## June 13, 2010

### This Week’s Finds in Mathematical Physics (Week 299)

#### Posted by John Baez

In week299 of This Week’s Finds, hear about the school on Quantum Information and Computer Science that was recently held in Oxford, and also the workshop on Quantum Physics and Logic. Watch videos of the talks! Learn how classical structures give Frobenius algebras, and how complementary classical structures almost give bialgebras:

Admire the program called “Quantomatic”, which automatically carries out string diagram computations. Hear about Aaron Fenyes’ no-cloning theorem for classical mechanics. And finally, learn about “pre-Lie algebras” — algebraic gadgets with deep connections to trees, operads, and the work of Connes and Kreimer on renormalization in quantum field theory.

From Coecke’s paper Quantum picturalism:

Experiment

Consider ten children of ages between six and ten and consider ten high-school teachers of physics and mathematics. The high-school teachers of physics and mathematics will have all the time they require to refresh their quantum mechanics background, and also to update it with regard to recent developments in quantum information. The children on the other hand will have quantum theory explained in terms of the graphical formalism. Both teams will be given a certain set of questions, for the children formulated in diagrammatic language, and for the teachers in the usual quantum mechanical formalism. Whoever solves the most problems and solves them in the fastest time wins. If the diagrammatic language is much more intuitive, it should in principle be possible for the children to win.

Posted at June 13, 2010 4:34 PM UTC

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### Re: This Week’s Finds in Mathematical Physics (Week 299)

Alain Connes and Dirk Kreimer, Hopf algebras, renormalization and noncommutative geometry…

I’d like to add that the Connes-Kreimer theory is explained in the book “Noncommutative Geometry, Quantum Fields and Motives” that is available for free here.

André Joyal calls it “a quantum leap towards the mathematics of the future”.

So “Quantum leap” is now a phrase that means a big step, instead of the “smallest possible”? :-)

…these trees have their root on top - I hear that’s how they grow in Europe.

They do :-), one omnipresent kind that grows this way is the B-tree. (I heard a talk of its inventor Rudolf Bayer a couple of years ago, and he confirmed that “B” does not have a specific meaning. It’s a little bit like the M in M-theory).

In physics we say position and momentum are complementary because if you know everything about one, you know nothing about the other.

Nitpicking: This may be confusing to someone who did not take a QM class, because taken at face-value this is true for the spacecoordinates (x ,y, z) of a particle, too (if you know one you don’t know anything about the others). Something along the lines “complementary means that a Heisenberg uncertainty relation holds” would avoid this.

Posted by: Tim van Beek on June 14, 2010 9:51 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

Thanks for reminding me that Connes and Marcolli’s book is free online! This will increase the chance that I learn all this stuff. It’s a very readable book.

So “quantum leap” is now a phrase that means a big step, instead of the “smallest possible”? :-)

That slippage has been widely noted. But I thought maybe Joyal knew that, and was making a pun on the relation between combinatorics, category theory, and quantum theory!

Nitpicking: This may be confusing to someone who did not take a QM class, because taken at face-value this is true for the space coordinates $(x,y,z)$ of a particle, too (if you know one you don’t know anything about the others).

What I meant is that if you know everything about position, you are required to know nothing about momentum. Roughly: when a particle has a known position, the probability of measuring it to have some particular momentum is completely independent of the momentum.

This the basic idea behind mutually unbiased bases — but the idea is technically simpler for finite-dimensional Hilbert spaces, since my ‘roughly’ true statement above involves ‘eigenstates’ that aren’t really in the Hilbert space (the Dirac delta is not in $L^2$) and ‘probability measures’ that aren’t really finite (Lebesgue measure).

In the finite-dimensional case we can quite precisely say that two orthonormal bases are unbiased iff a state in one basis has equal probabilities of being found in all the states in the other basis.

Something along the lines “complementary means that a Heisenberg uncertainty relation holds” would avoid this.

I’m not sure people who haven’t taken a quantum mechanics class would find this clearer! I think I’ll say:

In physics we say position and momentum are complementary because if you know everything about one, you cannot know anything about the other.

Posted by: John Baez on June 14, 2010 4:14 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

Thanks for reminding me that Connes and Marcolli’s book is free online!

It’s got a very nice appendix about operator algebras, and it explains Feynman graphs to mathematicians, and Connes’ work about the standard model.

Ugh, when I took a look for the first time my estimate was “math buzzwords I know nothing about” $\approx$ one per page $\approx$ 700, time to understand one of those $\approx$ 1,5 $\pm$ 0,5 man days, ergo estimated time for a full reading 3 $\pm$ 1,5 years. Thomas Thiemann wrote in his lecture notes about canonical quantum gravity that the number of people working on these ideas (standard model from noncommutative geometry) is about $10^1$. No wonder that there aren’t more…

Posted by: Tim vB on June 14, 2010 5:07 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

Sorry, it was silly of me to say:

I have spent decades of hard work learning the material needed to understand this stuff! What I meant is that it starts exactly where I am and goes from there in a nice way.

Thomas Thiemann wrote in his lecture notes about canoncical quantum gravity that the number of people working on these ideas (standard model from noncommutative geometry) is about $10^1$. No wonder that there aren’t more…

But I don’t think there really should be more. There are lots of places where we urgently need more smart people working on something, but it’s not clear this is one.

Posted by: John Baez on June 14, 2010 5:37 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

Sorry, it was silly of me to say…

No problem :-)

In fact it is still a long term goal of mine to understand this a little bit better - it’s just a little observation of mine that, if you need three years at least to be able to contribute to the topic, that pretty much excludes it as a topic for a PhD. Gerd Faltings once said in an interview that it took him 2 years of hard work to prove the Mordell conjecture, and that he was able to do it because he already had tenure (at the age of 27, which is still a record for mathematics in Germany, I think). For someone without tenure it is a considerable risk to invest all the time in one project that may not lead to any publications for years (or at all).

BTW: Has anybody done some interesting physics with the Connes-Kreimer theory yet? Something like: “Hey, now that we have a better grip at the combinatorics of Feynman graphs, we can calculate the next 5 digits of the anomalous magnetic dipole moment of the electron”?

Posted by: Tim van Beek on June 14, 2010 7:16 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

In the latest Newsletter from the Oxford Math Inst, David Craven comments on precisely a three year success, though year 4 was needed for the rewrite.
“By the end of the second year, you’re usually on your way to having at least some idea of what you are doing.’

Posted by: jim stasheff on June 15, 2010 12:48 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

Tim wrote:

it’s just a little observation of mine that, if you need three years at least to be able to contribute to the topic, that pretty much excludes it as a topic for a PhD.

Well, maybe it means those topics are too deep to make good PhD theses. The Mordell Conjecture, for example, would have made a very risky PhD topic. I’m convinced that there are lots of things that are quite straightforward to do — if you’ve been told what they are — yet make good PhD theses.

As for your other point: I don’t know of the Connes-Kreimer work contributing to actual physics. That’s a good question! But people have certainly done some big industrial-strength calculations:

• D. J. Broadhurst and D. Kreimer, Renormalization automated by Hopf algebra, preprint available as arXiv:hep-th/9810087

Abstract. It was recently shown that the renormalization of quantum field theory is organized by the Hopf algebra of decorated rooted trees, whose coproduct identifies the divergences requiring subtraction and whose antipode achieves this. We automate this process in a few lines of recursive symbolic code, which deliver a finite renormalized expression for any Feynman diagram. We thus verify a representation of the operator product expansion, which generalizes Chen’s lemma for iterated integrals. The subset of diagrams whose forest structure entails a unique primitive subdivergence provides a representation of the Hopf algebra HR of undecorated rooted trees. Our undecorated Hopf algebra program is designed to process the 24,213,878 BPHZ contributions to the renormalization of 7,813 diagrams, with up to 12 loops. We consider 10 models, each in 9 renormalization schemes. The two simplest models reveal a notable feature of the subalgebra of Connes and Moscovici, corresponding to the commutative part of the Hopf algebra HT of the diffeomorphism group: it assigns to Feynman diagrams those weights which remove ζ values from the counterterms of the minimal subtraction scheme. We devise a fast algorithm for these weights, whose squares are summed with a permutation factor, to give rational counterterms.

Posted by: John Baez on June 14, 2010 7:33 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

But people have certainly done some big industrial-strength calculations…

Yes, that’s the kind of work I had in mind - although I should have looked at it earlier: it is mentioned in Dirk Kreimer’s book Knots and Feynman diagrams. It’s on my reading list but so are many other books, too…

Christan Brouder has written an expository paper that is on the arXiv (I don’t have access to Springer journals and therefore no access to the paper that Eugene Lerman mentioned): Quantum field theory meets Hopf algebra.

Runge-Kutta methods, really? Everything is connected in mathematics…

Posted by: Tim van Beek on June 15, 2010 8:56 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

The wikipedia article on the Butcher group is another interesting source of information.

I find it remarkable that John Butcher came across the Hopf algebra of trees some 26 years before Connes and Kreimer.

By the way, Brouder’s “Trees, Renormalization and Differential equations” is available here.

Posted by: Eugene Lerman on June 15, 2010 5:37 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

Eugene wrote:

I find it remarkable that John Butcher came across the Hopf algebra of trees some 26 years before Connes and Kreimer.

Yeah, but I’d say it a bit differently, like:

I find it remarkable that information in mathematics is so poorly organized that 26 years after Butcher discovered the Hopf algebra of trees, Connes and Kreimer found it easier to redo his work than to look it up!

Things are getting better, though. Now, thanks to Google, Wikipedia, MathOverflow and the nLab, if someone gets interested in a Hopf algebra of trees, they should just do a Google search under Hopf algebra tree.

Thanks to the link to Brouder’s paper! I added it to week299.

Posted by: John Baez on June 15, 2010 7:36 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

John wrote

Now, thanks to Google, Wikipedia, MathOverflow and the nLab, if someone gets interested in a Hopf algebra of trees, they should just do a Google search under Hopf algebra tree.

And no papers by John C Butcher will come up. Because he didn’t actually use the words “Hopf algebra;” he just saw the structure. OK, I am shooting off the hip here. But the index of the second edition of his Numerical Methods for Ordinary Differential Equations does not have the word “Hopf” in it, for example. And flipping through the book, I can’t find any Hopf algebras.

Posted by: Eugene Lerman on June 15, 2010 9:34 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

Okay, that helps explain it. But did nobody else read his work and make the connection? There were 26 years for someone to notice that he’d found a Hopf algebra…

By the way, while we’re pondering missed opportunities, I guess it’s worth re-emphasizing that Cayley found a relation between trees and vector fields here:

• Arthur Cayley, On the theory of the analytical forms called trees, Phil. Mag. 13 (1857), 172-176.

Apparently his work can now be nicely understood using the fact that every torsion-free flat connection gives a pre-Lie algebra, while trees are a basis of the free pre-Lie algebra on one generator. Apparently he was looking at the standard torsion-free flat connection on vector fields on the real line. But I haven’t looked at this paper!

Posted by: John Baez on June 15, 2010 9:51 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

Another relatively unknown ‘ancient’ paper:

MR0025715 (10,53c) Otter, Richard. The number of trees. Ann. of Math. (2) 49, (1948), 583–599.

Otter and I were colleagues at Notre Dame, but I didn’t discover this relation to my own work until after I’d left ND!

Posted by: jim stasheff on June 17, 2010 1:52 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

For a simple elaboration of Cayley’s ideas related to some of the thoughts in this thread, you might look at the pdf file “Mathemagical Forests” at http://tcjpn.spaces.live.com/default.aspx

Posted by: Tom Copeland on July 12, 2010 3:05 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

John said;

I find it remarkable that information in mathematics is so poorly organized that 26 years after Butcher discovered the Hopf algebra of trees, Connes and Kreimer found it easier to redo his work than to look it up!

Eugene objected:

And no papers by John C Butcher will come up…And flipping through the book, I can’t find any Hopf algebras.

Affirmative, I just skimmed the first edition from 2003 and he seems to deliberately avoid the term, probably because he does not want to scare all the people who are mainly interested in solving ODE for their applications.

After all, this is a book for applied scientists, with much emphasis on algorithms and implementation issues. He does mention a paper by Brouder though, so he knows about the work of Connes-Kreimer.

I would never have guessed that this Hopf algebra structure could have been discovered in the context of numerical ODE. Have a look at his book: He derives algebraic relations of the coefficients of the Butcher tableaus of Runge-Kutta methods, using rooted trees, that have to hold in order for the method to have a certain order! Could be that many people in numerical analysis don’t know about this.

(But admittedly MathOverflow is already pretty good for the “is this structure known” type of question. If Eugene had not blown it we could have used this topic for an evaluation test run :-).)

Posted by: Tim van Beek on June 15, 2010 10:35 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

My background IS numerical analysis and I didn’t know that, but I am not at all surprised. One reason I hang around this place is that I believe category theory, homotopy theory, etc have deep connections to numerical analysis that has yet to be explored.

I’ll surely get the labels wrong, but I’ve suspected that $A_\infty$ or $E_\infty$ algebras are secretly keeping track of “error terms”.

The paper I wrote with Urs, I have described as a “meta algorithm”, i.e. an algorithm for generating numerical algorithms.

The behavior of error terms in finite difference approximations is very interesting, but instead of thinking of finite differences, I think it is better to think in terms of coboundary $d$. Given a graph, we can construct a coboundary $d$ that is a finitary approximation to the exterior derivative. We can define a product, but it is impossible to “have everything”. We can have $d$ be nilpotent and satisfy graded Leibniz, but we cannot have the finitary product be BOTH graded commutative AND associative.

If you want an associative product, the way that graded commutative fails is interesting and is the subject of Urs and my paper.

If you want a graded commutative product, the way associativity fails is interesting and is the subject of work by Sullivan (it is either $A_\infty$ or $E_\infty$ or something, I can never keep it straight :))

In both cases, the “failure” is small and vanishes in the continuum limit, e.g. Urs and my stuff becomes graded commutative in the continuum limit. Sullivan’s stuff becomes associative in the continuum limit. Prior to the continuum limit, you have “error terms”. These error terms have interesting properties I think have not been properly studied within the framework of category theory. I’m trying to learn enough category theory to do it myself, but we all know how little progress I’m making :)

Posted by: Eric Forgy on June 16, 2010 3:17 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

I never thought $A_\infty$-algebras were all that secretive about tracking error terms: that’s what the whole “homotopies of homotopies of the associator” is about - once you set everything up correctly…

Posted by: Mikael Vejdemo-Johansson on June 16, 2010 4:02 AM | Permalink | PGP Sig | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

Thanks Mikael. From an numerical analyst’s/computational physicist’s/quant’s perspective, hearing that “tracking error terms” is related to “homotopies of homotopies” sounds very interesting.

I wish I could understand that. Is there a fairly pedestrian reference describing how that works?

Posted by: Eric Forgy on June 16, 2010 4:37 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

It’s a little bit more subtle than just “tracking error terms” - it’s error terms and higher order error terms for a particular kind of error; for $A_\infty$ it’s the higher order errors measuring failure for the base operation to be associative.

This, in some sense, is visible already in Stasheff’s original $A_\infty$ papers; and again in Keller’s survey papers, in Lu-Palmieri-Wu-Zhang’s survey papers and in my own thesis.

Posted by: Mikael Vejdemo-Johansson on June 16, 2010 4:57 AM | Permalink | PGP Sig | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

Thanks for the thesis :) I think it would great to add a reference to it somewhere on the nLab, e.g. group cohomology or A-infinity algebra, but I’m probably not the best person to add it since I don’t understand it well enough.

I’ve read the first few pages only, but like it a lot so far. Already, I can’t help but to start asking questions though.

I can’t put my finger on it, but at the heart of the the definition of $A_\infty$ algebra seems to be the relation

$m_2(a,b) = (-1)^{|a||b|} m_2(b,a).$

Is that correct? I could be (and probably am) way off the mark, but is it possible that the “Stasheff identities” are derived in order to enforce this relation (at the possible cost of associativity)? If only “someone” around here knew Stasheff’s original motivation :)

I wonder what would happen if, instead of forcing skew commutativity and deriving identities to handle associativity, you forced associativity and derived identities to handle skew commutativity? As I said, generally, you can’t have both so you need to choose one or the other.

How were the Stasheff identities derived? It would be fun to try to reproduce the alternate set of “skew” identities as outlined above. I’m sure we’d end up rediscovering something interesting.

Posted by: Eric Forgy on June 16, 2010 6:29 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

No, it’s perfectly feasible to have skew commutative associative algebras. Where $A_\infty$ comes in is in trying to transfer the algebra structure across functors - specifically across taking the homology of a dg-algebra.

You start out with a dg-algebra - i.e. it’s associative, graded, has a differential, and the multiplication obeys the Leibniz rule.

Taking the homology of this, you get a bilinear operation induced from the multiplication map. This map, however, is not necessarily associative! We get the next best thing, though: it’s associative up to a homotopy - and this is the “error term” showing up: for any a,b,c, $m_3(a,b,c)$ is the homotopy from $m_2(m_2(a,b),c)$ to $m_2(a,m_2(b,c))$.

Introducing these homotopies though, higher order relations between them show up - according to MacLane’s pentagonal axiom and other diagrams like it - and for dg-algebras in general, we have no guarantee that these diagrams will actually commute. However, with dg-algebras we are guaranteed that there will be, in fact, a homotopy that ‘fills in’ each of these diagrams - and that homotopy forms the corresponding higher multiplication.

So this, specifically, is what we mean by $A_\infty$-algebras capturing error terms: the higher multiplications capture failures of equality in these resulting higher relations, and thus the corresponding errors from their diagrams.

Posted by: Mikael Vejdemo-Johansson on June 16, 2010 7:12 AM | Permalink | PGP Sig | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

M V-J:

Where A comes in is in trying to transfer the algebra structure across functors - specifically across taking the homology of a dg-algebra.

That’s one way - the special case is due to Kadeishvili.

M V-J:

This map, however, is not necessarily associative!

Oh yes it is (the map on homology) but that doesn’t mean the A-structure is trivial.

Posted by: jim stasheff on June 17, 2010 2:18 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

JS: That’s one way - the special case is due to Kadeishvili.

And as most people who have discussed this with me are aware by now, Kadeishvili and group cohomology is how I arrived at $A_\infty$. Thank you Jim for filling in the gaping blanks I left open.

JS: Oh yes it is (the map on homology) but that doesn’t mean the $A_\infty$-structure is trivial.

Oh right - associativity carrying over but bringing higher $A_\infty$-structures happens for all dg-algebra homologies. Somehow, when I wrote this, I had managed to convince myself that this was special for the group case.

Posted by: Mikael Vejdemo-Johansson on June 17, 2010 7:50 PM | Permalink | PGP Sig | Reply to this

### homotopy-commutativity

No, it’s perfectly feasible to have skew commutative associative algebras.

Eric expressed himself a bit imprecisely: of course there are skew commutative associative algebras, but the natral algebra structures on functions on a cell complex (which is what he is thinking about) are not of this form.

One way to answer Eric’s question here in the affirmative is this:

take a simplicial set and form the cochain complex of functions on the simplicial set. Using the cup product this becomes a dg-algebra which is associative, but not graded-commutative. But it is always graded-commutative up to coherent homotopy, i.e. it is an $E_\infty$-algebra!

That’s what Eric was asking for: strict associativity, but skew-commutativity only up to coherent “error terms”.

In fact, Eric is thinking of functions not on a simplicial set, but on a cubical set. I suspect these form an $E_\infty$-algebra in cochain complexes just as well under a suitable cup product, but to my shame I have to admit that I am not as certain on the details for this as I should be.

Posted by: Urs Schreiber on June 16, 2010 8:07 AM | Permalink | Reply to this

### Re: homotopy-commutativity

Thanks again Mikael and thanks Urs for coming to the rescue.

I would consider cases where you have both graded commutativity AND associativity to be very special. I think (but could be wrong) that the only time this can happen is when you’re dealing with a continuum space, e.g. differential forms on smooth manifolds. When you’re dealing with a finitary representation of some continuum space, e.g. a cell complex, or even just an abstract complex not necessarily associated to some space, I think you won’t always be able to have both graded commutativity and associativity. I’m pretty sure (based on proof by exhaustion) that there is some no-go theorem to this effect, but I’ve never seen it. That’s what I meant (as Urs pointed out).

Given a (finitary) cell complex with a differential, you can define a product of cochains that is either graded commutative or associative, but not both. Based on the responses I’ve seen (and not on actual first-hand knowledge), it seems like if you choose graded commutativity, then you’re dealing with an $E_\infty$ algebra. If you choose associativity, you’re dealing with an $A_\infty$ algebra.

Which is more fundamental?

I know that if we have an associative product and differential that satisfies Leibniz, but is not graded commutative, we can always “antisymmetrize” it to get a product that still satisfies Leibniz (due to linearity), is graded commutative, but in the process you sacrifice associativity. In other words, antisymmetrization is not associative.

Conversely, if we have a graded commutative product and a differential that satisfies Leibniz, but is not associative, I can imagine there might an “associatization” process that makes the product associative, but somehow loses graded commutativity.

I’ve always thought this “failure” was very interesting and somehow very important. You can have graded commutativity or associativity but not both except in certain continuum limits. But prior to the limit, these failures are very interesting and relate to all kinds of deep maths and physics.

Posted by: Eric Forgy on June 16, 2010 10:01 AM | Permalink | Reply to this

### Re: homotopy-commutativity

Eric Forgy wrote:

it seems like if you choose graded commutativity, then you’re dealing with an E algebra. If you choose associativity, you’re dealing with an A algebra.

See my earlier comments on special kinds of E. What you wrote is correct if you meant: IF YOU CHOOSE TO RELAX graded commutativity and keep associativity, then you’re dealing with an E algebra.

Posted by: jim stasheff on June 17, 2010 2:43 PM | Permalink | Reply to this

### Re: homotopy-commutativity

I would consider cases where you have both graded commutativity AND associativity to be very special.

I know what you have in mind, but you should be careful with making such statements, because it will make raise people’s eyebrow’s: it’s easy to write down plethora of examples of garded-commutative associate algebras.

But apart from language, I think you are right with what you actually have in mind, and we discussed it before. I like to put it this way:

if you form the Chevalley-Eilenberg-algebra of an $\infty$-groupoid and that turns out to be graded-commutative on the nose, then the original $\infty$-groupoid was one with infinitesimally small morphisms, hence was an $\infty$-Lie algebroid.

That’s maybe not the language that you would be using, but this is precisely the phenomenon that you are thinking of. I claim. :-)

Posted by: Urs Schreiber on June 16, 2010 11:00 AM | Permalink | Reply to this

### Re: homotopy-commutativity

Yep yep. I think this point is important enough to mention whenever the subject veers closely enough to it though.

So I guess “$\infty$-groupoid” is now the word we use to describe “space” and I think I’m ok with this by now (although I think “spacetimes” are not $\infty$-groupoids).

By “special”, I guess I mean “have infinitesimally small morphisms” :)

I understand that $\infty$-Lie algebroids encompass a large swath of maths and physics, but I still think these are relatively special and are merely a small niche of a much larger arena.

I wonder if, give an $\infty$-groupoid that is NOT an $\infty$-Lie algebroid, if there is a procedure that amounts to taking a “continuum limit” so that morphisms become infinitesimally small and the $\infty$-groupoid becomes an $\infty$-Lie algebroid.

In the case of cell complexes approximating a smooth manifold, this continuum limit would be a limit of cell refinements. The cell complex itself can probably be thought of as an $\infty$-groupoid but it would not have infinitesimally small morphisms so would not correspond to an $\infty$-Lie algebroid. However, if you refined the cell complex into smaller and smaller cells in an appropriate way, you eventually would end up with an $\infty$-Lie algebroid.

In physics and particularly computational physics (and I would argue in quantum gravity) then we do not necessarily want to go to the extreme and focus exclusively on $\infty$-Lie algebroids.

In this larger arena of $\infty$-groupoids that are not $\infty$-Lie algebroids, then we cannot have both graded commutativity and associativity on the nose. It is remarkable to me that we can even have one of the two and the fact that the failure of the other is interesting is itself very interesting :)

Posted by: Eric Forgy on June 16, 2010 11:57 AM | Permalink | Reply to this

### Re: homotopy-commutativity

Urs wrote:

if you form the Chevalley-Eilenberg-algebra of an ∞-groupoid and that turns out to be graded-commutative on the nose, then the original ∞-groupoid was one with infinitesimally small morphisms, hence was an ∞-Lie algebroid.

Surely a Chevalley-Eilenberg-algebra is by definition graded commutative and associative - what you really meant is ???

Posted by: jim stasheff on June 17, 2010 2:26 PM | Permalink | Reply to this

### Re: homotopy-commutativity

Urs wrote:

Using the cup product this becomes a dg-algebra which is associative, but not graded-commutative. But it is always graded-commutative up to coherent homotopy, i.e. it is an E-algebra!

Again true but misleading since it is strictly commutative.

What subtlety do you expect in the cubical case?

Posted by: jim stasheff on June 17, 2010 2:20 PM | Permalink | Reply to this

### Re: homotopy-commutativity

Using the cup product this [the cochain complex of functions on a simplicial set] becomes a dg-algebra which is associative, but not graded-commutative. But it is always graded-commutative up to coherent homotopy, i.e. it is an E∞-algebra!

Again true but misleading since it is strictly commutative.

No, the cup product on cochains is not graded commutative in general. On cohomology it is. On cochains it is $E_\infty$-though.

As you know.

Posted by: Urs Schreiber on June 17, 2010 8:09 PM | Permalink | Reply to this

### Re: homotopy-commutativity

Sorry - should read what I write
meant it is strictly associative hence a very special kind of Einfty

don’t know if the is a name for it - AEinfty??

Posted by: jim stasheff on June 18, 2010 12:52 PM | Permalink | Reply to this

### Re: homotopy-commutativity

Sorry - should read what I write meant it is strictly associative hence a very special kind of Einfty

Oh, I see.

don’t know if the is a name for it - AEinfty??

Isn’t it $C_\infty$?

Posted by: Urs Schreiber on June 18, 2010 3:58 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

Eric Forgy wrote:

I can’t put my finger on it, but at the heart of the the definition of A algebra seems to be the relation

Is that correct? I could be (and probably am) way off the mark, but is it possible that the ‘Stasheff identities’; are derived in order to enforce this relation (at the possible cost of associativity)? If only “someone” around here knew Stasheff’s original motivation :)

I wonder what would happen if, instead of forcing skew commutativity and deriving identities to handle associativity, you forced associativity and derived identities to handle skew commutativity? As I said, generally, you can’t have both so you need to choose one or the other.

How were the Stasheff identities derived?

“someone”:

His identities were derived at the spatial level first, then transferred to the chains:

Classic: In a fibration F → E → B, if F is contractible in E, then F admits an H-space structure.

Then Sugawara generalized this for ‘group-like spaces’ which involved homotopy inverses. I was able to avoid inverses. The identities were forced on me in trying to build the appropriate fibrations/’projective spaces’. See SLNM 161.

Since typewriters were used in those days, multiple indices were used instead of trees.

Posted by: jim stasheff on June 17, 2010 2:13 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

Literally visible - see Springer Lecture Notes in Mathematics 161.

Posted by: jim stasheff on June 17, 2010 2:37 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

No access :(

Posted by: Eric on June 18, 2010 4:46 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

If the following interests you sufficiently

Stasheff, James
H-spaces from a homotopy point of view.
Lecture Notes in Mathematics,Vol. 161
Springer-Verlag, Berlin-New York 1970 v+95 pp.

An H-space X is a space equipped with a map m: X ×X → X and a point e ∈ X such that m(x, e) = e = m(e, x). These notes present the author’s present view of the theory of H-spaces. One begins by ruling out local pathologies with the assumption that X have the homotopy type of a CW-complex. The initial developments concern various characterizations (i.e., equivalent conditions) of H-spaces. This is followed by what has historically been a strong influence in the theory, the study of constructions on H-spaces analogous to those available for topological groups. The main ingredient here being the classifying space, its numerous variants and approximations. The conditions required for X to have a classifying space lead to the study of homotopy associative H-spaces and their generalizations the An-spaces. It is here in the study of homotopy associativity that the vast portion of the notes lie. There is an excellent survey of various alternative approaches and simplified proofs of many results. On the debit side some theorems are not proved at all and several definitions are only sketched. The recent work of Zabrodsky, Hilton and Roitberg, Curtis and Mislin et al. on finite-dimensional H-spaces comes in for mention at several points but is peripheral rather than central to developments. Infinite loop spaces à la Boardman are outlined, and Araki-Kudo/Dold-Lashof operations are introduced in outline fashion. Their use in the calculation of H(BSF;Zp) by J. P. May and R. J. Milgram is explained, and on this point of recent advance the notes come to a close.

I have a few Xerox copies left

jim

Posted by: jim stasheff on June 19, 2010 1:18 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

It looks very interesting, but presumably over my head. Nevertheless, I’d love to have a look. Has anyone scanned it?

Posted by: Eric on June 19, 2010 1:32 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

Eric Forgy wrote:

If you want a graded commutative product, the way associativity fails is interesting and is the subject of work by Sullivan (it is either A or E or something, I can never keep it straight :))

Many have thought that symmetrizing a homotopy commutative one, one would get ‘commutative cochains’, but the associativity fails, to their consternation - though only up to strong homotopy (built from the commuting homotopy) - i.e. A.

Note the nomenclature: A for ‘associative’, E for ‘everything’, implying homotopy commutative etc. so A and strictly commutative is in a misleading sense E. More relevant is C aka balanced C in which the A structure maps are suitably symmetric.

For further relations at the H-space level, see Adams: The ten types of H-spaces.

Posted by: jim stasheff on June 17, 2010 2:02 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

Now, thanks to Google, Wikipedia, MathOverflow and the nLab, if someone gets interested in a Hopf algebra of trees, they should just do a Google search under Hopf algebra tree.

And no papers by John C Butcher will come up. Because he didn’t actually use the words “Hopf algebra;”

But conversely, with Google, Wikipedia, MO and the $n$Lab around, Butcher might have noticed that the structure he is studying is already known and called “Hopf algebra”. This works in both directions.

We used to discuss here at some length the curious case of K. T. Chen, who spent a good deal of his career studying sheaves and optimizing his definition of a sheaf over the years – without ever realizing (at least not in print) this, and that the notion whicch he was after was well known and had a huge theory supporting it. Not that he didn’t accomplish a bunch of impressive deeds, but in hindsight, if he had realized the knowledge that was available at his time, how much further might he have been able to proceed?

Posted by: Urs Schreiber on June 15, 2010 9:52 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

I find that paragraph about Chen very dubious. Perhaps you are referring to his generalized smooth structures? That was a mere technical device to prove his constructions did what he claimed. Since he was a protege of Andre Weil. I doubt he was ignorant of sheaves. Indeed, he did accomplish ‘a bunch of impressive deeds’. How much further might he have proceeded if he had lived long enough for the world to appreciate what he had accomplished!

Posted by: jim stasheff on June 17, 2010 2:32 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

Yeah, but I’d say it a bit differently, like:

I find it remarkable that information in mathematics is so poorly organized that 26 years after Butcher discovered the Hopf algebra of trees, Connes and Kreimer found it easier to redo his work than to look it up!

Indeed, I keep thinking this a lot. The diffusion of information through the mathematical community is less optimal than one might hope it is.

We need to change this. Google, Wikipedia and MO are a start, but all three still don’t quite force the diffusion the way one could force it. Wikipedia has the archiving functionality but lacks discussion and cutting-edge input. MO has the discussion and cutting-edge input, but leaves the archiving to Google and good luck.

But eventually we’ll get to the point that not every new generation has to waste lots of time to dig out what the former generation dropped on the floor without putting it into the shelf.

I remember a recent incident where I asked a senior big-shot for a reference, was ridiculed for not knowing the answer myself, and upon insistence was provided with a reference. Which was very beautiful. But unpublished.

Posted by: Urs Schreiber on June 15, 2010 8:48 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

Urs wrote:

The diffusion of information through the mathematical community is less optimal than one might hope it is. […] But eventually we’ll get to the point that not every new generation has to waste lots of time to dig out what the former generation dropped on the floor without putting it into the shelf.

Let me mention again the FMathL project on creating a computer-readable Formal Mathematical Language. This should be a step towards easier processing of math information. (An attempt to discuss it in the n-category café turned instead into a discussion of categorial foundations….) We still have far too little funding to realize the big vision, but we are making progress on a small scale. Next week I am going to give a talk in Cambridge, UK, called ”Natural mathematical language for the computer”, summarizing our progress so far.

Posted by: Arnold Neumaier on June 16, 2010 5:48 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

Please teach the young’uns under your tutelage to expand biblio references to include arXiv listings even after hard copy has been published; many institutions do NOT have access to the published version.

If unpublished but available on someone’s web page
(e.g. Deligne’s famous letter to Millson) please list that also.

Posted by: jim stasheff on June 17, 2010 2:35 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

Please don’t forget to add such lists of references to the $n$Lab. I added the ones provided above now to renormalization: references

Posted by: Urs Schreiber on June 15, 2010 1:32 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

Tim wrote:

… one omnipresent kind that grows this way is the B-tree. (I heard a talk of its inventor Rudolf Bayer a couple of years ago, and he confirmed that “B” does not have a specific meaning.)

Yeah, yeah. Just like the “Q” in Quillen’s Q construction has no specific meaning. Just like Kasparov happened to call his version of K-theory “KK-theory”. And just like Euler happened to use “$e$” to stand for the fifth number in a paper he wrote: the base of the natural logarithms.

Pure coincidences, all!

My own plan for everlasting fame is to write an important paper on JB-algebras.

Posted by: John Baez on June 23, 2010 7:58 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

Not to mention K-theory itself and before that the J-homomorphism.

On the other hand, there is purported to be a paper by Alpher, Bethe and Gamov

jim

Posted by: jim stasheff on June 24, 2010 1:19 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

But K is for German “Klasse”, at least I thought so, and Rudolf Bayer did a convincing show of his humility (I don’t remember if he proposed a different name for his invention, which is always the best alibi).

And the J in JB-algebra is already taken (Jordan), and wouldn’t you like something with your whole name on it? (I guess we are talking about a theorem, and that “hypothesis”, “the Baez crackpot index” or “the Baez column” does not count [I like that better than TWF which my subconciousness always twists into WTF]).

With regard to JB-algebras: Harald Hanche-Olsen pointed out a book at mathoverflow here: M. Alfsen and Frederic W. Schultz: State spaces of operator algebras (ZMATH, there is a follow-up completing the theorey: ZMATH). I’m looking at the foreword right now which says that the state spaces of JB-algebras where characterized in 1978, and that this was somehow the starting point of a characterization of state spaces of operator algebras - completely different from Connes’ approach, using geometric requirements on the facial structure instead.

Maybe you can find something interesting here :-)

Posted by: Tim van Beek on June 24, 2010 2:13 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

Tim wrote:

And the J in JB-algebra is already taken (Jordan), and wouldn’t you like something with your whole name on it?

Of course the B is also already taken! For those not in the know: ‘JB-algebra’ is short for ‘Jordan–Banach algebra’. But there’s nothing like the pleasure of ill-gotten gains. So I’ve always wanted to write a paper on JB-algebras, to fool people into thinking these algebras are named after me.

I guess we are talking about a theorem, and that “hypothesis”, “the Baez crackpot index” or “the Baez column” does not count…

I prefer to have my name attached to hypotheses, like this and this. Theorems are too much work!

I’m looking at the foreword right now which says that the state spaces of JB-algebras where characterized in 1978, and that this was somehow the starting point of a characterization of state spaces of operator algebras - completely different from Connes’ approach, using geometric requirements on the facial structure instead.

Maybe you can find something interesting here :-)

Okay, I’ll consider that. But I’d really like to think about infinite-dimensional JB-algebras related to the octonions! Unfortunately there doesn’t seem to be much more than the exceptional Jordan algebra $H_3(\mathbb{O})$ going on here — for example, continuous $H_3(\mathbb{O})$-valued functions on a compact Hausdorff space form an infinite-dimensional JB-algebra.

But surely the exceptional Jordan algebra was put there for a reason…

Jim Stasheff wrote:

on the other hand, there is purported to be a paper by Alpher, Bethe and Gamov…

Yes indeed! It’s a very famous paper explaining the abundance of hydrogen and helium in terms of Big Bang nucleosynthesis. Gamow added Bethe’s name to the paper as a joke! Alpher, then a grad student, resented this, because Bethe was much more famous, so Alpher felt this big name overshadowed his own contribution.

Little did he realize that this joke would make his name go down in history.

Posted by: John Baez on June 24, 2010 6:46 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

So I’ve always wanted to write a paper on JB-algebras, to fool people into thinking these algebras are named after me.

Then there is no problem with JC-algebras, too, but we’ll have to do something about the JBW algebras.

BTW, I just noticed that the classic book Harald Hanche-Olsen and Erling Stormer: Jordan operator algebras is available for free because the publisher has returned all rights to the authors!

Posted by: Tim van Beek on June 25, 2010 10:33 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

Mentioned this already in the The Quantum Whisky Club entry, but worth repeating: thanks to Andrei Akhvlediani, Philip Atzemoglou, Bill Edwards, Aleks Kissinger, Ray Lal, Shane Mansfield, Alex Merry, Johan K. Paulsson, Jakub Zavodny, Jamie Vicary and Chris Heunen of the OUCL Quantum Group for doing all filming!

Posted by: bob on June 14, 2010 12:55 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

Glad to hear pre-Lie algebras have reached the west coast ;-)

By assuming Hilbert spaces, you are disquising the Frobenius pairing axiom?

p.16 f(a) versus f_x(a)

typo:p.16 is a derivations

Posted by: jim stasheff on June 14, 2010 1:32 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

Thanks for catching those problems!

Alas, I’m probably not the first person on the west coast to have understood pre-Lie algebras.

I was not using Hilbert spaces to disguise the Frobenius pairing axiom; I only needed Hilbert spaces to define dagger-Frobenius algebras, where the counit and comultiplication are the Hilbert space adjoints of the unit and multiplication.

Indeed you cannot use the inner product on a complex Hilbert space as the Frobenius pairing, since the inner product is not linear in both arguments!

The point is that every special commutative dagger-Frobenius algebra structure on a finite-dimensional Hilbert space arises from an orthonormal basis of that Hilbert space:

• Bob Coecke, Dusko Pavlovic and Jamie Vicary, A new description of orthogonal bases, available as arXiv:0810.0812.
Posted by: John Baez on June 14, 2010 4:27 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

I see in the paper you mention by Aguiar and Mahajan they write:

There is another approach to combinatorial models for Fock spaces due to Baez and Dolan. It involves a generalization of the notion of species called stuff type…We do not pursue the connections between this interesting approach and the ideas presented here. (p. 600)

Elsewhere, Fiore, Gambino, Hyland and Winskel write:

Baez and Dolan considered further generalisations of these structures leading to the concepts of sorted symmetric set-operad and of stuff types… The former, though not the latter, can be directly recast in our setting. (p. 2)

For stuff types to become part of the scenery, shall we again have to bite the homotopical bullet:

Baez-Dolan ‘stuff types’ (a categorification of the harmonic oscilator) is a special case of homotopical species. (Joyal theorems for homotopical species)?

Posted by: David Corfield on June 14, 2010 2:53 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

I don’t really consider homotopy theory a ‘bullet’ — indeed, as you know, in our first paper on stuff types, Jim and I discussed cardinality not just for groupoids (= homotopy 1-types) but also for $n$-groupoids (= homotopy $n$-types). And all the weak pullbacks we use are special cases of homotopy pullbacks. So, almost everything about stuff types can be easily generalized to $n$-stuff types, with the help of homotopy theory.

That means it’s mainly a question of what people find most congenial: working with groupoids, or working with homotopy types. I’ve been carrying out the ‘groupoidification’ program with my grad students based on the assumption that, starting from scratch, they can learn about groupoids faster than they can learn about homotopy theory. But all of this will eventually be subsumed by ‘homotopification’… and indeed some already is.

Personally I think that it’s nice to see things done with groupoids and also done with homotopy types.

It may be difficult to persuade combinatorists that homotopy theory is a good thing, since many of them are committed to working with finite sets. I thought finite groupoids would be a smaller leap. I also thought that the ability to form a composite of stuff types $F \circ G$ without the ridiculous restriction that $G(0) = 0$ would make them obviously better than species. However, I haven’t put energy into selling stuff types to combinatorists. And one combinatorist I know who might expected to enjoy stuff types said he found them ‘scary’. So, it may be that people who don’t consider themselves combinatorists will take the lead here.

Posted by: John Baez on June 14, 2010 4:47 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

Speaking of avoiding overcaution with composites, remember those halcyon days when we could discuss the Bernoulli numbers cropping up in species/structure types?

$X/(1 - e^X)$ as a species is constructed by multiplying $X$ with the composite of Permutations (i.e., $1/(1 - X)$) acting on Set (i.e., $e^X$). Following your rules in Week 190, let’s give it a 2-element set $\{a, b\}$. First split this set into a one-element set and its complement. For the complement, find an ordered set of unordered sets whose union is that complement.

So, if the $X$ picks up $\{a\}$, the $1/(1 - e^X)$ will start listing: $\langle \{b\}\rangle, \langle \emptyset, \{b\}\rangle, \langle \{b\}, \emptyset \rangle, \langle \emptyset, \emptyset, \{b\} \rangle$, …

Clearly an infinite number of such things, but all is not lost.

Decategorified, the number concerned gives us: $1 + 2 + 3 + 4 +...$, or $\zeta(-1)$, which we know to be -1/12.

So the coefficient of $(X^2/2!)$ in $X/(1 - e^X)$ is twice this, i.e., -1/6, minus the 2nd Bernouilli number, as one would hope.

Do we now, seven years on, have a better idea about the remark made by Connes that you described?

He points out that if $H$ is the Hamiltonian for some sort of particle in a box and $\beta$ is the inverse temperature,

$1/(1 - e^{-\beta H}) = 1 + e^{-\beta H} + e^{-2 \beta H} + ...$

is the operator you take the trace of to get the partition function of a collection of an arbitrary number of particles of this sort. And he claims that pondering this explains all the appearances of $X/(1 - e^X)$ and the Bernoulli numbers in topology! See Milnor and Stasheff’s book “Characteristic Classes” for an introduction to that - but this book was written before quantum theory invaded topology, so we’re left to fit Connes’ clues together for ourselves.

Posted by: David Corfield on June 15, 2010 9:23 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

I’m still catching up with the Bernoulli-number discussion from 2008, but all those $\emptyset$s are unsettling me. Isn’t dealing with the vacuous species $e^X$ a bit problematic, since the vacuous structure can be put on the empty set? By contrast, the Bell numbers count the ways to partition n elements into non-empty boxes, so their generating function is given by the composition of the species “being a finite set” and “being a non-empty finite set”, $e^{e^X - 1}$.

If we tried to compose with the vacuous species instead of the uniform species, we’d naively get $e^{e^X}$; however, because we could interleave an arbitrary number of empty sets into our structure, the number of ways to build a “finite set of finite sets” would blow up in our faces. So, we have to move beyond ordinary combinatorial species into stuff types.

I’m basically copying this out of Section 4.2 of Jeffrey Morton’s “Categorified Algebra and Quantum Mechanics” (2006), which everybody else here has probably read and understood more thoroughly than I have.

Decategorified, the number concerned gives us: $1+2+3+4+\ldots$, or $\zeta(-1)$, which we know to be $-1/12$.

Recently, Terry Tao advised interpreting that $-1/12$ as the constant term in the asymptotic expansion of the regulated sum

$\sum_{n = 1}^\infty n \eta(n/N) = -\frac{1}{12} + \int_0^\infty dx\,x \eta(x) N^2 + O\left(\frac{1}{N}\right).$

More generally,

$\sum_{n = 1}^\infty n^s \eta(n/N) = -\frac{B_{s+1}}{s+1} + \int_0^\infty dx\,x^s \eta(x) N^{s+1} + O\left(\frac{1}{N}\right),$

where $B_s$ is the $s$th Bernoulli number, which arises because the expansion in the smoothing parameter $N$ is computed using the Euler–Maclaurin formula. (A physicist-y explanation of why the Bernoulli numbers appear in the Euler–Maclaurin formula starts with using the derivative operator $\partial_x$ as the generator of translations; the combination of exponentials in the sum considered by Euler–Maclaurin becomes the generating function for $B_n$.)

Regulating the infinite sum $1 + 2 + 3 + 4 + \ldots$ like this is a good physics thing to do. For example, in the Casimir effect, we might start by trying to sum up all the photon modes which are allowed to exist between two conductive plates, which — shock and horror! — leads to an infinite result, until we remember that the real world doesn’t provide us with ideal surfaces. Electrons only move so fast, so fields can only be cancelled so quickly. Instead of summing over all modes, we should introduce a cutoff which suppresses the contributions of those above the plasma frequency. The leading term in the Casimir force is universal, given by the $-1/12$, while the next-order corrections are material-specific.

Luckily for those of us who love abstruse formulas, it works out that when we have a sum which looks at first like $\sum_n n^s$ (but which we sneakily know we should regulate), we can get the constant term of the regulated answer by plugging in the value of $\zeta(-s)$ established via analytic continuation.

Posted by: Blake Stacey on June 20, 2010 6:29 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

Blake wrote:

I’m still catching up with the Bernoulli-number discussion from 2008, but all those $\emptyset$s are unsettling me. Isn’t dealing with the vacuous species $e^X$ a bit problematic, since the vacuous structure can be put on the empty set?

Yes! More precisely: if $F$ is a species, then $F(e^X)$ is typically not a species — but it is a stuff type.

I’m glad you’re studying this — it’s a lot of fun.

So, we have to move beyond ordinary combinatorial species into stuff types.

Right. Indeed, near the beginning of this very discussion, I emphasized that if $F$ and $G$ are species but $G(0) \ne 0$, the composite $F \circ G$ is typically not a species, but it is a stuff type.

So, let’s see. Naively,

$\frac{1}{1 - e^X}$

is the composite of the species

$F(X) = \frac{1}{1 - X}$

and the species

$G(X) = e^X$

So, we should naively expect it to be a stuff type. And naively, this is the stuff type ‘being a finite set written as the union of a possibly empty list of possibly empty subsets.’ Where ‘list’ means ‘linearly ordered set’.

But this is naive, since if we think of $F$ and $G$ as formal power series, the composite $F \circ G$ isn’t a well-defined formal power series! It blows up at $X = 0$.

Nonetheless, we could still hope to make sense of

$\frac{X}{1 - e^X}$

And there is where David’s remark comes in. I think you’re worried that he’s calling $\frac{X}{1 - e^X}$ a species instead of a stuff type. But really, there are much worse problems to worry about here! No?

Posted by: John Baez on June 23, 2010 8:37 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

I think you’re worried that he’s calling $\frac{X}{1−e^X}$ a species instead of a stuff type. But really, there are much worse problems to worry about here! No?

Indeed. I wasn’t really worried about the terminology, per se; it just seemed that the blowing-up to infinity seen in the Bernoulli-number case was a specific instance of a more general problem.

Posted by: Blake Stacey on June 23, 2010 10:42 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

Blake wrote:

Indeed. I wasn’t really worried about the terminology, per se; it just seemed that the blowing-up to infinity seen in the Bernoulli-number case was a specific instance of a more general problem.

Right. But I didn’t express myself very clearly, since I was pretty confused. Let me try again, at the risk of being wrong.

In the case of

$e^{e^X}$

the coefficient of $X^n$ is proportional to the number of ways of chopping an $n$-element set into a (possibly empty) finite set of (possibly empty) finite sets. And this is infinite unless we remember the permutation symmetries acting on a finite set of empty sets, and use groupoid cardinality.

But in the case of

$\frac{1}{1 - e^X}$

the coefficient of $X^n$ is proportional to the number of ways of chopping an $n$-element set into a (possibly empty) ordered finite set of (possibly empty) finite sets. Here the ordering means there are no permutation symmetries. So, groupoid cardinality does not save us: the coefficient is really infinite.

And maybe that makes sense, since

$\frac{1}{1 - e^X}$

blows up at $X = 0$.

We still could hope for a sensible answer for

$\frac{X}{1 - e^X}$

But this seems to require zeta-function regularization!

So in short, I don’t think that the passage from species to stuff types is the main problem here.

How can we understand this zeta-function regularization combinatorially? I don’t know. But here’s something I’m reminded of:

There are some infinite groups like $SL(2,\mathbb{Z})$ where if you count the elements in a clever way, and use zeta-function regularization to extract a finite answer from the resulting sum, you get an answer that correctly matches the results of other curious calculations.

In particular, the reciprocal of the cardinality of $SL(2,\mathbb{Z})$ works out to be

$1 + 2 + 3 + 4 + \cdots = \zeta(-1) = - \frac{1}{12}$

But this is also the orbifold Euler characteristic of the upper half-plane mod the usual action of $SL(2,\mathbb{Z})$, as shown here! This thing is also called the ‘moduli stack of elliptic curves’.

Since the upper half-plane is contractible, we’d naively expect the orbifold Euler characteristic of this space mod $SL(2,\mathbb{Z})$ to be the reciprocal of the number of elements of $SL(2,\mathbb{Z})$. So, something is ‘working’ here — but in a very strange way.

I explained some of this back in week213. I have no idea how it’s related to what we’re talking about now, but I thought I’d throw it into the pot….

Posted by: John Baez on June 23, 2010 11:21 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

David wrote:

… remember those halcyon days when we could discuss the Bernoulli numbers cropping up in species/structure types?

Yes. And that’s a strikingly nostalgic way of putting it! “Remember the good old days when we could discuss…” Why can’t we discuss things like we used to?

Well, one reason is that I got busy. With 5 grad students, every spare moment for discussion got soaked up. And my pile of half-written papers built up to the point where I had to discipline myself and stop chatting so much online.

I would like to become less busy. With my grad students gradually graduating, my pile of half-written papers slowly shrinking, and 2 years to loll around Singapore with no teaching duties, I hope that happens. But I’m afraid my interest in ecological issues will make me more busy. And unfortunately, the main reason I want to become less busy is that I want to have time to do more things. That doesn’t really work, does it?

Another reason is perhaps that you got busy with a permanent job?

Other reasons? Urs is off doing $(\infty,1)$-stuff, but I don’t think he was ever very active in discussing combinatorics, so I don’t think that’s a reason.

Do we now, seven years on, have a better idea about the remark made by Connes that you described?

Do you? I don’t.

And that’s pathetic! It’s probably something very simple. These days you could fire off a question like this to MathOverflow and ten smart people would answer it in hours. No?

Posted by: John Baez on June 15, 2010 7:58 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

Looking again at Connes’ comments, there’s plenty to go on.

The passage from the tangent bundle to the normal bundle is represented on the level of formulas by the passage to the inverse. The mystery of the generating function of the Atiyah-Singer formula, containing the exponential function in the denominator is completely dispelled if we understand this passage to the inverse.

But there’s a more ‘satisfying answer with respect to Planck’s formula’:

It is a strict parallelism…between the local proof of the index theorem using the heat equation and Planck’s formula.

This parallelism shows how this surprising function appears in exactly the same way as in Planck’s formula. This function is simply a sum of exponentials, the geometric series $1/(1 - exp(-\beta))$. It appears in the same way when one calculates the trace of the heat operator using supersymmetry, as Witten has shown. (p. 61)

Posted by: David Corfield on June 17, 2010 11:33 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

I wish I understood those remarks of Connes better, so I could extract some sort of combinatorial essence.

Posted by: John Baez on June 19, 2010 4:22 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

Connes’ remarks made a little more sense to me after I read P. Windey’s “Supersymmetric quantum mechanics and the Atiyah–Singer index theorem” [Acta Physica Polonica, Vol. B15, No. 5 (1984)].

Posted by: Blake Stacey on June 20, 2010 12:13 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

Well, since Blake has been thinking about the index theorem, maybe it’ll pay for me to say exactly what I’m confused about! Connes writes:

The passage from the tangent bundle to the normal bundle is represented on the level of formulas by the passage to the inverse.

I understand this when it comes to Chern classes. Every vector bundle $E$ over a space $M$ gives rise to an element $c(E)$ in the cohomology of $M$, called the total Chern class of $E$. Taking the direct sum of bundles corresponds to multiplying their total Chern classes:

$c(E_1 \oplus E_2) = c(E_1) \cup c(E_2)$

where $\cup$ stands for the product in the cohomology ring. The total Chern class of a trivial vector bundle is just 1.

Any manifold $M$ has a tangent bundle $T M$, and we can find a vector bundle over $M$, called the ‘stable normal bundle’ $N M$, such that $T M \oplus N M$ is a trivial vector bundle.

So, from what I’ve said, the total Chern class of the stable normal bundle is the inverse of the total Chern class of the tangent bundle:

$c(T M) \cup c(N M) = 1$

At first this sounds what Connes could be talking about. But then:

The mystery of the generating function of the Atiyah-Singer formula, containing the exponential function in the denominator is completely dispelled if we understand this passage to the inverse.

The formula for the topological index involves not the total Chern class, but rather the Chern character and the Todd class.

He must be talking about the Todd class, since the definition of that involves Bernoulli numbers and the generating function

$\frac{x}{1 - e^{-x}}$

Hmm, I see the Todd class is multiplicative too!

$td(E_1 \oplus E_2) = td(E_1) \cup td(E_2)$

So, my remarks about the total Chern class for the stable normal bundle being the inverse of the total Chern class for the tangent bundle would also apply to the Todd class.

However, the ‘mystery’ that Connes wants to ‘completely dispel’ is the appearance of the exponential function in the denominator of

$\frac{x}{1 - e^{-x}}$

I don’t understand how this is explained by the stuff about normal bundles.

Posted by: John Baez on June 21, 2010 7:32 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

Did you see Paul Siegel’s explanation for why Todd classes appear in the index formula in this very good question + answers at Math Overflow?

The Todd class arises because we are trying to convert what ought to be a K-theory statement into a cohomological statement, not for a reason that is truly intrinsic to the index theorem.

Posted by: David Corfield on June 24, 2010 8:48 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

All this stuff about ‘normal bundle as inverse of tangent bundle’ and multiplicativity for characteristic classes is making me — yet again! — want to understand the splitting principle more deeply. I understand it well enough in a textbook way, but there really should be a nice category-theoretic way of precisely stating the essential idea, namely: while it’s not true that every vector bundle is a direct sum of 1-dimensional vector bundles, for the purposes of defining characteristic classes you can pretend that it is true.

There’s something very general going on here. The category of vector bundles $Vect(X)$ on some space $X$ is a symmetric monoidal category with direct sums. This lets us define the concept of a line object: namely, an object $E$ that has an inverse with respect to tensor product, where the canonical ‘symmetry’ map $E \otimes E \to E \otimes E$ is the identity.

It’s not true that every object in $Vect(X)$ is a direct sum of line objects, but there’s a full embedding of $Vect(X)$ in a larger symmetric monoidal category with direct sums where this is true. And this embedding gets along with the symmetric monoidal structure. I think that’s roughly the idea of the splitting principle.

There should be some very broad class of symmetric monoidal categories with direct sums for which this idea works! Does someone know a theorem to this effect?

And when this idea works, another idea comes along for the ride:

What kind of a category is symmetric monoidal, with direct sums, where every object is a direct sum of line objects? Well, the classic example is the category of representations of an abelian group!

So I’m getting the feeling that any category where the splitting principle works can be embedded in the category of representations of some abelian group. Maybe there’s not always a specific ‘best choice’, but it would be really cool if there were a best choice.

This idea is somehow implicit in the concept of Adams operations and their link to the Galois group of the maximal abelian extension of the rationals!

It seems that whenever we apply the splitting principle to embed our symmetric monoidal category with direct sums into the category of representations of an abelian group, it’s natural to do it in a way so that this abelian group contains $Gal(\mathbb{Q}^{ab}/\mathbb{Q})$.

Can someone help me out? This stuff should be well-understood by someone out there…

Posted by: John Baez on June 21, 2010 8:26 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

Sorry, I was getting carried away above. The splitting principle as usually stated lets us split a single vector bundle, not all of them. More precisely, given a vector bundle $E$ over $X$, the splitting principle lets us find a map $f : Y \to X$ such that:

• the vector bundle $f^* E$ is a direct sum of line bundles over $Y$, and
• the map $f^* : K(X) \to K(Y)$ is injective

where $K(X)$ is the Grothendieck group of $Vect(X)$.

We can also split any finite collection of vector bundles using this trick. I’m not sure if we can split infinite collections!

Posted by: John Baez on June 22, 2010 2:31 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

I had a few thoughts about this a while ago. Let me see if I can remember something. I believe that whatever I’ll write here is known to someone, but I wouldn’t be surprised if nothing it written down.

In some sense what you’re trying to do is (I) categorify the definition of lambda-ring and (II) find a precise formulation of the splitting principle for such categories. Actually, the axioms for lambda-rings are modeled on the structure you get from applying the Grothendieck-group construction to the category of vector bundles on a given space, or finite-dimensional representations of a given group, or … So maybe a better way of putting (I) is that you’re trying to figure out what a lambda-ring is the decategorification of.

I think the basic idea of (I) should be something like this. You don’t just want tensor products and sums, you also want exterior powers, symmetric powers, and so on, because you want to be able to prove things about Chern class of exterior powers of vector bundles using the splitting principle. So, you really want all tensor constructions, in some sense. I think people can write these down using partitions and Schur functors (or something), but there should be an immediate conceptual way using representations of $\mathrm{GL}_n$ directly.

Here’s what I mean. Giving a vector bundle of rank $n$ on $X$ is equivalent to giving a $\mathrm{GL}_n$-torsor on $X$, which (up to isomorphism) is equivalent to giving an element of $E\in H^1(X,\mathrm{GL}_n)$. Now if you have a representation $\rho\colon\mathrm{GL}_n\to\mathrm{GL}_m$, you can apply $\rho$ to $E$ and get an element of $\rho(E)\in H^1(X,\mathrm{GL}_m)$. For example, for usual determinant map $\mathrm{det}\colon\mathrm{GL}_n\to\mathrm{GL}_1$, the line bundle $\rho(E)$ should be the usual determinant of $E$. Observe that this should all work for a ringed topos $X$, and in particular the classifying topos of your favorite group $G$. So this all includes the case of categories of representations, not just usual vector bundles. (Further, the semi-ring structure on the category of vector bundles should also come about this way: direct sum would come from the sum representations $\mathrm{GL}_m\times\mathrm{GL}_n\to\mathrm{GL}_{m+n}$ and the tensor product should come from the tensor representations $\mathrm{GL}_m\times\mathrm{GL}_n\to\mathrm{GL}_{mn}.$ But I’ll ignore this because I only want to work with unary operations, as you’ll see below.)

So I think the definition of the kind of category you want to consider ($\lambda$-category?, Riemann-Roch category?) should be a category together with an action, in some precise sense, of the collection of all representations of all general linear groups. There will also be relations. For instance, the $n$-th exterior power of a sum is the usual formula involving the exterior powers of the summands. But these should come automatically from the analogous relations in the representation theory of $\mathrm{GL}_n$.

This is all basically a categorification of the usual definition of a $\lambda$-ring (actually, I think ‘anti-decategorification’ might be a more accurate way of putting it :)). In that situation, everything above is more or less done. The category of $\lambda$-rings is monadic (and comonadic) over the category of rings. The free $\lambda$-ring on one generator, denote it $\Lambda$, should then be thought of as the collection of all natural unary operations on $\lambda$-rings. This is just like in module theory—-the free $R$-module on one generator, in other words $R$, should be thought of as the collection of natural unary operations on $R$-modules. Then it shouldn’t be a surprise that $\Lambda$ is (more or less) the representation ring of the family of general linear groups. Further, just as the $R$-module $R$ carries extra structure (=ring structure) with the property that an action of $R$ on an abelian group is the same as an $R$-module structure, the $\lambda$-ring $\Lambda$ carries extra structure with the property that an action (in a certain precise sense) of $\Lambda$ on a ring is the same as a $\lambda$-ring structure on the ring. (This is all made precise in my paper Plethystic Algebra with Ben Wieland.)

So, here are my predictions: (1) There is a good categorification of the notion of ‘ring’, or maybe ‘semi-ring’. This probably exists already. Let’s call it a ring-category. There’s probably a good notion of the 2-category of ring-categories. (2) There is a 2-monad $2\Lambda$ on the 2-category of ring-categories such that the free $2\Lambda$-ring-category on one object is the category of representations of the general linear group. In other words, the category of such representations admits certain extra structure which allows us to make sense of an action of it on a ring-category. (3) The category of vector bundles on a ringed topos should naturally have an action of the 2-monad $2\Lambda$. (4) The 2-category of algebras for this 2-monad $2\Lambda$ is the right abstract context for talking about the splitting principle and, more generally, undecategorified $K$-theory.

Everything in the previous paragraph already exists at the decategorified level. This is essentially Grothendieck’s theory $\lambda$-rings (see the book by Fulton-Lang) plus the language of monads, as in Plethystic Algebra. So I don’t think it’s a stretch to expect it to work at the undecategorified level.

Whew. I hope that was understandable! Just like the old days around here!

(One more thing. I’ve completely glossed over two issues. One is that maybe it’s better to look only at representations of $\mathrm{GL}_n$ that don’t involve having to divide by the determinant. I think people call these ‘polynomial’ representations. So the unary operation that dualizes a bundle is out. Once you do this, you can stabilize things in $n$. This comes up in symmetric functions where you look at infinite symmetric functions in infinitely many variables, such as $x_1x_2+x_1x_3+x_2x_3+\dots$, but where all but a finite (indeterminate) number of the variables are zero. Then all identities become independent of the ranks of the bundles involved.)

Posted by: James on June 22, 2010 11:37 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

After your (re)categorification of $\lambda$-rings, do we see categorified versions of all the good qualities of Symm, the ring of symmetric functions in a countably infinite number of indeterminates, appearing in some free $\lambda$-ring category on one object? As Michiel Hazewinkel explained:

Symm, the Hopf algebra of the symmetric functions is a truly amazing and rich object. It turns up everywhere and carries more extra structure than one would believe possible. For instance it turns up as the homology of the classifying space BU and also as the cohomology of that space, illustrating its self-duality. It turns up as the direct sum of the representation spaces of the symmetric group and as the ring of rational representations of the infinite general linear group. This time it is Schur duality that is involved. It is the free $\lambda$-ring on one generator. It has a nondegenerate inner product which makes it self-dual and the associated orthonormal basis of the Schur symmetric functions is such that coproduct and product are positive with respect to these basis functions…Symm is also the representing ring of the functor of the big Witt vectors and the covariant bialgebra of the formal group of the big Witt vectors (another manifestation of its auto-duality)…

As the free $\lambda$-ring on one generator it of course carries a $\lambda$-ring structure. In addition it carries ring endomorphisms which define a functorial $\lambda$-ring structure on the rings $W(A) = CRing(Symm, A)$ for all unital commutative rings $A$. A sort of higher $\lambda$-ring structure. Being self dual there are also co-$\lambda$-ring structures and higher co-$\lambda$-ring structures (whatever those may be).

Of course, Symm carries still more structure: it has a second multiplication and a second comultiplication (dual to each other) that make it a coring object in the category of algebras and, dually, (almost) a ring object in the category of coalgebras.

The functor represented by Symm, i.e. the big Witt vector functor, has a comonad structure and the associated coalgebras are precisely the $\lambda$-rings.

All this by no means exhausts the manifestations of and structures carried by Symm. It seems unlikely that there is any object in mathematics richer and/or more beautiful than this one, and many more uniqueness theorems are needed. (Witt vectors. Part 1: 7)

Posted by: David Corfield on June 24, 2010 11:16 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

I would expect that the ‘plethory’ structure on $\Lambda=\mathbf{Symm}$ recategorifies—in other words, the structure that makes $\Lambda$ represent a comonad on the category of rings. Where does that structure sit in Hazewinkel’s list, exactly? First, in my opinion much of the structure Hazewinkel lists is redundant. For example, if I understand him correctly, the following are all category-theoretic consequences of the last bit of structure, number (vi): (i) It is the free λ-ring on one generator, (ii) Symm is also the representing ring of the functor of the big Witt vectors, (iii) carries a λ-ring structure, (iv) it carries ring endomorphisms which define a functorial λ-ring structure on the rings W(A)=CRing(Symm,A) for all unital commutative rings A, (v) it has a second multiplication and a second comultiplication (dual to each other) that make it a coring object in the category of algebras, (vi) The functor represented by Symm, i.e. the big Witt vector functor, has a comonad structure and the associated coalgebras are precisely the λ-rings.

He does give some other structure, but I’ve only ever been interested in the plethory structure myself. I wouldn’t be surprised if the positivity defined by the Schur basis is closely related to the ‘rig’ structure you’d get from decategorifying categories of vector bundles and to some still-undefined notion of $\lambda$-rig.

Posted by: James on June 24, 2010 2:33 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

When you say ‘redundant’ is this in the same sense as one might say it is redundant to specify that $\mathbb{Z}$ has a multiplication compatible with its addition, i.e., is a ring, from its characterisation as the free group on one generator? Or might the connections be more trivial?

Posted by: David Corfield on June 24, 2010 4:31 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

Yes, that’s exactly the sense in which I mean it.

Posted by: James on June 24, 2010 11:21 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

Just like the old days around here!

A lot of the original Café energy has dispersed into the nLab. Regarding the current conversation, I hope you will take a look at this nForum discussion between John and Todd (and maybe join in).

Posted by: David Corfield on June 29, 2010 8:52 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

James wrote:

In some sense what you’re trying to do is (I) categorify the definition of lambda-ring and (II) find a precise formulation of the splitting principle for such categories.

Right, exactly. Thanks for splitting the task into two pieces like that.

So maybe a better way of putting (I) is that you’re trying to figure out what a lambda-ring is the decategorification of.

Yes. Later you mention the term ‘anti-decategorification’. I don’t like to use this term, because it has all the elegance and grace of ‘anti-disestablishmentarianism’, but I’ve always been most interested in categorification when it corrects a mistake we already made, namely decategorification… and thus reveals mathematical structures that were already staring us in the face.

And that’s clearly what we need to do with $\lambda$-rings, since they describe what’s left after we decategorify things like categories of vector bundles or categories of group representations.

But I thought I already knew how to anti-decategorify the concept of $\lambda$-ring: I thought the answer was something like ‘symmetric monoidal $k$-linear abelian categories where tensoring with any object is right exact.’

Why? Because it’s these categories whose Grothendieck groups become $\lambda$-rings!

Also: it’s these categories that are acted on by Schur functors!

We’ve discussed this before, here.

So, I was hoping, and I guess I’m still hoping, that this will allow us to shortcut some of the program you outline, and achieve most of the same goals.

There is a good categorification of the notion of ‘ring’, or maybe ‘semi-ring’. This probably exists already. Let’s call it a ring-category.

These guys call it a ring category:

• M. Laplaza, Coherence for distributivity, Lecture Notes in Mathematics 281, Springer Verlag, Berlin, 1972, pp. 29-72.
• G. Kelly, Coherence theorems for lax algebras and distributive laws, Lecture Notes in Mathematics 420, Springer Verlag, Berlin, 1974, pp. 281-375.

I would prefer to call it a ‘rig category’. They define it as a category equipped with a symmetric monoidal structure called $\oplus$ and a monoidal structure called $\otimes$, with all the usual rig axioms holding up to natural isomorphism, and these isomorphisms satisfying a set of coherence laws.

We can talk about ‘symmetric rig categories’, where the tensor product $\otimes$ is a symmetric monoidal structure….

However, in the project I’m dreaming of here, I prefer to work with symmetric monoidal $k$-linear abelian categories where tensoring with any object is right exact.

(There are plenty of fine points that I’m worried about. For example, I don’t like the appearance of the field $k$ here. In our previous discussions we came close to eliminating it, or replacing it by $\mathbb{Z}$. Jim Dolan has also argued that instead of symmetric monoidal abelian categories where the tensor product is right exact, I should be using symmetric monoidal with finite colimits where tensoring with any object distributes over those colimits. But these are details that can be hammered out in the process of proving theorems, perhaps.)

It’s possible that my shortcut will only be useful for the ‘orthodox’ $\lambda$-rings, not the ‘heterodox’ ones you like. So maybe it’s just part of some bigger story…

Whew. I hope that was understandable! Just like the old days around here!

Understandable just like the old days, or ununderstandable just like the old days?

Seriously, thanks for all the help. Your plan for antidecategorification was not ununderstandable!

Posted by: John Baez on June 22, 2010 4:34 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

Maybe *re*categorification? Like returning to the Garden of Eden or something…

Anyway, maybe you’re right that it would be enough to work with ‘symmetric monoidal k-linear abelian categories where tensoring with any object is right exact’ or something very similar. One thing I’m a little worried about is that these categories don’t seem vector bundley to me. For instance, if $k=\mathbf{Z}$, then the tensor category of all abelian groups would work, and I would be a little surprised if the splitting principle worked there. But if these categories are the right ones, the category of representations of the general linear group should act, in some sense, on any such category. It would be nice to see a precise formulation of that.

Also, I like the emphasis on rigs rather than rings. I’ve wondered for a while about how much of the theory of lambda-rings can be extended to rigs, but I’ve never pursued it. This is much more in the ‘orthodox’ direction than the ‘heterodox’ one, because Adams operations (which are what I’m really interested in) only exist once you adjoin formal additive inverses of vector bundles. So rigs are probably good for many things but not everything. As you’d say, you’d hope that the recategorified approach would clarify this.

Someone should work this out! (Famous last words…)

Posted by: James on June 23, 2010 1:31 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

James wrote:

Maybe *re*categorification? Like returning to the Garden of Eden or something…

Right: if equations are evil, then decategorification was the original sin (perhaps when Adam and Eve counted the animals?), and recategorification is some sort of return to innocence. All very religious. I hope nobody takes this too seriously.

Anyway, maybe you’re right that it would be enough to work with ‘symmetric monoidal $k$-linear abelian categories where tensoring with any object is right exact’ or something very similar. One thing I’m a little worried about is that these categories don’t seem vector bundley to me.

Yeah. I got a bit carried away (again) and momentarily forgot that the category of vector bundles doesn’t have all finite colimits! Not every map between vector bundles has a cokernel that’s a vector bundle. This is why god created coherent sheaves.

Lately Jim and I have been thinking about categories of coherent sheaves, and ‘symmetric monoidal $k$-linear abelian categories where tensoring with any object is right exact’ are supposed to be an abstraction of those categories.

(Digression: could you tell me some fun basic stuff about the Grothendieck group of a category of coherent sheaves? Like: what do people do with its $\lambda$-ring structure? Or: does the splitting principle apply here? Apparently it does, at least for a variety or something.)

So, for a category of vector bundles, say $Vect(X)$, I need some other story. The category $Vect(X)$ doesn’t have all finite colimits, but it has enough to get Schur functors to act on this category… and that’s what we really need to make its Grothendieck group into a $\lambda$-ring.

But what is ‘enough’, exactly?

Perhaps I now see a nice answer. In the nLab, Todd Trimble worked with symmetric monoidal $\mathbb{Q}$-linear categories where every idempotent splits. This is enough to get Schur functors to act! And $Vect(X)$ has this property.

(I prefer all these tricks to directly categorifying the concept of $\lambda$-ring, since I prefer tricks that look like ‘recategorification’: seeing what was always there.)

For instance, if $k=\mathbf{Z}$, then the tensor category of all abelian groups would work, and I would be a little surprised if the splitting principle worked there.

Hmm. Before I think about that, I can’t resist mentioning a fun puzzle that Jim raised (and solved). Everyone should take a crack at it!

What’s the Grothendieck group of the category of finitely generated abelian groups?

We all know the classification of finitely generated abelian groups…

Posted by: John Baez on June 23, 2010 5:16 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

I’m pretty rusty when it comes to $K$-theory (I’ve mostly tried to forget my orthodox past), but I think that the Grothendieck groups of vector bundles and coherent sheaves agree for nonsingular varieties, in some sense (maybe regular noetherian schemes?). This is probably because coherent sheaves admit resolutions by vector bundles. I guess I think that $\lambda$-ring structure on $K$-groups ought to come from the splitting principle, because $\lambda$-operations are all about replacing a sum of line bundles with symmetric functions in those line bundles. I don’t know if the Grothendieck group of coherent sheaves always has a $\lambda$-structure, in other words, whether the $\lambda$-structure comes for free from general considerations or because under certain assumptions, the Grothendieck group happens to agree with something else that has a $\lambda$-structure.

I just had a quick look at the nLab page you link to, and I have to admit I like it more than I expected to! I thought Schur functors were defined in terms of certain combinatorial data (which is why I liked representations of GL_n better), but I see you all take a nice conceptual definition (which no doubt I once heard and then forgot). Maybe the abstract version works for coherent sheaves, and the combinatorial definition (or the GL_n one) works for vector bundles.

The nLab page is getting close to the right thing. Someone needs to bring it home!

Posted by: James on June 24, 2010 9:32 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

Urs is off doing (∞,1)-stuff, but I don’t think he was ever very active in discussing combinatorics, so I don’t think that’s a reason.

Not that I disagree that I wasn’t involved in these discussions, but don’t blame the $(\infty,1)$-category theory for it. Given what we heard they should serve to pave the royal road to this subject.

$(\infty,1)$-category theory is not taking one away from the phenomena, but towards them. ;-)

Posted by: Urs Schreiber on June 15, 2010 10:06 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

Does anyone have a reference for the work by David Gepner and Joachim Kock on “homotopical species” , the kind of work to which the abstract that David C. pointed to refers to? Anything?

John writes:

It may be difficult to persuade combinatorists that homotopy theory is a good thing, since many of them are committed to working with finite sets. I thought finite groupoids would be a smaller leap.

What about finite simplicial sets? Every combinatorist should love them.

Posted by: Urs Schreiber on June 15, 2010 11:12 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

I wrote:

Does anyone have a reference for the work by David Gepner and Joachim Kock on “homotopical species” , the kind of work to which the abstract that David C. pointed to refers to? Anything?

By private email somebody kindly points out this article, which may or may not be relevant:

I haven’t read this. Not sure what to make of it. With a little luck Mike finds five minutes and can tell us more… ;-)

Posted by: Urs Schreiber on June 15, 2010 1:44 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

I can’t see that it’s got anything directly to do with homotopical species.

The abstract runs

In 1967 Quillen introduced model categories “to cover in a uniform way” “a large number of arguments [in the different homotopy theories encountered] that were formally similar to well-known ones in algebraic topology”. We show the same formalism “covers in a uniform way” a number of arguments in (naive) set theory. We argue that the formalism is curious as it suggests to look at a homotopy-invariant variant of Generalised Continuum Hypothesis which has less independence of ZFC, and first appeared in PCF theory independently but with a similar motivation. Techically, we show how a naive, diagramme chasing homotopy theory approach to set theory leads to a construction of a model category (in the sense of Quillen) modelling some invariants in set theory. These invariants, the covering numbers of PCF theory, appear, in homotopy theory, as values of (minor variations of) the derived functor of cardinality. Homotopy theory suggests to look at a homotopy-invariant version of Generalised Continuum Hypothesis (hGCH) replacing cardinality by its homotopy invariant approximation, the derived functor, and we observe that ZFC proves strong bounds towards hGCH for many cardinals, either by PCF theory or trivially.

and a final hope

homotopy theory provides a non-trivial analogy between Poincare’s continuous and Cantor’s infinite.

Gromov’s Structures, Learning and Ergosystems is quoted frequently. Is there some distinctly Russian drive to speculate about the larger place of mathematics? My friend Alexandre Borovik has written ‘Mathematics under the Microscope: Notes on Cognitive Aspects of Mathematical Practice’ and ‘Shadows of the Truth’. Yuri Manin has several such papers.

Interestingly, both Manin and Gromov point to n-categories as the mathematics of the future, Manin in Georg Cantor and his heritage and Gromov in the above-mentioned paper

Probably, the full concept of “ergo-brain” can be expressed in a language similar to that of n-categories which make an even more refined (than categories/functors) tool for dealing with “arbitrariness and ambiguity” (p. 53)

Posted by: David Corfield on June 15, 2010 3:23 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

David wrote:

Is there some distinctly Russian drive to speculate about the larger place of mathematics?

Ah yes, the distinctive features of the Slavic soul. This reminds me of Frank Quinn’s remark:

On page 206 he [David Corfield] quotes Gromov as attributing a lack of appreciation for his bookon the $h$-principle in partial differential equations to resistance to conceptual work. I suspect Gromov is using ‘conceptual’ here as ‘lacking technical detail’: he is a product of the Russian school that puts more emphasis on ideas than precision…

I get the feeling he does not altogether approve….

Posted by: John Baez on June 15, 2010 8:15 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

With a little luck Mike finds five minutes and can tell us more… ;-)

It would take more than that. It’s basically impossible for me to read, written in a very opaque and roundabout style (plus unfinished), and most of it is scanned at low resolution. The most I can make out is that he’s trying to put a model structure on some large poset of sets where the relation $X\le Y$ is something like “for all $x\in X$ there is a $y\in Y$ with $x\subseteq y$.” Perhaps weak equivalence models something like equipotence, possibly up to finiteness or countability? One problem is that this category has no terminal object, so it can’t actually support a model structure, although perhaps it satisfies some of axioms.

Posted by: Mike Shulman on June 15, 2010 9:54 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

I did not read much (lack of time and eye-strain with that pdf),
but I am puzzled by the description of the trivial cofibrations
in 1.0.1 (3) on page 4: they do not seem to be closed under transfinite
composition.

Posted by: Marc Olschok on June 17, 2010 7:35 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

I have been lurking at the n-cafe for a while, and was now rather quite pleased to find my paper mentioned here…I wish to add a link to a 15-page summary of the paper discussed above which is shorter and therefore might be easier to read.

I can’t see that it’s got anything directly to do with homotopical species.

Neither do I (although I do not know much about homotopical species).

but I am puzzled by the description of the trivial cofibrations
in 1.0.1 (3) on page 4: they do not seem to be closed under transfinite
composition.

Thank for your remark and I am sorry for the confusion but I am not sure I understand your remark. The definition of the trivial cofibrations may be found on page 25 (Def.2(wc)) and its set-theoretic characterisation at page 25, claim 3(wc): A –> B is a trivial cofibration iff every element of B is a subset of an element of A up to finitely many elements. And thus it seems to be closed any transfinite induction.

Posted by: misha gavrilvich on June 30, 2010 11:40 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

Hi Misha! I hope you didn’t take offense at my comments, which were written in haste while busy with something else; but I did find the development in terms of the “not-too-bright homotopy theorist” quite opaque and hard to follow. It would be a lot clearer to me if you stated up-front exactly what notion of weak equivalence / homotopy between sets (or sets of sets?) this model category is trying to capture.

Am I correct that the category (poset) in question doesn’t have a terminal object? As far as I can tell, the argument for the existence of limits only works for limits of nonempty diagrams, and the category fails to have a terminal object for the same reason the large poset of all sets under inclusion does – such a terminal object would have to be essentially a set of all sets. But perhaps I have misunderstood the definition of the category.

Posted by: Mike Shulman on July 1, 2010 3:07 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

My new summary exposition does start with the definitions, and I hope you will find it easier to follow. Indeed, I am no longer sure who was the intended reader of the longer text…

Am I correct that the category (poset) in question doesn’t have a terminal object? …. – such a terminal object would have to be essentially a set of all sets. But perhaps I have misunderstood the definition of the category.

This is very true, and I do ignore this sort of set theoretic difficulties as they wouldnt be illuminating and are easy to resolve in a variety of ways: one may assume that we work either within a Grothendieck universe, within a model of ZFC or a set theory with a universal set. Another way to put everything in ZFC is perhaps to define objects as classes of sets, or to treat the set of all sets as a formal object.

Posted by: misha gavrilovich on July 1, 2010 2:24 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

I’m not talking about set-theoretic difficulties or universes – I mean that a category that lacks a terminal object can’t be a model category, since by definition a model category has at least all finite limits.

Posted by: Mike Shulman on July 1, 2010 5:19 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

Thank you for your comment. I am sorry if my answer was misleading. What I meant to say it was that existence of the terminal object (that is, the set of all sets) depends on the precise set theoretic setup used, and that one can easily ensure that the terminal object exists. I believe many set theory with a universal set would be able to prove that QtNaamen has the terminal object (and is a model category).

In ZFC there appear the following two ways : (a) make the categories Qt/StNaamen small in some way, e.g consider only objects which are families of subsets of a fixed small set (b) modify the definition of StNaamen so as that its objects are classes of sets; the class of all sets exists and therefore the terminal object exists. i believe it also should not be hard to see that all finite limits exist with this definition.

Posted by: misha gavilovich on July 4, 2010 12:02 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

Okay, considering subsets of some fixed set should work. (There’s only one set theory with a universal set that I know of, namely NF(U), and it’s pretty bizarre—did you have something particular in mind?)

Unfortunately, I don’t have time to figure out what the definition of QtNaamen means (can you make it more explicit?). But I did remember one reason why people don’t usually talk about model structures on posets. Namely, the homotopy theory presented by a model category is determined by the full subcategory of fibrant+cofibrant objects, and the “homotopies” and higher homotopies between such. But in a poset, there are no nontrivial homotopies since there are no parallel arrows. So in particular, the homotopy category of a model poset is just the full sub-poset on the fibrant+cofibrant objects, as is the simplicial localization. Moreover, in a model poset the fibrant objects are a reflective subcategory and the cofibrant objects a coreflective one, and so the fibrant+cofibrant objects are a coreflective subcategory of a reflective subcategory and thus inherit all completeness and cocompleness properties of the ambient category. So there doesn’t seem to be much need to do homotopy theory, since we’d really just be talking about some other poset which is just as well-behaved as the one we started with.

What are the fibrant+cofibrant objects in QtNaamen?

Posted by: Mike Shulman on July 5, 2010 7:26 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

(There’s only one set theory with a universal set that I know of, namely NF(U), and it’s pretty bizarre—did you have something particular in mind?)

Yes, NF and NF(U) seem to be possibilities but
I do not understand them well enough to be able to claim
that they work. But it appears that the proof (of QtNaamen
being a model category) uses very little, and set theory
with a good enough notion (approximating that) of
a universal set would do, e.g. the class of
all sets in ZFC is a “good enough” notion of
the universal set in ZFC.

can you make [the defnition of QtNaamen] more explicit?

I am not sure; the paper has all the explicit reformulations of the definition of QtNaamen that I know of. But perhaps I could point out that cofibrant objects in QtNaamen and StNaamen are the same (meaning if 0–c–>X is a cofibration in StNaamen, then X is an object of QtNaamen), and that each object of StNaamen is weakly equivalent to a cofibrant object
which is necessarily in QtNaamen.

What are the fibrant+cofibrant objects in QtNaamen?

X is cofibrant iff every element of X is a countable set. X is fibrant iff for every finite set a and every element x of X it holds that a\cup x is a subset of an element of X.

So there doesn’t seem to be much need to do homotopy theory, since we’d really just be talking about some other poset which is just as well-behaved as the one we started with.

Yes, this argument does seem to imply that not much of homotopy theory can be used, and it is a big problem. However, in the case of QtNaamen
it appears rather impossible to work directly with the “some other poset”, i.e. QtNaamen_cf; and the subposet of cofibrant objects (i.e. families
of countable sets) also appears to be difficult to work with. Also, notice that in the application to Continuum Hypothesis it is
it appears rather impossible to work directly with the “some other poset”, i.e. QtNaamen_cf; and the subposet of cofibrant objects (i.e. families
of countable sets) also appears to be difficult to work with. Also, notice that in the application to Continuum Hypothesis it is
important not to pass to the poset of cofibrant and fibrant objects (what is the cardinality of the family is this case), and it is also important that we derive something which is not a functor.

Posted by: misha on July 8, 2010 11:44 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

Okay, I think you’ve lost me. Unfortunately, I don’t have time right now to figure out why it appears that the “some other poset” is hard to work with, or what it means to derive something that isn’t a functor. Good luck; maybe when you get something more polished written I’ll take a look at it again.

Posted by: Mike Shulman on July 9, 2010 3:25 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

maybe when you get something more polished written I’ll take a look at it again.

I have just put a preprint arXiv:1102.5562 on arxiv describing the construction.

Posted by: misha gavrilovich on March 12, 2011 5:04 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

Here’s something that just occurred to me. In your categories there is an arrow $X\to Y$ iff $\forall x\in X, \exists y\in Y, x\subseteq y$. Now for any set $X$ let $\downarrow(X) = \{ x' | \exists x\in X, x'\subseteq x\}$. We have $X\subseteq \downarrow(X)$, hence $X\to \downarrow (X)$, while everything in $\downarrow(X)$ is by definition a subset of something in $X$, so $\downarrow(X)\to X$. Hence $X\cong \downarrow(X)$ in the category. But $\downarrow(X)$ is down-closed under $\subseteq$, so every object is isomorphic to such a down-closed set, and we may as well assume all objects are down-closed. But if $X$ and $Y$ are down-closed and $X\to Y$, then for all $x\in X$ we have $y\in Y$ with $x\subseteq y$, hence $x\in Y$ since $Y$ is down-closed; thus $X\subseteq Y$. So the category in question is equivalent to its full subcategory whose objects are $\subseteq$-down-closed, and in that subcategory the arrows are just set-containment.

Assuming I’ve got that right, this starts to look much more familiar to a category theorist. Suppose we have some big universe set $U$ and we’re looking at subsets of $U$ here; then these down-closed sets of subsets of $U$ are the same as downsets in the poset $P(U)$, a.k.a. “sheaves of truth values” or subterminal objects in the presheaf topos $Psh(P(U))$.

Now my argument from before implies that the homotopy theory is equivalent to the full subcategory of fibrant+cofibrant objects. You say the cofibrant objects are those each of whose elements are countable. These are evidently just the downsets in the poset $P_c(U)$ of countable subsets of $U$, so the cofibrant objects are the subterminals in $Psh(P_c(U))$. And you say $X$ is fibrant if for all $x\in X$ and finite $a\subseteq^{fin} U$ we have $a\cup x\subseteq x'$ for some $x'\in X$ – hence $a\cup x \in X$ if $X$ is a downset. In other words, $X$ is fibrant if whenever $x\in X$ and $x\subseteq x'$ with $x'\setminus x$ finite, also $x'\in X$. This just says that $X$ is a “sheaf” for the “coverage” on $P_c(U)$ in which the covering families are the singleton inclusions with finite complement. (Although these aren’t stable under pullback, so they aren’t an actual coverage in the sense of sheaf theory.) These form a reflective subcategory of the cofibrant objects, and it seems to me that they are really fairly nice.

Posted by: Mike Shulman on July 9, 2010 8:59 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

Thank you; your reply shall take me some time to understand. Your point of view is novel to me, although i am somewhat familiar with the terminology you use.

So the category in question is equivalent to its full subcategory whose objects are ⊆-down-closed, and in that subcategory the arrows are just set-containment.

Assuming I’ve got that right, this starts to look much more familiar to a category

Yes, that is right.

These form a reflective subcategory of the cofibrant objects, and it seems to me that they are really fairly nice.

I presume you mean that they are very nice category-theoretically, and this descriptions tells a lot about them. But it is a question is whether this description is good enough for the qestions we could be asking, as these questions might be set-theoretic and therefore somewhat unexpected.

However, this is a pure speculation on my part, as I do not understand well what you said. I shall try to say more once I understand some more…

what it means to derive something that isn’t a functor

This is in paragraphs 3.3 (and 3.2, 3.1) on page 6 of the note. What I meant to say is that the only application is that of deriving cardinality, and there it appears important that cardinality is not a functor (and, by the way, therefore the derived functor cannot be defined using on the “sheaves” for the “coverage” on P_c(U) , or at least now I do not see how).

Posted by: misha gavilovich on July 10, 2010 11:03 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

I am a bit confused now about what model structure you introduce; let’s clear up the point where i might have misunderstood something. In the current version paragraph 1.2.1 states

“Further we define an inclusion $A \subseteq B$ to be both a weak (homotopy) equivalence and a cofibration iff $B \setminus A$ is finite.”

This condition is not stable under transfinite composition (simply take th chain $0 \subseteq 1 \subseteq \cdots$ with colimit $\omega$).

What did I miss?

P.S.: I do not think that a missing terminal object is a big problem, on can still use such model structures.

Posted by: Marc Olschok on July 10, 2010 6:30 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

I do not think that a missing terminal object is a big problem, one can still use such model structures.

Maybe for some things. But most of the theory of model categories that I’m familiar with depends on having a notion of “fibrant object” and “fibrant replacement,” and as far as I can see that depends on having a terminal object so you can ask whether $A\to 1$ is a fibration to know whether $A$ is fibrant, and factor it into an acyclic cofibration / fibration in order to replace $A$ by a fibrant object.

Posted by: Mike Shulman on July 11, 2010 12:07 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

That is why I prefer to speak of ”a model structure $(\mathcal{C},\mathcal{W},\mathcal{F})$” without further assumptions on the base category and reserve the term ”model category” for a bicomplete category equipped with a model structure.

If only the terminal object is missing and the category $\mathcal{K}$ in question is otherwise sufficiently well behaved, you can still do quite a bit with it. For instance, in a model category an object is fibrant iff every trivial cofibration from it is a split monomorphism. This characterization does not mention the terminal object any more and can be taken as definition whenever pushouts (along trivial cofibrations) are available.

So the only missing bit is fibrant replacement in $\mathcal{K}$, but you still have it in every slice $\mathcal{K}/B$.

Posted by: Marc Olschok on July 14, 2010 2:29 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

Interesting. Do fibrant objects thusly defined share the good properties of fibrant objects in the usual setup? In particular, is it still true that you can construct the homotopy category as the quotient of the full subcategory of fibrant+cofibrant objects by a homotopy relation? I would guess not, if you don’t have fibrant replacements.

Posted by: Mike Shulman on July 14, 2010 6:15 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

Eugene Lerman pointed me to this helpful review article:

• Ch. Brouder, Trees, renormalization and differential equations, Numerical Mathematics 44 (2004), 425-438.

Let me quote:

Abstract. The Butcher group and its underlying Hopf algebra of rooted trees were originally formulated to describe Runge-Kutta methods in numerical analysis. In the past few years, these concepts turned out to have far-reaching applications in several areas of mathematics and physics: they were rediscovered in noncommutative geometry, they describe the combinatorics of renormalization in quantum field theory. The concept of Hopf algebra is introduced using a familiar example and the Hopf algebra of rooted trees is defined. Its role in Runge-Kutta methods, renormalization theory and noncommutative geometry is described.

Introduction

This paper tells the story of a mathematical object that was created by John Butcher in 1972 and was rediscovered by Alain Connes, Henri Moscovici and Dirk Kreimer in 1998. Butcher wanted to set up a theory of Runge-Kutta methods in numerical analysis, Connes and Moscovici were working at an index theorem in noncommutative geometry, Kreimer was looking for the mathematical structure behind the renormalization method of quantum field theory, and all these people hit upon the same object: the Hopf algebra of rooted trees. The appearance of an object relevant to so widely different fields is not common. And the fact that a computer scientist discovered it 26 years in advance shows the power of inspiration provided by numerical analysis. Connes and Kreimer themselves noted: “We regard Butcher’s work on the classification of numerical integration methods as an impressive example that concrete problem-oriented work can lead to far-reaching conceptual results”.

I wish I really understood how all these pieces fit together. But I feel I’m getting closer, because at least I understand the Hopf algebra of trees now. Before it always seemed a bit mysterious, but somehow the pre-Lie algebras make it seem very fundamental to me — and it makes them feel very fundamental. Now I just need to understand better what this stuff has to do with ‘renormalization’ and ‘Runge-Kutta methods’.

Posted by: John Baez on June 14, 2010 9:05 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

The Hopf algebras of rooted trees can be categorified!

Posted by: David Corfield on June 15, 2010 11:53 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

Spooky…

Posted by: Bruce Bartlett on June 15, 2010 12:20 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

Hmm. Joachim is really getting into combinatorics! And he knows enough category theory to be unafraid of stuff types…

As you’ll note, his categorified Hopf algebra only gets an antipode after he formally introduces minus signs. This is unsurprising, since an antipode is like an inverse and inverses often bring minus signs into the picture. And it’s typical: Aguiar and Mahajan categorify a vast range of famous Hopf algebras, including Hopf algebras of rooted trees — but they need to use, not ordinary $Set$-valued species, but $Vect$-valued species to do this.

I wonder how Kock’s construction relates to Aguiar and Mahajan’s.

Posted by: John Baez on June 15, 2010 2:58 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

I often wonder how effective This Week’s Finds is at explaining certain ideas. I often worry that your eyes have glazed over by the time you get to the really cool stuff.

So: will anyone admit to understanding — or not understanding — the calculation which led up to this revelation?

This calculation reveals the secret meaning of pre-Lie algebras. The secret is that pre-Lie algebras are all about attaching two things by connecting a special point of the first to an arbitary point of the second!

I should have emphasized that Bill Schmitt showed me this calculation. I think it’s wonderful. If you understand it, you will see that pre-Lie algebras are everywhere.

Posted by: John Baez on June 15, 2010 7:46 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

Makes perfect sense to me despite there being no way you could say this was my field.

I wonder if there is some computational interpretation. Trees are just recursive data types (aka F-algebras) and other recursive data types can also have “special points” and “arbitrary points”.

Posted by: Dan Piponi on June 15, 2010 9:50 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

Okay, it’s nice to hear that at least one person found that passage comprehensible.

What’s an $F$-algebra? Maybe every $F$-algebra gives a pre-Lie algebra. You need to be able to ‘attach’ or ‘glue together’ $F$-algebras, with the ‘head’ or ‘root’ of one $F$-algebra getting attached to some ‘more general point’ of another, and get a new $F$-algebra.

A list sounds like an example of a recursive data type to me. The main way to glue together lists is to concatenating. This is like attaching the ‘head’ of one list to ‘tail’ of the other. If we do this, the ‘more general point’ mentioned above is not actually more general! We can’t glue lists together head to head — at least, not by concatenation.

So, some terms in the calculation I did drop out… and we get an obvious fact. Namely: concatenating lists is associative:

$(a b)c = a (b c)$

This is much stronger than the pre-Lie property:

$(a b)c - a (b c) = (a c) b - a (c b)$

Posted by: John Baez on June 15, 2010 10:33 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

The notation is unfortunate but it’s standard in the CS world: F-algebra or F-algebra. Lists and trees can both form F-algebras.

Posted by: Dan Piponi on June 16, 2010 1:40 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

Oh, so an $F$-algebra is just an algebra for an endofunctor! Why didn’t you say so? I always complain about my calculus students who call the method of doing integrals by substitution “$u$-substitution”, because it amounts to taking the name of a variable too seriously. But I see they’re not alone.

Okay… well, I don’t see how we can ‘glue together’ algebras of an arbitrary endofunctor in the manner I described. However, there should be some conditions on an endofunctor that let us do this, turning the free vector space on the set of isomorphism classes of algebras of that endofunctor into a pre-Lie algebra. This sounds like a nice puzzle: find those conditions!

Posted by: John Baez on June 16, 2010 2:00 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

Lists and trees aren’t just algebras for an endofunctor, they’re initial algebras for an endofunctor. And if F is a polynomial endofunctor, which are exactly the sort of endofunctors whose initial algebras model recursive data types, then its initial algebra always does consist of “labeled rooted well-founded trees.” So I would guess that the free vector space on any recursive data type is a pre-Lie algebra.

More explicitly, the input data for a polynomial endofunctor is a function $f\colon A\to B$, where $B$ is the set of constructors and the fiber $A_b$ over $b\in B$ is a set specifying the arity of the constructor $b$. So for instance natural numbers are the initial algebra for the polynomial endofunctor described by $1\to 2$, since there are two constructors, one nullary (the natural number zero) and one unary (successor). The data type “lists of elements of $X$,” for some fixed set $X$, corresponds to $X\to X+1$, where each $x\in X$ is interpreted as the unary constructor “add $x$ in front” (aka “cons x”) and the extra nullary constructor is the empty list “nil”. And trees of arbitrary finite branching correspond to $N_{\lt} \overset{\pi_2}{\to} N$, where $N_{\lt} = \{ (x,y) \mid x\lt y \}$ and $\pi_2(x,y)=y$; that is there is one constructor of arity $n$ for each natural number $n$.

The polynomial endofunctor itself is given by the composite $Set \overset{A^*}{\to} Set/A \overset{\Pi_f}{\to} Set/B \overset{\Sigma_B}{\to} Set$ where $A^*$ means pullback, $\Sigma_B$ means forget the map to $B$, and $\Pi_f$ is a dependent product. We can write this out as $F(X) = \sum_{b\in B} X^{A_b}$ whence the name “polynomial.”

The initial algebra for this endofunctor is the set of well-founded rooted trees in which every node is labeled by some element of $b$ and the edges connecting it to its children are labeled by the elements of $A_b$ (so in particular a node labeled by $b$ must have $|A_b|$ children). If we have two such trees $x$ and $y$, perhaps we can put them together in some way like this: choose a node of $x$, labeled by some constructor $b$, change its label to some other constructor $b'$ of arity one more than $b$, and stick in $y$ as the additional argument to $b'$. That seems to be what is happening for trees and lists.

However, I have to admit that while I can sort of see how gluing trees together gives us a pre-Lie algebra (and generalizes to recursive data types in this way), I don’t understand why we should expect trees to form the free pre-Lie algebra on one generator.

Posted by: Mike Shulman on June 16, 2010 3:44 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

Cool!

If you want to see why the free vector space on the set of isomorphism classes of rooted trees gives the free pre-Lie algebra on one generator, maybe the best way for you to do it is to figure out the universal property of the species of rooted trees, and then decategorify that!

(I can’t do it that way, but maybe you can.)

This stuff about polynomial functors reminds me strongly of material in this book:

• F. Bergeron, G. Labelle, and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, Cambridge U. Press, 1998.

which starts out as a general introduction to Joyal’s species (and thus, implicitly, polynomial functors), but then morphs into a discussion of tree-like structures and how solutions to certain differential equations can be found by decategorifying species of tree-like structures on finite sets.

So, I suspect that your remarks will help tie together a number of threads we’ve been seeing here: trees, pre-Lie algebras, species of tree-like structures, algebras of monads, recursive data types in computer science, differential equations, and the work of Butcher on the Runge-Kuttta method!

(Why am I posting so much, and so ebulliently? It’s because I’ve finished grading my finals, and I won’t have to give any more for one or two years!)

Posted by: John Baez on June 16, 2010 4:33 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

In CS we think of many types of “container” data types as functors, initial F-algebras being good examples. Examples of containers are ordered pairs, ordered triples, (ordered) lists, binary trees and so on. This is very similar to combinatorial species.

We can compose containers. So if F is the functor for pairs (ie. $PX=X \times X$) and L is the functor for lists, then $P\circ L$ is pairs of lists and so on.

There is also the notion of a container with an element removed. Eg. a list with an element removed is the same as a pair of lists. Intuitively, this is because when you remove an element from an ordered list you leave behind two pieces. It’s known that removing an element corresponds to taking the derivative, in some sense.

So the list example gives us L’(X)=P(L(X)). Or in less rigorous notation, with a renaming of the functors, we have $f'=f(y)$.

So when we have the situation where removing an element from a container gives a container full of containers of the original type, we have a first order differential equation.

The trees in the Butcher group now start appearing naturally as soon as you start expanding in Taylor series. Here they immediately get a combinatorial interpretation.

I’ve already been thinking about this for a while, though I can’t see if it actually leads anywhere interesting. Is there any reason why anyone would be interested in containers with that element removal property?

Posted by: Dan Piponi on June 16, 2010 7:24 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

It’s known that removing an element corresponds to taking the derivative, in some sense.

I’m sure John’s been teaching us over the years that taking the derivative is like adding an element.

$L'$ applied to a set forms lists of set members + $\{*\}$. These lists map to ordered pairs of lists of the original set members by taking the lists either side of $*$.

Posted by: David Corfield on June 17, 2010 9:42 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

When thinking about data types it makes more sense to think of $F'$ as the type of $F$-containers with an element removed. We then have natural maps like one $X\times F'(X)\rightarrow F(X)$ which can be thought of as the “putting the removed element back in” map.

Posted by: Dan Piponi on June 17, 2010 2:55 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

I thought you could edit comments. Oh well.

Anyway, the duality is easy to explain. JB “adds” a hole where CS people “remove” an element.

Posted by: Dan Piponi on June 17, 2010 3:40 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

John wrote:

If you want to see why the free vector space on the set of isomorphism classes of rooted trees gives the free pre-Lie algebra on one generator, maybe the best way for you to do it is to figure out the universal property of the polynomial functor that gives rooted trees, and then decategorify that!

(I can’t do it that way, but maybe you can.)

Actually, maybe I can get you started.

If I understand your definition of ‘polynomial endofunctor’ (which seems suspiciously similar to Joyal’s notion of ‘analytic functor’), maybe rooted trees are the initial algebra for the terminal polynomial endofunctor.

Why? Since rooted trees seem related to the polynomial endofunctor that has one constructor of each arity… and ‘terminal’ tends to go along with ‘one thing of each sort’.

Am I close?

Posted by: John Baez on June 16, 2010 4:00 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

Joachim Kock has cropped up once or twice on this thread, and he does so again here as author of Polynomial functors and trees, and other papers and a 420 page draft – Notes on polynomial functors – listed here.

When you write

If I understand your definition of ‘polynomial endofunctor’ (which seems suspiciously similar to Joyal’s notion of ‘analytic functor’)…

Kock writes in those Notes

Analytical functors can be seen as polynomial functors with symmetries. The theory of species and analytical functors became a foundation for power series theory in combinatorics… Possibly, because of the success of analytic functors and species, the simpler theory of polynomial functors was never written down, and did not become mainstream. (p. 7)

Posted by: David Corfield on June 16, 2010 4:10 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

Thanks for the tips about polynomial functors. I guess they’re part of a familiar set of analogies:

monoidal category : symmetric monoidal category

PRO : PROP

structures on totally ordered finite sets : structures on finite sets

ordinary generating function : exponential generating function

polynomial functor : analytic functor

Posted by: John Baez on June 16, 2010 5:22 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

It dawned on me that pre-Lie algebras make perfect sense in the context of recursive data types. In particular, containers. A container is just a functor, F, on the category of types and functions. Examples are F(X)=X2, the container of pairs of the same type, or F(X)=1+XF(X), the list container.

We can make a hole in a container F by differentiating the functor using the Leibniz rule to give F’. Pointwise multiplication of two functors gives us the product of containers, ie. a container consisting of one instance of one container and one instance of the other. Given two functors, F and G, define F*G = F G’, where the product on the right is pointwise. This has the combinatorial interpretation of a G container with one element excised and replaced by an F container. Ie. it’s just like sticking in an F as a child of a G. It’s pretty easy to show that * gives a pre-Lie structure. The only catch is that we can’t subtract types. So to make sense of it we need to rearrange the definition of a pre-Lie algebra so that the subtractions become additions.

Posted by: Dan Piponi on July 13, 2010 11:27 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

Dan wrote:

Given two functors, $F$ and $G$, define $F \ast G = F G’$, where the product on the right is pointwise. This has the combinatorial interpretation of a $G$ container with one element excised and replaced by an $F$ container. I.e. it’s just like sticking in an $F$ as a child of a $G$. It’s pretty easy to show that $\ast$ gives a pre-Lie structure.

Hey, that’s nice!

I guess this makes sense whenever $F$ and $G$ are ‘polynomial functors’ in the sense of Joachim Kock, or even the more general ‘analytic functors’ of André Joyal.

If we like, we can think of each such functor $F$ as giving a ‘first-order differential operator’ $F \partial_x$ which acts on other functors of this type. Each of these differential operators can also be used to differentiate other differential operators of the same type. So, we can define your $\ast$ operation by:

$(F \partial_x) \ast (G \partial_x) = (F \partial_x G) \partial_x$

All this is a categorification of Cayley’s realization that polynomial coefficient vector fields form a pre-Lie algebra, with the pre-Lie operation $\ast$ given by

$(F \partial_x) \ast (G \partial_x) = (F \partial_x G) \partial_x$

Posted by: John Baez on July 14, 2010 12:17 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

The operation that Dan points to is IMHO rather important and much underutilized. (I think I mention this operation in one of the emails on the bar construction, part A, that you have on your website.) We have

$(F \ast G)[S] = \sum_{T \subseteq S} F[T] \times G[S/T]$

where $S/T$ is the result of contracting the subset $T$ to a point. (Note how reminiscent this is of the Green convolution product studied by Joyal and Street in their approach to modular representation theory!) This operation is not associative, but is at least lax associative in that we have an inclusion

$\alpha: F \ast (G \ast H) \to (F \ast G) \ast H$

and $F \ast G$ has the virtue of being cocontinuous in each of its arguments, unlike the substitution product of species.

One reason for its importance is that the notion of operad can be recast as a funny kind of monad whose structure is given by a map

$m: F \ast F \to F, \qquad u: X \to F$

where $X$ plays the role of a lax unit. (I won’t make this precise now, but it can be made precise.) The cocontinuity in separate arguments makes this operation especially suitable for bar constructions. In particular, it is interesting to consider this approach to bar constructions as applied to the Lie operad.

There is much more that could and should be said about this. There is some nice combinatorics here!

Posted by: Todd Trimble on July 14, 2010 2:43 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

Just to add one gloss on what I said above: one frequently sees a (possibly non-unital) operad defined to be a species (a representation of the permutation groupoid in some category, given by a sequence of $S_n$-representations $M(n)$) equipped with maps

$\circ_i: M[m] \otimes M[n] \to M[m+n-1]$

(thought of as the plugging of an $n$-ary operation into the $i^{th}$ argument of an $m$-ary operation) satisfying some associativity axioms. That’s exactly what this operation $\ast$ is supposed to convey: a map

$m: M \ast M \to M$

considers all such pluggings (over $1 \leq i \leq m$) simultaneously, but without a debauch of indices.

Posted by: Todd Trimble on July 14, 2010 4:35 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

Very nice stuff, Todd! The construction of pre-Lie algebras from linear operads is well-known — to a handful of super-genius mad scientists — but it sounds like you’re way ahead of the rest when it comes to figuring out its inner meaning.

I must admit I’m also fond of the interpretation that Jim and Bill and I guessed over dinner, as explained in week299. If the math gods smile on us, that could work out to be just another aspect of what you’re saying.

Just in case anyone forgets how that went:

Suppose $O$ is a linear operad and $A$ is the free $O$-algebra on one generator. We can visualize elements of $A$ as short little trees — ‘sprouts’ — corresponding to the operations of $O$, but where we don’t care about the ordering of the inputs. We can multiply two sprouts like this by summing over all ways of attaching the root of one to a leaf of the other and composing them using our operad. And, thanks to the “secret meaning of pre-Lie algebras”, this makes $A$ into a pre-Lie algebra!

This is nice. But over dinner, James Dolan, Bill Schmitt and I came up with an even slicker construction which seems to give the same multiplication on $A$.

$A$ is the free $O$-algebra on one generator, say $x$. So, for any element $a \in A$, there’s a unique $O$-algebra endomorphism

$f(a): A \to A$

sending $x$ to $a$. Note that $f(x)$ is the identity. By the general philosophy that “an infinitesimal endomorphism is a derivation”, the operator

$\frac{d}{dt} f(x + t a)(b) \vert_{t = 0}$

is a derivation of $A$.

(For certain familiar sorts of algebras, you may aready know what a derivation is. These are just special cases of a general concept of derivation for $O$-algebras. I leave it as an exercise to reinvent this general concept.)

Now, we can define a multiplication on $A$ by

$a \ast b = \frac{d}{dt} f(x + t a)(b) \vert_{t = 0}$

And this is the same as the multiplication I just described. Can we use this slick description to more efficiently prove that $A$ is a pre-Lie algebra? I don’t know.

Posted by: John Baez on July 14, 2010 6:53 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

John wrote:

short little trees — ‘sprouts’ — corresponding to the operations of O

aka corollae to operadchiks if you mean n-ary ops

Posted by: jim stasheff on July 14, 2010 1:03 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

Yes, I meant “corollae”. I prefer Todd’s term for them — “sprouts” — because it’s more humble and Anglo-Saxon, less elaborate and Latinate.

Posted by: John Baez on July 15, 2010 5:20 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

I agree - one of the dangers of the initial attempt at nomenclature becoming set in stone.

Posted by: jim stasheff on July 15, 2010 2:34 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

I personally prefer “shrubs.” It’s shorter than “sprouts” and it’s tree-like.

Posted by: Eugene Lerman on July 15, 2010 4:23 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

I can see why you might say that, but for me the term would have the disadvantage of continually bringing to mind this guy.

Posted by: Todd Trimble on July 15, 2010 5:56 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

I concede that “bush” is even shorter, but shrubs have subobjects! :)

Posted by: Eugene Lerman on July 15, 2010 6:31 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

This is all so very interesting! I hadn’t looked closely at week 299 before, but the material on pre-Lie algebras definitely strikes a chord.

I haven’t thought much yet about your derivations, but it reminds me of some ruminations on a categorified notion of derivation that I was considering in my notes here. There I didn’t quite put it in the language of the operation Dan gave above, but this can surely be done. In fact, why don’t I do this here?

So, there is a notion of derivation on an operad. Let $A$ be a linear operad (or just an operad enriched in commutative monoids), and let’s follow Joyal in thinking of the underlying species as a functor

$A: FinBij \to CMon$

from the groupoid of finite sets and bijections to the category of commutative monoids.

Then, the operad structure on $A$ gives maps of the form

$sub_T: A[S/T] \otimes_{\mathbb{N}} A[T] \to A[S]$

for each subset $T \subseteq S$. Is it clear how this goes? The picture is of plugging one sprout (or corolla) of type A into one of the arguments of another. So if we call an element of $A[T]$ a “$T$-ary operation”, then the operation $sub_T$ plugs a $T$-ary operation into the argument $T/T$ (a point of the set $S/T$) of an operation of arity $S/T$, to obtain an $S$-ary operation.

A derivation $d: A \to A$ is a linear species map such that

$d(sub_T(f, g)) = sub_T(d f, g) + sub_T(f, d g)$

for every subset $T \subseteq S$. Or, if you want to package this in terms of Dan’s operation, the equation amounts to the commutative diagram

$\array{ & & A \ast A & \stackrel{m}{\to} & A \\ \langle d \ast 1_A, 1_A \ast d \rangle & \swarrow & & & \downarrow d \\ (A \ast A) \oplus (A \ast A) & \stackrel{\nabla}{\to} & A * A & \stackrel{m}{\to} & A }$

where $\nabla$ is the codiagonal, or summing operation, and $m: A \ast A \to A$ is the operad multiplication, obtained by collating all the $sub_T$ operations together.

More fun can be had if one is willing to go out on a limb and exponentiate the derivation, i.e., consider

$\exp(d) = \sum_{n \geq 0} \frac{d^n}{n!}: A \to A$

provided this infinite sum can be given a precise meaning. (In the particular application I had in mind in my notes referred to above, this was unproblematic: I was considering operads enriched in sup-lattices, where infinite sums exist.)

Just as the exponential of an algebra derivation is an algebra map, so the exponential of an operad derivation is an operad map… Does this type of thing have similar resonances when applied to the derivations you guys were considering?

[As could be appropriate for the Café, I feel as though I’m indulging here in a bit of “coffeehousing”. Most people who like to play chess have heard of that style of chess known as “coffeehouse”: one should picture two guys in a Viennese café who, fueled by strong coffee and tobacco and absinthe, engage in wildly speculative gambits full of sacrifices and slashing attacks, like those classic examples of the Evans Gambit from turn-of-the-century chess, amid a throng of kibitzers commenting loudly. So, I’m engaging not in the kind of sober reflection ready for the nLab, but more speculatively here, partly for the fun of it, and partly in reminiscence.]

Posted by: Todd Trimble on July 14, 2010 7:33 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

Sadly, even though you’re talking about “Dan’s operation”, you’re using language that’s just outside my working vocabulary. It seems tantalisingly close to making sense to me so I guess if I stare at it long enough it’ll eventually make sense. I’m used to think about datatypes rather than more general categories.

BTW In Aguiar’s paper, Infinitesimal bialgebras, pre-Lie and dendriform algebras, he shows how to get a pre-Lie algebra from what he calls an $\epsilon$-bialgebra. The same construction goes through almost without change for types. Again, you have to generalise the definitions to work in the context of not having subtraction.

In particular, the divided difference bialgebra (or at least the “non-negative” part of it) makes sense for types (and I guess more generally for polynomial and maybe analytic functors too). The divided difference operator (in fact, comultiplication) for types is precisely the dissection operator described here. (Curiously, the author was unaware of the connection with classical divided differences, but managed to accidentally choose to represent dissection using the same symbol used in many papers on divided differences, except tuned upside-down.) If you apply Aguiar’s theorem 3.2 on this operation you get the pre-Lie algebra I described. It’s a bit surprising that divided differences make sense in this context, after all they appear to involve subtraction and division. But divided differences, at least for polynomials, can be defined without reference to either of these operations.

I guess this is connected to the finite differences in Dold-Kan but I don’t understand that stuff yet.

It’s quite amazing that the divided difference comultiplication describes the details of how to convert a recursive function into an iterative one by turning the data implicitly stored on the stack into an explicit data type.

Posted by: Dan Piponi on July 14, 2010 10:36 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

Here’s to coffeehousing! reminds me of Steenrod’s seminar titled `Small semianr for 1/2 baked topolgoical ideas’. It was fun and productive!

Posted by: jim stasheff on July 15, 2010 2:37 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

A CLASSICAL STRUCTURE ON A FINITE-DIMENSIONAL HILBERT SPACE IS THE SAME AS A GENUS-INDEPENDENT UNITARY 2D TQFT

Just a wild guess: If one tries to associate a physical interpretation to this mathematical fact (I’m not sure it makes any sense to try to do that for any 2d TQFT), doesn’t it mean that the time evolution of the described system is trivial?

This reminds me of the “absence of classical fields” in AQFT, to be more precise: In the Haag-Kastler approach this is a theorem that says that any causal net of local commuting algebras fullfilling the spectrum condition necessarily transforms trivially under the representation of the translation subgroup of the Poincare group. The physical interpretation of this would be that “nothing ever happens and everything looks the same everywhere”.

Reference: H.J. Borchers: “Translation Group and Particle Representations in Quantum Field Theory”, theorem IV.6.2.

I don’t know if there is any interesting connection here, it’s only an association.

Posted by: Tim van Beek on June 16, 2010 10:25 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

A CLASSICAL STRUCTURE ON A FINITE-DIMENSIONAL HILBERT SPACE IS THE SAME AS A GENUS-INDEPENDENT UNITARY 2D TQFT

I tend to be sceptical about a deeper meaning of this. After all, this is just rephrasing that matrix algebras are Frobenius algebras. Lots of things are given by matrix algebras. That all these things correspond to 2D TFTs seems to me to be rather a coincidence of low dimensions.

Compare to the situation even one dimensionl lower: 1d TFTs are equivalent to finite dimensional vector spaces. Does this make us want to exclaim each time we see a fine dimensional vector space: “See, this is a 1d TFT!”?

Posted by: Urs Schreiber on June 16, 2010 11:07 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

Urs wrote:

Does this make us want to exclaim each time we see a finite dimensional vector space: “See, this is a 1d TFT!”?

Not each time, but I did it often enough that I’ve built that viewpoint into my thinking. I feel I understand its significance and use. It basically means that we can reason with finite-dimensional vector spaces using string diagrams where we have ‘caps’

$\mathbb{C} \to V \otimes V^*$

and ‘cups’

$V^* \otimes V \to \mathbb{C}$

which act formally like the creation and annihilation of particle-antiparticle pairs in a theory without conservation of energy. In other words: we can reason wiht finite-dimensional vector spaces using Feynman diagrams of a simplified sort.

I also feel I’ve fully incorporated into my thinking the idea that finite-dimensional Hilbert spaces are 1d unitary TQFTs.

But I don’t feel as comfortable with the idea that a finite-dimensional Hilbert space with a ‘classical structure’ (a chosen decomposition into 1d subspaces) is the same thing as a 2d unitary genus-independent TQFT. Now our simplified Feynman diagrams are getting a bit more interesting:

It seems we could be seeing the beginning of a more interesting story here. But I agree, it could also be just a trivial coincidence. It seems to be teetering at the brink of significance…

But then comes another fact, which I learned at Oxford and sketched here: a finite-dimensional Hilbert space equipped with a pair of complementary classical structures gives a richer structure, tying together two commutative Frobenius algebras in a Hopf-like structure. Will this help me figure out what’s going on?

I still don’t know…

Posted by: John Baez on June 16, 2010 4:38 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

You know, genus-independence isn’t so important — a unitary 2d TQFT corresponds to an orthogonal basis in the genus-sensitive case. This still gives you a decomposition into 1d subspaces, it’s just that each 1d subspace now comes equipped with a real-valued weighting. Genus-independence makes this basis orthonormal.

Posted by: Jamie Vicary on June 16, 2010 10:51 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

Good point, Jamie. But now I think I’ve been a bit sloppy about ‘decompositions into orthogonal subpspaces’ versus ‘orthonormal bases’. The map from {orthonormal bases} to {decompositions into orthogonal subpspaces} is many-to-one, with fiber $\U(1)^n$ if we’re working with an $n$-dimensional Hilbert space.

In week299, I had said that a genus-independent unitary 2d TQFT gives a decomposition of the Hilbert space $Z(S^1)$ into orthogonal subspaces. But it gives more: it gives an orthonormal basis, right? $Z(S^1)$ is a special commutative Frobenius algebra, so it gets a basis of minimal idempotents… which turns out to orthonormal. So I should fix what I said.

Similarly, there’s a difference between ‘decompositions into orthogonal subspaces where each subspace has a positive-real-valued weighting’ and ‘orthogonal bases’ — but you seem above to be saying they’re the same.

You know all this stuff, of course. In fact, I guess you’ve busy categorifying all this stuff…

I just want to find the cleanest relation between some sort of 2d TQFT and some sort of ‘classical structure’. I had defined a classical structure to be a decomposition into orthogonal subspaces, but now I realize that a 2d TQFT really gives more. Getting the details exactly right may help me understand the philosophical big picture (if there is one).

Posted by: John Baez on June 19, 2010 4:16 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

A CLASSICAL STRUCTURE ON A FINITE-DIMENSIONAL HILBERT SPACE
IS THE SAME AS A GENUS-INDEPENDENT UNITARY 2D TQFT

I wonder if this has anything to do with the fact that if you expand a classical field theory into Feynman diagrams, you do not get loop diagrams, whereas for a quantum field theory, you do.

Not that I understand that, but I would like to. Feynman diagrams with loops seem to correspond to classical structures with holes.

Gerard

Posted by: Gerard Westendorp on June 30, 2010 11:52 PM | Permalink | Reply to this

### quantum mechanics in dagger-categories

A CLASSICAL STRUCTURE ON A FINITE-DIMENSIONAL HILBERT SPACE IS THE SAME AS A GENUS-INDEPENDENT UNITARY 2D TQFT

Gerard Westendorp says:

I wonder if this has anything to do with the fact that if you expand a classical field theory into Feynman diagrams, you do not get loop diagrams, whereas for a quantum field theory, you do.

Maybe care needs to be exercised with taking that slogan too far.

First, as Jamie Vicary mentioned, the genus-independence is not crucial for this statement.

Second, notice that the notion “classical structure” used here is far, far from the notion “classical field theory”.

What is being called a “classical structure here” is nothing but a choice of basis for a Hilbert space. Nothing else.

If somebody asked me for the relevance of the work by Bob Coecke, Jamie Vicary and others on “classical structures”, I’d say its not 2dQFT, but the following:

these people are following a grand program where they try to rephrase as much as one can about quantum mechanics using just the abstract language of $\dagger$-compact categories. Eventually somebody will find the time to review this im more detail at quantum mechanics in terms of $\dagger$-compact categories, but the basic idea is that of the category $Hilb$ of Hilbert spaces – that is traditionally the context in which quantum mechanics is set up – only the abstract property of being $\dagger$-compact is actually relevant, whereas the concrete technical details of what a Hilbert space is are just there to model these axioms.

For our context, for instance, the observation that these authors were promoting in Coecke, Pavlovic, Vicary, A new description of orthogonal bases was that specifying an orthonormal basis on a finite-dimensional Hilbert space – something induced say from a self-adjoint operator and having the interpretation of the possible classical measurement outcomes of that operator – is something that can be phrased without referring to the concrete details of what a Hilbert spaces is, in particular without a concept of vector space at all: it can be phrased just abstractly in terms of $\dagger$-category language.

So in particular if you followed the authors’ philosophy that quantum mechanics is something that makes sense in any $\dagger$-compact category, this should serve to reassure you that such old friends as the notion of possible classical measurement outcomes of a quantum observable still make sense in this full generality.

I think that’s the point of this result. I think to appreciate this one should look at the rest of their work on quantum mechanics in terms of $\dagger$-abstract theory, it is one puzzle piece in that story.

There are some other noteworthy puzzle pieces that have been found. For instance in Completeness and the complex numbers Jamie Vicary shows how under mild assumptions, the intrinsic quantum mechanics inside any $\dagger$-compact category is necessarily one based on the field of complex numbers .

Well, the axioms of a $\dagger$-category immediately imply that the “monoid of scalars” in it is a $*$-monoid, but looking just a little more closely Jamie finds that this *-monoid has trouble not being the complex numbers. Which is kind of noteworthy, it seems.

Posted by: Urs Schreiber on July 1, 2010 7:39 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

Hello all,

the information era is scary — people are commenting on a talk before it has even been given… (Or perhaps it is completely normal, but this is the first time it happened to me, and I don’t know exactly how to react. I could of course just pretend I don’t read this blog, while in reality I am refreshing the page every 20 minutes to see if somebody has guessed my whole talk!)

John wrote:

I wonder how Kock’s construction relates to Aguiar and Mahajan’s

I don’t want to ‘give away’ the talk, but a short answer is given below, after a few comments on related issues.

Re universal property

Connes and Kreimer discovered a universal property: the linear operator B+ on H that adjoins a common root to a forest is the universal 1-cocycle for Cartier-Hochschild cohomology. This amounts to the fact that

Δ B+ = (B+ ⊗ Id + e ⊗ B+) Δ

and that any Hopf algebra with a linear operator satisfying this equation admits a unique Hopf algebra map from H compatible with the linear maps. Moerdijk (1999) soon found a nice conceptual explanation: for an operad P, let P[t] denote the operad obtained by freely adjoining a unary operation. If P is a Hopf monad, then P[t](0) becomes a Hopf algebra, and t itself becomes a 1-cocycle. The Connes-Kreimer Hopf algebra H is nothing but Comm[t](0), and B+ is t.

In other words,

(H,B+) is the initial Comm[t]-algebra.

Concerning trees as initial algebra: polynomial functors cartesian over M (the free-monoid monad) are precisely nonsymmetric operads. The terminal object in Poly/M is of course just M itself, and its initial algebra is the set of planar trees. In other words, planar trees are M-trees. Nonplanar trees are not P-trees for any polynomial functor, and in fact the category of polynomial functors does not have a terminal object. The terminal object that ‘ought’ to exist is the free-commutative-monoid monad which is not polynomial (for example, it does not preserve pullbacks). It is instead analytical — it is the exponential! (It is analogous to saying that the terminal G-set is not free.)

(Over infinity-groupoids instead of sets, the difference between ‘polynomial’ and ‘analytic’ evaporates, as you can imagine. This is a main point of the theory of homotopical species — which at the moment has been written down but not written up.)

On the other hand, while nonplanar trees cannot be characterised as initial algebra for a polynomial endofunctor, they are themselves polynomial endofunctors: the polynomial endofunctor corresponding to a tree is A ← M → N → A, where A is the set of edges, N is the set of nodes, and M is the set of nodes with a marked input edge. (The first map returns the marked edge, the second forgets the mark, and the third map returns the output edge.) This makes it very convenient to talk about decorated trees (like opetopes) in terms of polynomial functors. (In my paper ‘Polynomial functors and trees’ I argue that this viewpoint on trees is in fact a practical definition of tree.)

Re the role of polynomial functors

Polynomial functors correspond to flat species, i.e. those for which the symmetric-group actions are free. Analytical functors are more expressive than polynomial functors. But many interesting analytical functors happen to be flat (for example the sequence of opetope monads of Baez-Dolan, at least the version given in Leinster’s book). In such cases the polynomial viewpoint is powerful because it exploits representability. Handling polynomial functors in terms of their representing diagrams I ← E → B → J was advocated by Joyal. Polynomial functors allow many interesting intrinsic characterisations (e.g. as local right adjoints), whereas the ‘bridge’ diagram is just a presentation. Perhaps many category theorists or geometers will feel that the intrinsic viewpoint is better. But in combinatorics a ring is not just a ring: typically, specific bases carry the combinatorial information, and change-of-bases can amount to deep combinatorial identities. Polynomial functors in the bridge-diagram viewpoint is very much about combinatorics. A more combinatorial approach to data type theory also seems to be in the air — cf. scattered remarks in the recent literature.

Re the typicalness of minus signs in antipodes

Combinatorial bialgebras B are most often graded connected: graded because they have bases given by combinatorial structures, and these have sizes (or otherwise simply because B is a polynomial algebra), and connected (i.e. B0=k) because there is only one zero-size structure. It is a general fact that a connected graded bialgebra has an antipode — provided k is a ring! The formula is given by recursion by degree, and it always involves minus signs (-1)n where n is the degree. Since an antipode is unique if it exists, a connected graded bialgebra over N never has an antipode. On the other hand, combinatorial bialgebras ought to be defined over N. Perhaps one can say (sorry if this is superficial) that the availability of additive inverses is the major trick of algebraic combinatorics?! Additive inverses are crucial for getting multiplicative inverses, as in Möbius inversion, Lagrange inversion, Birkhoff decomposition, etc. — the basic example is really matrix inversion!

Re categorification of Hopf algebras

Aguiar and Mahajan have found a very good way to put combinatorics into vector spaces, lifting the constructions to a categorical level. Extremely elegant. But the outcome still has the magic of algebra — the minus signs are not really explained, they are just part of the magic. To understand the minus signs, I would like to go as far as
possible just working with finite sets. The recent paper by Lawvere and Menni on the Hopf algebra of Möbius intervals is a good expression of the philosophy. The categorified antipode does not exist on its own, but it can be written as a difference between an even-degree antipode and an odd-degree antipode. Part of the game is the strive for introducing the minus sign as late as possible. And of course it becomes more and more urgent to understand what are negative sets, for example in the sense of Schanuel. Unfortunately this is very hard.

The categorification of the Connes-Kreimer Hopf algebra that I will talk about at the CT is mostly an example computation to showcase the combinatorial content of polynomial functors. But I hope it will develop into a more well-organised theory that should be a multi-version of Lawvere-Menni theory: a tree should be seen as a multi-interval. I hope there will soon be things like ‘Möbius multicategories’, in analogy with Leroux’s Möbius categories studied by Lawvere and Menni, and so on. In fact, since multicategories are just polynomial functors with a double-categorical monad map to M, it may be convenient to go instead for ‘Möbius polynomial functors’…

Re numerical analysis

There is an ongoing research trimester in Madrid on Combinatorics and Control, covering topics like pertubative quantum field theory, numerical integration theory, and control theory. The final conference of the programme is next week, and the speakers include Butcher, Ebrahimi-Fard, Hairer, Kawski, Munthe-Kaas.

I participated in the School earlier this Spring — it was a
fantastic experience, it opened a new world for me. It is very comforting for a pure mathematician to see advanced algebraic structures in use in very applied mathematics. What they are doing in control theory is still quite obscure to me (although Kawski gave a very entertaining example computation, showing how the differential equations and controls involved in parking a car (steering wheel and gas pedal) lead to nested integrals, brackets and pre-Lie structure!). But the numerical analysis has impressed me very much, and I want to learn more about it.

An alternative to Butcher’s book is Hairer, Lubich, and Wanner’s Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations, Springer, 2002. It is somewhat more algebraic, and does not avoid Hopf algebras and Lie theory.

Cheers,
Joachim.

Posted by: Joachim Kock on June 17, 2010 11:22 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

Joachim, from what it’s worth from my my inexperienced and lowly grad-student position I think it’s very useful for an audience to have a good idea of the content of a talk before they see it.

When I attended undergrad lectures I would always try to know what would be said during each lecture before I attended it. This way I understood a lot more.

Posted by: Tom Ellis on June 17, 2010 11:28 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

I agree it is helpful to study the material for a talk before listening to it. However, I think live mathematics like conferences should have some priority. Perhaps I am oldfashined, but imagine

“I was in the audience when Prof. X explained his theory…”

“Ah, I couldn’t go to the conference, but his talk was transmitted live on the internet two days before the conference”.

(Not that I think I will ever become Prof. X.)

Cheers,
J.

Posted by: Joachim Kock on June 17, 2010 1:51 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

Especially for colloquium talks, some places have a pre-talk
to provide the background.

Posted by: jim stasheff on June 17, 2010 2:41 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

Joachim wrote:

An alternative to Butcher’s book is Hairer, Lubich, and Wanner’s Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations, Springer, 2002.

Aha, thanks! There is a second edition already (ZMATH).

Chapter III explains the relation of the order of Runge-Kutta methods and rooted trees, table 1.1 is particularly enlightning. Here is an appetizer: We take an initial value ODE $y^{(1)} = f(y), y(t_0) = y_0$, with $f: \mathbb{R}^n \to \mathbb{R}^n$ real analytic (let’s pretend we are physicists for the moment), and we compare the Taylor series of the exact solution (at $t_0$) to the Taylor series of the approximation from the Runge-Kutta method, the derivatives of the exact solution are

$y^{(1)} = f(y)$ $y^{(2)} = f^{(1)} y^{(1)}$ $y^{(3)} = f^{(2)} (y^{(1)}, y^{(1)}) + f^{(1)} y^{(2)}$ $...$ Using the differential equation itself we can substitute derivatives of $y$ with derivatives of $f$ and get

$y^{(1)} = f(y)$ $y^{(2)} = f^{(1)} f$ $y^{(3)} = f^{(2)} (f, f) + f^{(1)} f^{(1)} f$ $...$ Now each summand on the right hand side can be represented by a rooted tree (see the book), same can be done for the approximation and demanding that the first n terms agree leads to constraints on the coefficients of the Runge-Kutta method that can be calculated from rooted trees…

BTW: geometric numerical integration is concerened with numerical approximation methods for systems with symmetries such that the approximation exhibits the same symmetries. This is a very important topic, example: Numerical approximations to systems with energy conservation and bound states will fail in the long run if they violate energy conservation.

Posted by: Tim van Beek on June 17, 2010 1:21 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

The polynomial functor of a tree turned out to
contain some sort of html comment tag! It should have displayed as

A \leftarrow M \rightarrow N \rightarrow A

Posted by: Joachim Kock on June 17, 2010 1:37 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

To display the itex, just enclose within dollar signs and choose an itex setting from the Text Filter menu.

So

$A \leftarrow M \rightarrow N \rightarrow A$.

Posted by: David Corfield on June 17, 2010 1:56 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

I fixed Joachim’s post as well as I could, given that he did not choose a text filter that allows for the use of TeX. I wish Jacques would make such a text filter the default.

Posted by: John Baez on June 17, 2010 5:36 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

Thanks for a truly amazing blog-comment, Joachim. You are pulling together a lot of my favorite things. I hope you meet Bill Schmitt and talk to him for a long time someday.

I’ll probably comment more after I recover.

Posted by: John Baez on June 17, 2010 5:30 PM | Permalink | Reply to this

### Other news in combinatorial Hopf stuff

Hi! Let me add my approval to John’s suggestion about talking to Bill Schmitt, and extend that same recommendation with respect to his student Forest Fisher. At the Discrete Mathematics conference in Austin (SIAM) we just heard from the latter in place of the former. The session on combinatorial Hopf algebras (parts one and two) was organized by Frank Sottile.

Forest’s talk described a cocommutative Hopf algebra of graphs where the coproduct decomposes as the sum of two non-coassociative coproducts. They call this coZinbiel – try reading it in reverse!

Bill Schmitt’s point of view, using any species with some notion of restriction (as in restricting a set structure to a subset) in order to define Hopf algebras, is really starting to be appreciated. Several of the talks focused on that point of view. One of the key motivators of this surge of interest is the work of Marcelo Aguiar and Swapneel Mahajan, and their work with Federico Ardila. They are working on a Hopf monoid with basis the species of Postnikov’s generalized permutahedra. Since so many combinatorial objects have corresponding generalized permutohedra, their antipodes recapture lots of famous identities.

Here is a talk given by Marcelo on the subject: I’m hoping to get a copy of Federico’s talk from Tuesday soon!

Which brings me to question for Joachim–Is your categorification a Hopf monoid as described by these guys? A Hopf monoid is a species with coproduct and product parameterized by the decompositions of a set into disjoint unions. Then the main axiom says that the order in which a set is decomposed into four subsets will not matter–reminiscent of the Eckman-Hilton picture. The third and fourth slide I link to above show it nicely.

Posted by: Stefan on June 17, 2010 11:17 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

Thanks, John, for the improved typography — and for your kind words! Indeed I hope to get the opportunity to meet Schmitt one day. I am a big fan of his two papers “Incidence Hopf algebras” and “Hopf algebras of combinatorial structures”.

Thanks, Stefan, for the pointers. They all look very interesting.

Cheers,
J.

Posted by: Joachim Kock on June 18, 2010 1:48 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

Joachim, from a physical point of view the utility of operads (and generalisations) is clearer in twistor techniques, such as BCFW recursion, than it is in local QFT. See this paper by Kreimer and Suijlekom. Cheers, Marni.

Posted by: Kea on June 18, 2010 11:38 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

Hi Marni,

I have actually been trying to read Suijlekom’s papers, but it is
difficult for me because I am quite ignorant about the background.
So I just skip the physics in the papers and start reading again
when there is a sentence that contains the words ‘graph’ or ‘tree’.

Can you say in a few words what ‘twistor techniques’, ‘BCFW recursion’,
and ‘local QFT’ mean?

Thanks!
Joachim.

PS: I don’t blame you for freely using such words — after all
it is (also) a physics blog!

Posted by: Joachim Kock on June 19, 2010 9:42 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

Hi Joachim. By local QFT I mean the standard techniques of renormalisation that you are discussing. These hinge on Feynman diagrams associated to (momentum) flat Lorentzian spaces. Let us call Minkowski space M. As you know, the one dimensional nature of the trees signifies point particle interactions.

Twistor spaces are alternative classical projective geometries which are obtained through a (span) correspondence with M, by (i) complexifying M (using SL(2,C) and introducing spinors) and (ii) interpreting, for instance, light rays as basic elements rather than points in M. The old basic theorem by Penrose was that solutions to the massless Dirac equation on M are in one to one correspondence with (Cech) cohomology classes on the twistor side. A ‘twistor’ is just a convenient geometric element of the new space, built from two spinors.

Now only in the last few years have twistor theorists managed to understand diagram techniques for twistor spaces, with which they always intended to replace Feynman techniques. This is in the (physics style) papers by Hodges, Arkani-Hamed, Mason and others. For certain theories of physical interest, the new techniques uncover hidden simplifications in complex Feynman sums, so they must be telling us something important about nature. There are 2-colored tree graphs known as ‘MHV diagrams’ (MHV is one special case, namely the maximal helicity violating one, where helicity is the color (or sign) on the tree). These are not at all the same as Feynman trees. BCFW recursion is a rule for recovering a tower of physical amplitudes for these trees from the basic trivalent pieces (roughly speaking). Despite the coloring, you will see that the combinatorics basically comes from the associahedra, since Arkani-Hamed discovers the Catalan and Narayana numbers for indexing terms. So when I say ‘operad’ I am thinking in particular of the topological ones associated to Batanin’s world of polytopes, but there must be many other operads lying around here, such as in the connections in Suijlekom’s papers.

Although pretty well nobody is looking at the category theory connection yet, the real experimentalists do appreciate the importance of the classical twistor techniques. To a theorist, the interest comes from beyond standard physics, because there should be an entirely different twistorial formulation of standard particle physics. Old twistor ideas about particle masses use higher cohomology groups. Newer ideas suggest more universal cohomologies, but the physical intuition still comes from the twistor picture, not the local QFT picture.

Posted by: Kea on June 19, 2010 11:05 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

polynomial functors cartesian over M (the free-monoid monad) are precisely nonsymmetric operads.

Is that really what you mean? I would expect nonsymmetric operads to be monads cartesian over M, not just functors. Where does the composition of the operad come from if you just have a functor?

Posted by: Mike Shulman on June 19, 2010 5:59 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

> I would expect nonsymmetric operads to be monads cartesian over M, > not just functors.

Sure, I did mean monads. Thanks for catching this slip.

By the way, as you may remember from Nicola Gambino’s talk at the Hyland-Johnstone fest (see the recent preprint ‘Monads in double categories’), the double-categorical viewpoint, and indeed the framed-bicategory viewpoint, is important in the theory. For example, multicategories are precisely monads cartesian over M in the double-categorical sense.

Furthermore, decorated trees (such as opetopes) are trees cartesian over a polynomial endofunctor in the double-categorical sense. In fact, the trees that occur in renormalisation are also decorated in this sense (although the physics papers do not formulate it exactly in this way): they are decorated by the polynomial endofunctor $I \leftarrow E \rightarrow B \rightarrow I$ where $I$ is the set of interaction labels for the theory, $B$ is the set of ‘primitive 1-particle-irreducible divergent’ Feynman graphs for the theory, and $E$ is the set of such graphs with a marked vertex. In this way the decorated tree can be seen as a recipe for building the graph, exactly as ‘composition trees’ serve to reconstruct ‘constellations’, in the terminology of the paper ‘Polyomial functors and opetopes’.

These two example stress that trees are an expression of nesting. This is of course is a banality, but passing back and forth between the two viewpoints can often have a great effect.

Here is the nesting reformulation of John’s explanation of the pre-Lie axiom: you want to nest three things, let’s say circles, and define an auxiliary binary operation * meaning ‘nest the second argument inside the first argument’. Now compute (A*B)*C and A*(B*C). In the first, after putting B inside A you have two options for where to put C: inside A but outside B, or inside B. When computing A*(B*C) of course only the second option occurs. So the difference between the two is the case where B and C are next to each other inside A. This is clearly symmetric in B and C.

Shit — when formulated this way, in prose and algebra, there is no difference with John’s explanation! You really have to draw the circles to see for yourself (sorry, I don’t know how to do it here). If you are too lazy to try, here is an attempt at describing the psycological effect:

1) nestings have a ‘God-given root’, namely the outermost element, whereas for trees you need some bookkeeping to know which vertex is the root.

2) The equivalence relation in force for the drawings of nestings is just isotopy, which is probably already hardcoded into the vision unit of the brain, whereas for the trees drawn on the paper you need to impose the extra equivalence relation of dividing out by symmetry, something that may have some overhead, even for a mathematician.

Not convinced? Maybe it is a bad idea to display your brain on the internet or to assume that other peoples’ brains work the same :-)

Cheers, Joachim.

Posted by: Joachim Kock on June 19, 2010 8:59 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

I’m not too lazy, but this picture might not be exactly what you are describing. However, it does show correspondence between trees and nestings! In my work it has come up often that different sorts of trees (here I label the nodes) have corresponding nestings. (I usually call them tubings, after Satyan Devadoss.)

It is Figure 8 of this paper.

These pictures allow us to perform Hopf algebra operations explicitly by nesting.

Posted by: stefan on June 19, 2010 10:31 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

Neat! I do remember seeing that double categories come up, and being happy about it. Presumably this point of view on operads/multicategories is related to the double-categorical approach to generalized multicategories.

Posted by: Mike Shulman on June 20, 2010 4:40 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

I would really like to read this paper on Fock space and Witt vectors … but I don’t have access to it. Would somebody be kind enough to email it to me?

Posted by: Kea on June 25, 2010 3:38 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

Posted by: Tim van Beek on June 25, 2010 9:19 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

I love how the appendix of that paper goes all the way from the definition of a commutative ring to fibre products of schemes in just a couple of pages.

Posted by: Dan Piponi on June 26, 2010 2:08 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

Thanks, Tim! Lol, yes Dan. But this is often necessary in real physics papers, where the readership can be very diverse.

Posted by: Kea on June 26, 2010 2:31 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

Gee, I sure wasted a lot of time reading Hartshorne when all the algebraic geometry I need is in that appendix.

Posted by: John Baez on July 1, 2010 12:17 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

OK, so this is now one of the most confusing comment threads ever written. The conversation seems to have gone: Connes-Kreimer trees $\rightarrow$ Hopf monads $\rightarrow$ species $\rightarrow$ Riemann gases $\rightarrow$ stuff types $\rightarrow$ characteristic classes $\rightarrow$ splitting principles $\rightarrow$ $\lambda$-rings … … but now I’m wondering about connecting $\lambda$-rings back to the topic of the post, namely mutually unbiased bases. Now James here is the expert on $\lambda$-rings and the field with one element, and MUBs may be viewed as representing the finite fields on sets as vector spaces over the field with one element. A naive analogue of a Witt vector then would be an infinite list of MUB generators. For example, in the $2 \times 2$ qubit case we could take $F_2$ (the Hadamard matrix), $R_2$ (the circulant with top row $(1,i)$) and $I$ (the identity). These could be labeled $X$, $Y$ and $Z$ since they correspond to the three Pauli operators. Then an infinite sequence of the $X$, $Y$ and $Z$ form a non-commutative path, as in a component of a non-commutative path integral. So the Witt process on these paths would be what allows the non-commutative path integral to be well defined. Very cool.
Posted by: Kea on June 26, 2010 4:04 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

short notice:

it seems the videos on the page:

http://www.cs.ox.ac.uk/quantum/content/events.html#QICS%20spring%20school%202010

are gone.

### Re: This Week’s Finds in Mathematical Physics (Week 299)

I don’t see a problem with the videos on that page. I just watched two of them: Abramsky’s and Barrett’s.

Posted by: John Baez on June 29, 2011 8:32 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

Thanks, it seems to be a firefox problem.
I even can’t see this comment here in firefox.
In Safari the videos seem to be there - that is I get the comment that a new Quicktime version has to be installed.

### Re: This Week’s Finds in Mathematical Physics (Week 299)

I always use Firefox (right now I’m using version 5.0 on a Windows machine), and I’ve never had trouble seeing the videos on that website, or reading the comments here. So, I’m not sure what problem you’re having. I’m glad that at least you can get things to work with Safari!

Posted by: John Baez on June 29, 2011 5:04 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 299)

I didn’t say that I got things to work with Safari.
I just said that I get a comment that a new Quicktime version has to be installed.

I have a mac firefox version that may be the problem.

If I would switch to Safari then I would need to transfer all my bookmarks, if this is possible at all.