this pseudofunctor can of course be regarded as a fibration. And if SS has finite products, then then the “fiberwise” monoidal structures A\otimes_A can also be “Grothendieckified” into an “external product” :C A×C BC A×B \boxtimes\colon C^A \times C^B \to C^{A\times B} defined by MN=π 2 *M A×Bπ 1 *NM\boxtimes N = \pi_2^\ast M \otimes_{A\times B} \pi_1^\ast N. This makes the total category of the fibration a monoidal category and the fibration itself a strict monoidal functor (when SS has its cartesian monoidal structure); I call this a monoidal fibration. Moreover, we can recover A\otimes_A from \boxtimes via M AN=Δ A *(MN)M\otimes_A N = \Delta_A^\ast (M\boxtimes N), so the two structures have the same information.

Here are some nice examples:

  • S=SetsS=Sets, C A=C^A= AA-indexed families of objects of CC, for any monoidal category CC.
  • S=GpdS=Gpd, C A=C^A= AA-diagrams of objects of CC
  • S=S= any category with pullbacks, C A=S/AC^A = S/A
  • S=TopS=Top, C A=C^A= spectra parametrized over AA
  • S=GrpS=Grp or TopGrpTopGrp, C A=C^A= sets or spaces with an action by AA
  • The homotopy category of any of the above equipped with a homotopy theory

In many cases, the reindexing functors f *:C BC Af^\ast\colon C^B \to C^A induced by a morphism f:ABf\colon A\to B in SS all have left adjoints f !f_!. If these left adjoints satisfy the Beck-Chevalley condition for all pullback squares in SS, then the indexed category is traditionally said to have indexed coproducts. For many applications, though, we only need condition for a few pullback squares, which coincidentally (?) happen to be those that are pullbacks in any category with finite products (whether or not it even has all pullbacks).

What’s especially interesting, though, is that the cases we care about for fixed-point theory are mostly homotopy categories, and in this case the Beck-Chevalley condition is usually not satisfied for all these pullback squares; only for those that are homotopy pullbacks. Fortunately, this includes most of the pullback squares built out of products; see section 3 of the paper for a list.

Back in August of last year, I asked whether there was a string diagram calculus for these things, and was pointed to this paper by Todd Trimble and Geraldine Brady, about representing predicate calculus categorically using C.S. Peirce’s “system beta”. This turned out to be just what we needed, except that we needed a three-dimensional “surface diagram” sort of calculus too in order to represent morphisms. (Predicate calculus is all about indexed posets, so there’s no need to keep track of morphisms, only \le relationships.) Then Daniel Schäppi suggested a “schematic” sort of surface diagram picture as being easier to draw and understand, and that led to the pictures we ended up with in the paper. (Daniel’s diagrams are actually even more schematic than ours; hopefully we’ll get a guest post from him soon when his thesis is finished.)

I’d like to show some pictures of the string diagrams, but it would be a lot of work to extract them into a form that I can post on the blog. So to get a taste of the string diagrams, I suggest you have a look at the first half of these slides from my talk last week at the Category Theory Octoberfest. (The second half of the slides gives an overview of the part of the paper that’s about duality, traces, and fixed-point theory. I’m still looking for a good answer to the question on the final slide, which is more or less the same question I asked here back in August.) Then, if you’re hooked, you can read the paper.

Posted at November 4, 2011 9:59 PM UTC

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18 Comments & 2 Trackbacks

Re: Traces in Indexed Monoidal Categories

Wowee! Congratulations on getting this finished — it looks really nice!

Posted by: Tom Leinster on November 5, 2011 2:13 AM | Permalink | Reply to this

Re: Traces in Indexed Monoidal Categories

Thanks! It’s exciting to finally finish something again; it feels like forever since I did that. (-:

Posted by: Mike Shulman on November 5, 2011 2:44 AM | Permalink | Reply to this

Re: Traces in Indexed Monoidal Categories

Hi Mike,

I haven’t had a chance to look at the pdf yet (am on the train, cell phone connection too slow for your graphics, it seems), so possibly the following questions finds plenty of reply there.

But here I am wondering: clearly the above list of examples wants to contain also monoidal derivators?

Posted by: Urs Schreiber on November 5, 2011 3:16 PM | Permalink | Reply to this

Re: Traces in Indexed Monoidal Categories

clearly the above list of examples wants to contain also monoidal derivators?

Yes, but. A monoidal derivator is, indeed, an indexed monoidal category, and its reindexing functors indeed have left adjoints. However, it fails to satisfy all the Beck-Chevalley conditions that we need for this paper. A derivator, by definition, satisfies the Beck-Chevalley condition for comma squares, but not all the pullback squares we need are comma squares in CatCat.

If we restrict the domain categories of our derivator to groupoids, of course, we do get all the conditions necessary, since for groupoids, comma categories and (pseudo) pullbacks coincide. It seems that the way of thinking presented in this paper really only applies to indexings by “groupoidal” objects.

However, you can still do many of the same things with a monoidal derivator. In particular, the essential construction of the current paper is the same as that of this old paper: from an indexed monoidal category we build a bicategory with Hom(A,B)=C A×BHom(A,B) = C^{A\times B}. (We then get refined notions of trace from Kate’s bicategorical traces in this bicategory.)

But clearly, if DD is a monoidal derivator, then the bicategory you want to build is a bicategory of profunctors, with Hom(A,B)=D(B op×A)Hom(A,B) = D(B^{op}\times A). Traces in this bicategory exhibit many of the same phenomena as in the bicategory constructed from a monoidal fibration, and I think we’ll have some other very interesting things to say about them in the near future. Stay tuned!

Posted by: Mike Shulman on November 5, 2011 4:52 PM | Permalink | PGP Sig | Reply to this

Re: Traces in Indexed Monoidal Categories

In skimming through figures 14 through 32, I am seeing these as possible relationships between foams. Is that what you have in mind, or is it just a manifestation of my wacko sense of visualization?

If the former, then I am interested in, and can assign homological information to certain relations among foams and higher dimensional analogues. I am pretty sure the foams that intrigue me are more limited than your axioms, but could be a common starting point for something.

Posted by: Scott Carter on November 5, 2011 4:15 PM | Permalink | Reply to this

Re: Traces in Indexed Monoidal Categories

Interesting question! What we have in mind, insofar as we have anything in mind beyond the schematic pictures you see, are surfaces with singularities (these being the “standard” surface diagrams for monoidal bicategories). For instance, figure 15(b) can be seen as a simple topological deformation between two such surfaces. But maybe this is the same thing; I don’t know what a “foam” is.

Posted by: Mike Shulman on November 5, 2011 4:55 PM | Permalink | Reply to this

Re: Traces in Indexed Monoidal Categories

A *2-foam* is a surface with branched curves. The Escher ping-pong table which is the standard dual to a tetrahedron or the deformation from
(Y ox I)(Y)=(I ox Y)Y gives a local model for any 2-foam. More specifically, a 2-foam is a compact topological space for which any point has a neighborhood that is homeomorphic to a point of the dual to the tetrahedron.

Such 2-foams can be embedded in 4-space in a way that is analogous to a knotted trivalent graph embedded in 3-space. Think of 2-dimensional analogue of your ipod headphones.

The Frobenius axiom between multiplication and comultiplication also describes the dual to a tetrahedron. Thus usually three sheets converge at a *branch arc* and a pair of branch arcs can cross. So your 15(b) has trivalent branch lines coming from the triangles.

Posted by: Scott Carter on November 5, 2011 5:55 PM | Permalink | Reply to this

Re: Traces in Indexed Monoidal Categories

Interesting. I’d like to hear what sort of homological information would be assigned to some of our diagrams. But it sounds like you’re right that your foams are more restricted than our surfaces: in theory, a singularity on one of our surfaces could join an arbitrary finite number of sheets.

Posted by: Mike Shulman on November 7, 2011 2:41 AM | Permalink | Reply to this

Re: Traces in Indexed Monoidal Categories

in theory, a singularity on one of our surfaces could join an arbitrary finite number of sheets.

Is there a name for surfaces of this kind? Perhaps you give a way to devise algebraic invariants for them.

Posted by: David Corfield on November 8, 2011 2:20 PM | Permalink | Reply to this

Re: Traces in Indexed Monoidal Categories

I don’t know of a name for exactly the kind of surfaces we care about, partly because I don’t think we have a formal description of them. Probably the right context in which to be talking about invariants of this sort is a monoidal bicategory. The putative surface diagram calculus for a monoidal bicategory would describe a precise sort of surface, and conclude that any monoidal bicategory could be used to give “invariants” of such a thing. (The cobordism hypothesis in 2 dimensions is a special case of this, where the inputs have no singularities.)

Posted by: Mike Shulman on November 8, 2011 5:07 PM | Permalink | Reply to this

Re: Traces in Indexed Monoidal Categories

Do your findings suggest anything interesting in the case of predicate logic and in general example 3.9 on hyperdoctrines? What about group or groupoid representations? We spoke about these in the context of the Beck-Chevalley condition back here.

Posted by: David Corfield on November 8, 2011 9:38 AM | Permalink | Reply to this

Re: Traces in Indexed Monoidal Categories

Do your findings suggest anything interesting in the case of predicate logic and in general example 3.9 on hyperdoctrines?

Hyperdoctrines are very interesting indexed monoidal categories. As you know, our string diagram calculus came from them. But I don’t think that duality and trace are going to have much to say in hyperdoctrines. A hyperdoctrine is cartesian monoidal, so there are no interesting dualities, and even if you looked at some non-cartesian “quantum hyperdoctrine”, traces in a poset aren’t going to be very interesting.

What about group or groupoid representations? We spoke about these in the context of the Beck-Chevalley condition back here.

Well, groupoid (hence also group) representations are an indexed monoidal category, as I mentioned above. And we found the same thing that Jeffrey Morton did – the Beck-Chevalley condition for groupoids (including groups, as a special case) only holds for pseudo pullbacks, not strict ones in general. We have a concrete counterexample, too – the strict pullback square G G G G×G\array{ G & \to & G \\ \downarrow & & \downarrow \\ G & \to & G\times G } is almost never going to satisfy the Beck-Chevalley condition. In fact, the failure of this Beck-Chevalley condition is crucial for the interesting fixed-point theory applications, since it means that the “shadow” of a groupoid is different from its homotopy quotient, and the shadow is where the refined fixed-point invariants like the Reidemeister trace live.

To reply to a few points raised at the discussion you linked to, which the participants probably know by now:

  • When looking for a Beck-Chevalley condition, there’s always a map in one direction; the question is precisely whether it is an isomorphism.
  • Any strict pullback of a fibration between groupoids is also (up to equivalence) a pseudo pullback. And any surjection of groups is a fibration between one-object groupoids. So the Beck-Chevalley condition holds for strict pullbacks of groups where one leg is a surjection.
  • There is a Beck-Chevalley condition for squares of categories, but it requires the square to be a comma square, not a pseudo pullback – it just so happens that the two coincide for groupoids.
Posted by: Mike Shulman on November 8, 2011 5:07 PM | Permalink | PGP Sig | Reply to this

Re: Traces in Indexed Monoidal Categories

Learning Peirce’s system, you spend a lot of time following his rules about ‘seps’, which designate negation. Does this correspond to a feature of the particular kind of hyperdoctrine that is classical logic that one might find in some more general indexed monoidal categories?

As I mentioned, this diagram of Osman is very Peircean.

Posted by: David Corfield on November 11, 2011 10:47 AM | Permalink | Reply to this

Re: Traces in Indexed Monoidal Categories

Perhaps I just mean are there interesting indexed involutive monoidal categories?

Posted by: David Corfield on November 11, 2011 11:00 AM | Permalink | Reply to this

Re: Traces in Indexed Monoidal Categories

Does this correspond to a feature of the particular kind of hyperdoctrine that is classical logic that one might find in some more general indexed monoidal categories?

Well, one thing we’ve mentioned here before is that a cartesian closed category in which A0 0 AA\cong 0^{0^A} (i.e. the categorified law of double-negation holds) must be a poset. So pretty much any example that’s “involutive” in that sense is going to have a “logical” flavor.

If you drop cartesianness, then you probably get something like an (indexed) ∗-autonomous category. The first two examples I mentioned above ought to be ∗-autonomous if the ordinary monoidal category C is. I don’t know whether there are other examples.

Finally, IIRC the idea of “sep lines” can also be weakened to some sort of boxes that merely denote exponentials, and of course there are plenty of interesting examples of closed indexed monoidal categories; many of the examples I mentioned above are closed.

Posted by: Mike Shulman on November 11, 2011 7:44 PM | Permalink | PGP Sig | Reply to this

Re: Traces in Indexed Monoidal Categories

I must say that when I first became interested in category theoretic diagrammatic notation around 10 years ago, I thought we’d see an explosion in its usage. Perhaps things are moving a little faster than the uptake of Peirce’s diagrams, but there still seems a way to go until it becomes instinctual to see which categorical setting one is in and then reach for the appropriate diagrammatic calculus.

Posted by: David Corfield on November 14, 2011 9:08 AM | Permalink | Reply to this

Re: Traces in Indexed Monoidal Categories

Yeah, there’s a lot of resistance. I’ve had a few people tell me flat-out that they don’t like string diagrams or find them ugly.

Posted by: Mike Shulman on November 14, 2011 5:45 PM | Permalink | Reply to this
Read the post Productive Homotopy Pullbacks
Weblog: The n-Category Café
Excerpt: Certain squares built out of cartesian products are always homotopy pullbacks, in any derivator.
Tracked: November 19, 2011 7:28 AM
Read the post The Multiplicativity of Fixed-Point Invariants
Weblog: The n-Category Café
Excerpt: A bicategorical point of view on fixed-point invariants implies a general multiplicativity theorem.
Tracked: March 6, 2012 4:43 AM

Re: Traces in Indexed Monoidal Categories

Your paper gets a mention here. Looks like something Cafe folk should understand.

Posted by: David Corfield on June 3, 2013 1:54 PM | Permalink | Reply to this

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