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March 27, 2008

Categorified Quantum Groups

Posted by David Corfield

Once in a distant blog, John was quick to pour cold water on the suggestion I made that the aims of those categorifying might differ sufficiently to merit distinguishing types of ‘categorification’:

I don’t like this “Frenkelian” versus “Baezian” distinction. Baez was inspired to work on higher categories thanks to the work of Crane and Frenkel. Frenkel’s student Khovanov cites Baez’s work on 2-tangles in his first paper on categorified knot invariants. Frenkel’s student Khovanov has taken on Baez’s student Lauda as a postdoc at Columbia starting next fall. Will their work on categorifying quantum groups and using these to get 2-tangle invariants be “Frenkelian” or “Baezian”?

Some results of the collaboration are now out. Aaron has just posted A categorification of quantum sl(2) to the arXiv.

Did the Geometric Representation Theory Seminar reach the point of having categorified quantum groups? Not that I’m after differences of approach, of course.

Posted at March 27, 2008 6:34 PM UTC

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Re: Categorified Quantum Groups

Aaron starts with:

It is quite natural to expect that a categorification of quantum groups should exist.

Personally, while I am acquainted with quantum groups, we have not become friends yet. Since their representation categories may coincide, I like to think of quantum groups as a certain incarnation of centrally extended loop groups.

These, in turn, can be thought of as being essentially Lie 2-groups, as described in From loop groups to 2-groups.

From that point of view, there is no question that you eventually want to categorify this.

I would have to spend probably a lot of time to see if the following is related to what Aaron is considering, but let me say it anyway:

the centrally extended loop groups are entirely controlled by a 3-cocycle on the underlying group. This becomes most strikingly manifest when we pass to the corresponding Lie 2-groups and then their Lie 2-algebras: these are the “String type” Lie 2-algebras g μg_\mu described by John and Alissa in HDA VI which are entirely determined by a Lie algebra and a 3-cocycle on it.

So probably we want to be looking at higher generalizations of that.

In L L_\infty-connections # we observe that there are actually large families of “higher String-like” Lie algebras of this kind:

there is a Lie nn-algebra g μg_\mu for every Lie kk-algebra with an Lie (n+1)(n+1)-cocycle on it. If this cocycle is in transgression with an invariant polynomial, this is of “String type” and generalized the String Lie 2-algebra which connects to centrally extended loop groups.

One would hence expect that integrating these higher String-type Lie nn-algebras up to nn-groups and looking at their representation theory relates to the categorification that Aaron is after.

(Every such Lie nn-algebra can be integrated to an ω\omega-group as I describe for instance at the end of space and quantity.)

Just an observation. I haven’t even started trying to think about whether this has any relevance for what Aaron is doing.

Posted by: Urs Schreiber on March 27, 2008 7:04 PM | Permalink | Reply to this

Re: Categorified Quantum Groups

These two points of view of categorification did not start out as being different. As Masahico and I were constructing the movie moves (early 1990s, TWF 1), Crane and Frenkel got a hold of a draft version. John Fisher contacted us to asked about them in the context of the Zamolodchikov equation.

Masahico was in Toronto at the time. I flew to Toronto and the two of us drove to New Haven to meet Fisher, Crane, and we had a Saturday afternoon discussion with Igor Frenkel about the problems of categorification.

I started at Yale in 1978. Frenkel started a couple of years afterwards as a grad. student. Before I had earned my degree Igor had his PhD and an Asssistant Professorship there.

In the Fisher, Frenkel, Carter, Saito, conversation Igor talked about categorification in terms of the double loop group. Masahico and I were interested in the Jones polynomial for knotted surfaces. Fisher’s thesis was finished. Everyone was happy. Except that the interpretation of the movie moves that Fisher used involved thinking equality when some things that were only natural isomorphisms.

In some of the original and subsequent formulations of KhoHo, the old school movie moves are used to show that it constructs an invariant of knotted surfaces. This construction works because there is a fixed frame of reference for all the knot diagrams.

One has to be careful! The misinterpretation lead to Langford’s dissertation. Masahico, Joachim Rieger, and I enhanced the movie moves to take into full consideration the height functions in the stills. These are the moves that Baez and Langford used to complete the proof that the free rigid braided monoidal 2-category with duals on one self-dual object generator was the 2-category of 2-tangles. Somewhere around this time, JB wrote out the tangle hypothesis; some of that was folklore or was being promulgated by Louis Crane or members of the east coast school.

Laurel Langford, Masahico Saito, and I met to try to use the cocycles of Neuchl that constructed a braided moniodal 2-category out of representations of a Hopf category. After that meeting, we were able to simplify the construction to use quandle 3-cocycles.

Meanwhile, Mikhail Khovanov was a subsequent student of Frenkel. (I think Pavel Etingof was before Khovanov, but I don’t feel like fact checking, now.) Khovanov had some interesting ideas about immersed curves (doodles), and other combinatorial questions that he was working on. But he began working on the representation theory questions.

Anyway, about 10 or 15 years ago, all these ideas were being born together and independently. They still are coming together. [Pretend I just linked to the Beatles’ song]. Recently we saw Tim Porter’s note on crossed things, (related to Yetter’s construction), Urs’s extensive posts, and the relation to Baez-Crans. Crans, Elhamdadi, Saito and I have used ideas from Hochschild homology to construct diagrammatic cocycles.

I am kind of thinking that there are a few different kinds of cohomology that are manifested to be the same in lower dimensions, but are controlling different aspects of categorification problems.

Posted by: Scott Carter on March 28, 2008 3:03 AM | Permalink | Reply to this

Re: Categorified Quantum Groups

It seems to me that the main ambiguity in “categorification” comes, roughly, from the dichotomy

internalizationenrichment. internalization \leftrightarrow enrichment \,.

Categorifying a concept XX is usually interpreted

a) either as realizing XX internally to nCatnCat

b) or as considering nCatn Cat internal to XX

(assuming all required pullbacks etc exist for this to make sense, of course).

Here the approach b) is the more systematic one. As long as the category of Xs has all the required pullbacks or whatever you need, it’s easy to go all the way to \infty-categorification using b): just pick your favored model of \infty-categories, internalize it in b) and done.

With a) it’s usually different. It’s systematic only if one internalizes XX strictly in nCatn Cat. In that case, a) tends to lead to something equivalent to b).

But usually the point is not to do that, but instead to make use of all the higher homotopies provided by nCatn Cat. That allows/requires making lots of structural decisions. And coherence theorems want to be proven then. So for a) then, categorification is not just a craft, but an art, if you know what I mean.

Recently I mentioned to somebody how I am working on a theory of \infty-connections. He gave me a look and then, very politely, remarked that there is no mystery here: just consider Kan complexes internal to the category of bundles with connection. And done.

That’s true. That’s strategy b). But it’s not, I think, what is sufficient for doing what would tend to want to do with an \infty-connection.

Just as an example.

Concerning knot theory and Khovanov homology, has anyone thought about: What is an \infty-knot?

Posted by: Urs Schreiber on March 28, 2008 11:26 AM | Permalink | Reply to this

Re: Categorified Quantum Groups

Noohi makes the point that Khovanov homology in turn can be categorified. Ingeneral, homtoloy is decategorification.

Posted by: jim stasheff on March 29, 2008 12:52 PM | Permalink | Reply to this

Re: Categorified Quantum Groups

Thanks for the history - we need more of that.

Posted by: jim stasheff on March 29, 2008 12:49 PM | Permalink | Reply to this

Re: Categorified Quantum Groups

People certainly use the word ‘categorification’ in different ways.

Many casual observers think that categorification is something to do with knot theory. That’s because knot theory is a context where categorification has had conspicuous success; people go to talks about it and come away with the impression that that’s what categorification is about. (I’ve actually met someone who thought that categorification was about one particular knot.)

Many of those working on categorification in knot theory and representation theory use the word in a much more restricted sense than I would. I’ve talked to one such person about this; for them, categorification is exclusively about linear categories — those where the hom-sets are vector spaces.

But for many people here, the word is not restricted to particular subjects (such as knot theory) or particular types of category (such as linear). Finding a bijective proof of a combinatorial identity is categorification. Transforming the theory of monoids into the theory of monoidal categories is categorification. In this usage, the word is free to roam just about anywhere.

Posted by: Tom Leinster on March 28, 2008 3:38 AM | Permalink | Reply to this

Re: Categorified Quantum Groups

Many of those working on categorification in knot theory and representation theory use the word in a much more restricted sense than I would.

This is definitely true. I remember last March when most people in a knot theory special session at an AMS meeting – many of whom had just presented “categorifications” of their own – were absolutely shocked to hear that there were categorifications that had nothing to do with homology.

Posted by: John Armstrong on March 28, 2008 4:55 AM | Permalink | Reply to this

Re: Categorified Quantum Groups

Apologies - the person I was semi-quoting
was John Armstrong and not Noohi. Successive recent visitors.

Posted by: jim stasheff on March 29, 2008 12:54 PM | Permalink | Reply to this

Re: Categorified Quantum Groups

I’d refine this to say that Khovanov homology is a decategorification. Khovanov defines a certain “combinatorial link covariant” – a functor from the groupoid of link diagrams/Reidemeister moves – and then decategorifies this by passing to homology to get an invariant. But the interesting stuff is actually in the covariant.

Posted by: John Armstrong on March 29, 2008 4:06 PM | Permalink | Reply to this

Re: Categorified Quantum Groups

I just looked back at David’s original entry and was reminded of this part:

In his recent post, Peter Woit wonders why “The math blogosphere seems to my mind somewhat weirdly dominated by those with an interest in category theory”, and Walt from Ars Mathematica comments:

What’s even odder is that it’s not just a fascination with categories, but with n-categories. I think part of it is that John Baez has always been such an effective advocate of his n-categorical point of view that he’s both attracted people to the subject and inspired them to follow his example and post about it on-line.

I think that, in actual fact, what is really weird is finding this nn-categorical emphasis weird.

You know, over at the Secret Blogging Seminar they recently mentioned a half-joking discussion over a conference dinner about the need to install an (,n)(\infty,n)-Category Café.

The point being that, given the recent developments in math, the blogosphere is not in fact overemphasizing higher categories, but is lagging behind.

Posted by: Urs Schreiber on March 28, 2008 12:51 PM | Permalink | Reply to this

Re: Categorified Quantum Groups

I think that, in actual fact, what is really weird is finding this n-categorical emphasis weird.

Well, you must admit that the portion of research-level mathematics weblogs dealing with categories at all, let alone higher categories, is a bigger portion of all research-level mathematics weblogs than mathematicians dealing with categories are of all mathematicians.

Posted by: John Armstrong on March 28, 2008 2:28 PM | Permalink | Reply to this

Re: Categorified Quantum Groups

you must admit

Probably. But on the other hand, we should really be counting properly: all of algebraic topology, homotopy theory, homological algebra, representation theory, algebraic geometry,… whenever any model category structure appears, whenever anything derived appears, it is \infty-categories.

Posted by: Urs Schreiber on March 28, 2008 4:24 PM | Permalink | Reply to this

Re: Categorified Quantum Groups

This is true, but I tend to see a lot of the practitioners avoiding this language. You know and I know that they’re really talking about categories, but many mathematicians wouldn’t be caught dead saying “category”.

But here on the weblogs the scattered cat-fanciers of the world manage to find a way to stay in contact and reinforce each other. It’s not exactly the first time a subculture has used the internet as a refuge as they build to full recognition in the wider culture.

Posted by: John Armstrong on March 28, 2008 5:06 PM | Permalink | Reply to this

Re: Categorified Quantum Groups

And some of us are not really intersted in categorification per se
but rather the relevant homotopy theory.
And some of us have foudn this blog by accidnet being unaware of many others, cf. the infinity semianr blog.

Posted by: jim stasheff on March 29, 2008 12:57 PM | Permalink | Reply to this

Re: Categorified Quantum Groups

My guess is that the people who find an emphasis on nn-categories ‘weird’ are under the impression that nn-categories are a specialized and esoteric topic in mathematics, rather than a new outlook that applies to all of mathematics.

This impression will take a while to go away. Right now nn-categories don’t even merit an entry in the official AMS mathematics subject classification.

The big question is: when they finally do show up, will nn-categories be listed as a special topic under categories… or will categories be listed as a special topic under them?

Posted by: John Baez on March 28, 2008 8:45 PM | Permalink | Reply to this

Re: Categorified Quantum Groups

Off-topic, but I’m curious: does anyone actually use those AMS classifications?

Posted by: Tom Leinster on March 30, 2008 3:44 PM | Permalink | Reply to this

Re: Categorified Quantum Groups

Sure, lots of journals and conference proceedings I publish in require those annoying MSC numbers — they get stuck on the first page of the paper. They’re required for all AMS publications, but also some others as well.

Their purpose is to help MathSciNet classify papers by subject. Nowadays that may not matter so much — how many people actually type something in “MSC Primary” when searching for a paper on MathSciNet? In the old days, I’d browse the paper equivalent, Mathematical Reviews, to see all the new papers on a given topic. But now I’d be more likely to browse the arXiv — to hell with people who don’t put there stuff there!

(Maybe you don’t use MathSciNet. But it’s got good summaries of pretty much every math paper since the early 1900s. This and other features make it much better than Google when you’re searching for math literature — and the fact that it’s not free helps keep the AMS afloat: it lets them charge less for journals and books than the LMS.)

The physicists have their own numbering system, run by the AIP.

Posted by: John Baez on March 31, 2008 2:12 AM | Permalink | Reply to this

Re: Categorified Quantum Groups

John rhetorically asked: how many people actually type something in “MSC Primary” when searching for a paper on MathSciNet?

I sometimes do, though for all I know I could be the only one. It can be useful if I’m searching for a word in the title or review text which is used to mean something different in another subfield. Using “MSC Primary/Secondary” together with “and” can also be a good way to search for those topics (that is, all of them) that don’t really fit any one MSC number and thus are likely to be described by some combination.

It can be interesting, by the way, to see how the MathSciNet reviewers’ classifications of your papers differ from your own. For example, I had no idea that one of my papers has to do with equilibrium statistical mechanics. (Nor have I figured out yet what the connection is - perhaps the authors of some of the papers I’ve reviewed would be similarly surprised.)

Posted by: Mark Meckes on March 31, 2008 4:13 AM | Permalink | Reply to this

Re: Categorified Quantum Groups

Yeah, I know you have to choose MSC numbers when you publish, but that’s not what I meant by using them. To me it’s simply an annoyance, and I wanted to know what purpose it serves.

(Whenever I choose those numbers, I feel like I’m lying. It’s a bit like when I had to go on a “New Lecturer Programme”, where they supposedly taught us new staff how to teach. Because it was across all subjects, I had to really stretch what the instructors said to make it applicable to mathematics — and even then it didn’t always make much sense. I get the same sensation when choosing MSCs. My only consolation is to flatter myself that what I’m doing is too cutting-edge to fit into their fuddy-duddy old system.)

I do use MathSciNet, a lot, but I’ve never used MSC numbers within it. I’ve also never used the paper version of Mathematical Reviews, and indeed it was always a mystery to me how it worked (where’s the search form?). So I guess you’ve solved two mysteries for me. Except that I don’t know why we still have to choose MSCs for our papers.

Posted by: Tom Leinster on March 31, 2008 3:02 AM | Permalink | Reply to this

In defence of the MSC

MSC is extremely useful and I would encourage everyone to use it properly. It is far more refined than the arXiv system (now there’s a system that I don’t use as a reader and just find annoying as an author).

The biggest use of MSC for me is to find papers in an area that I don’t know much about. Of course, the experts in any field know all the major papers in that field but I don’t. And if I don’t have a local expert who’s happy to be bugged by me asking for recommendations all the time then the MSC is a good way to find papers.

I don’t often use it, but when I do then I really need it.

And no, the arXiv is not a reasonable replacement. I regularly use papers that pre-date the arXiv (a quick grep through my BibTeX database reveals the earliest citation as being 1923, 12 as pre-1960, and about 150 in the 60s, 70s, and 80s, out of a total of just under 400; so not all, but a significant proportion).

Okay, so it’s not perfect (to misquote something, “If you find a perfect blog, don’t spoil it by posting to it.”). I don’t know how often it gets updated nor how to go about campaigning for an additional entry, but it’s presumably possible.

In the meantime, and to speed up the process of getting their own classification, maybe the n-categorickers should flame the MSC by listing their articles in as many sections as possible. Eventually the AMS will give you your own classification just to make you stop.

Posted by: Andrew Stacey on March 31, 2008 8:48 AM | Permalink | Reply to this

Re: In defence of the MSC

Andrew wrote:

The biggest use of MSC for me is to find papers in an area that I don’t know much about. Of course, the experts in any field know all the major papers in that field but I don’t.

How exactly do you do this?

Suppose you know almost nothing about symbolic dynamics, and want to find out what the major papers in the subject are. You start by discovering that, conveniently, symbolic dynamics has a classification of its own, 37B10. I don’t have access to MathSciNet from where I’m sitting now, but presumably if you type “37B10” into the form, you get a list of papers hundreds long. How do you identify the most important ones?

Posted by: Tom Leinster on March 31, 2008 11:29 AM | Permalink | Reply to this

Re: In defence of the MSC

I led with my chin:

The biggest use of MSC for me is to find papers in an area that I don’t know much about. Of course, the experts in any field know all the major papers in that field but I don’t.

Tom whammed back a left-hook:

How exactly do you do this?

Suppose you know almost nothing about symbolic dynamics, and want to find out what the major papers in the subject are. You start by discovering that, conveniently, symbolic dynamics has a classification of its own, 37B10. I don’t have access to MathSciNet from where I’m sitting now, but presumably if you type “37B10” into the form, you get a list of papers hundreds long. How do you identify the most important ones?

Yup, 1055 for both primary and secondary; 336 for primary alone. Actually, your question is very easy to answer: search for books. Then you get 13 and 2 respectively.

But your left-hook missed as I didn’t claim that I could use the MSC to identify the most important papers in an area. I suppose you could infer it from my subsequent sentence, but that was a feint. What I said was that it helped me find papers in an area, particularly one I don’t know much about.

As an example, over there there is/was a discussion about closed monoidal structures on locally convex topological spaces. Without looking anything up, we got quite far. Now suppose I want to see what’s in the literature. A MathSciNet search on “locally convex” and “monoidal” brings up 9 papers. Some seem interesting, but none seems to address the main question (can one classify all closed monoidal structures on lctvs?). “vector space” and “monoidal” brings up 102 papers, and most are clearly irrelevant to the question. “locally convex” and “tensor product” has 387 papers, and is a slightly dissatisfying search as, for all I know, there may be products that are not given by tensor products. On the other hand, “monoidal” and “46” brings up 56 papers, including one suggestively entitled:

On hom-functors and tensor products of topological vector spaces. MR0682968

Unfortunately, I don’t have electronic access to this (it’s actually a Springer Lecture Notes) and although we (finally!) have a lovely warm sunny day so I can make the trek to the library without skis, jeg må gå på norskkurs og lære mange norsk order.

Ha det!

Posted by: Andrew Stacey on April 1, 2008 10:11 AM | Permalink | Reply to this

Re: In defence of the MSC

Despite your boxing metaphor, my question wasn’t meant aggressively (and nor did I take your earlier comment to be aggressive). I’d like to learn new techniques for finding information.

Posted by: Tom Leinster on April 1, 2008 5:02 PM | Permalink | Reply to this

Re: In defence of the MSC

Tom lined up a roquet:

I’d like to learn new techniques for finding information.

Absolutely! I wish there was a reliable way to find information. Sadly, there doesn’t appear to be so, even in an academic subject with proper references and so on. I’ve been trying to track something down for a while now and each time I try a different search on MathSciNet I find something new - it’s very frustrating.

Something like the MSC that at least tries to be standardised is useful; otherwise one ends up trying to search on keywords and that can end up anywhere. After all, think of all the problems with just the name “Hopf algebra”! And that’s at least reasonable. Suppose I’ve invented something new and I’m searching on MathSciNet to see if someone’s done it before. How am I supposed to guess that they have and that they’ve called it a “hedgehog”???

Andrew

Posted by: Andrew Stacey on April 3, 2008 10:53 AM | Permalink | Reply to this

Re: In defence of the MSC

and that they’ve called it a “hedgehog”???

or Hopfish algebras in Poisson geometry? :P

Posted by: David Roberts on April 3, 2008 1:05 PM | Permalink | Reply to this

Re: In defence of the MSC

That’s a great point, Andrew, but it doesn’t really go far enough. What we really need is to prevent “hedgehogs” in the first place.

Some sort of systematized approach to mathematics so that every new concept will have a natural name (and more mathematical properties) that will be entirely predictable so that other people can tell what it means straight off, and any two people will get the same name for the same new concept.

Some sort of.. formal system..

Oh, wait…

Posted by: John Armstrong on April 4, 2008 7:02 PM | Permalink | Reply to this

Re: In defence of the MSC

As it happens the MSC is undergoing revision right now. The new version will go into use in 2010, and apparently is to be finished by the middle of this year. To see a draft and make comments, go to this website.

Posted by: Mark Meckes on March 31, 2008 11:36 AM | Permalink | Reply to this

Re: In defence of the MSC

The importance of changing classifications for algebra in the 1895-1930 period has been argued by Leo Corry in From Algebra (1895) to Moderne Algebra (1930): Changing Conceptions of a Discipline. A Guided Tour Using the Jahrbuch über die Fortschritte der Mathematik, in J. Gray and K.H. Parshall (eds.), Commutative Algebra and its History: Nineteenth and Twentieth Century, Providence, American Mathematical Society / London Mathematical Society.

The discipline of algebra underwent significant changes between the last third of the nineteenth century and the first third of the twentieth. These changes comprised the addition of important new results, new concepts and new techniques, but also deep changes in the very way that the aims and scope of the discipline were conceived by its practitioners.

Is there a case to be made that however the MSC changes for 2010 that it will miss something important of the contemporary conception of mathematics which cannot be captured by a tree-like classification?

Posted by: David Corfield on March 31, 2008 1:04 PM | Permalink | Reply to this

Re: In defence of the MSC

Is there a case to be made that however the MSC changes for 2010 that it will miss something important of the contemporary conception of mathematics which cannot be captured by a tree-like classification?

Absolutely. All classifications are highly limited in their scope and usefulness. But that doesn’t mean they aren’t useful. In particular, I doubt that anyone hopes the MSC will capture much of “the contemporary conception of mathematics” (my emphasis, of course). In my understanding at least, “MSC” is a misnomer which overstates its scope: it’s not really meant as a classification of mathematical subjects, just a classification of mathematical publications.

There used to be a page on either the arXiv or the Front which drew a similar distinction in explaining what are and are not good reasons to propose a new subject category, but I can’t find it now.

Posted by: Mark Meckes on March 31, 2008 2:07 PM | Permalink | Reply to this

Re: In defence of the MSC

In the meantime, and to speed up the process of getting their own classification, maybe the n-categorickers should flame the MSC by listing their articles in as many sections as possible. Eventually the AMS will give you your own classification just to make you stop.

No need to flame or be overly sectionned. The AMS periodically - like now - actively solicits suggestions for updating the MSC.

Posted by: jim stasheff on March 31, 2008 2:44 PM | Permalink | Reply to this

Re: In defence of the MSC

It is far more refined than the arXiv system (now there’s a system that I don’t use as a reader and just find annoying as an author).

I hope you mean you don’t use the arXiv classifying system (but then how do you get daily updates as to what has been posted?)
as opposed to not using the arXiv

Posted by: jim stasheff on March 31, 2008 2:50 PM | Permalink | Reply to this

Re: In defence of the MSC

Ah, now, the arXiv classifying system… there’s another rant waiting to happen.

I’m incensed by the fact that some arXiv administrators take it upon themselves to reclassify papers written by serious mathematicians. (OK, I’m not incensed right now. But I have been when people have recounted their experiences, and I’m sure I’ll get incensed if I think about it hard enough.) If my anecdotal evidence is anything to go by, this is happening increasingly often.

For example, I have a friend, X, who is a representation theorist. As it happens, X is an extremely good, prize-winning representation theorist with a prestigious position; in short, someone who knows what representation theory is about. Several months ago, X put a paper on the arXiv and selected Representation Theory as the primary topic. A few hours later, an email came back from an arXiv person saying that the paper had been reclassified as something else — metric geometry, I think.

A short email exchange ensued, involving some very flimsy arguments from the arXiv side: X didn’t cite any math.RT articles, the arXiv person making the decision was an “expert”, and the burden was on X to demonstrate that it really was a representation theory article, rather than the arXiv to show that it was not. I think my friend gave up.

Two things get my goat about this.

One is the sheer arrogance: how dare someone who looks over a paper for a few minutes think that they know better what it’s really about than someone who’s spent months of painstaking work on it and knows it inside out? And if you don’t happen to be at your computer at the right time, you don’t even get the chance to dispute the reclassification: it simply appears, reclassified, the next morning. It’s the kind of thing that makes me want to stop using the arXiv.

The other is the conservatism. Suppose I write a paper on complex analysis that uses some category theory. The paper’s not about category theory; it’s about complex analysis. I want the paper to be read by complex analysts, so I take great care to explain the necessary category theory. I submit it to the arXiv, classifying it as Complex Variables. What I fear is that whichever arXiv-appointed person has responsibility for vetting Complex Variables submissions will take a quick look at it, see that it’s full of categorical terminology and commutative diagrams, and reclassify it as Category Theory — thus defeating the whole object of communicating with complex analysts. In this way, subjects are kept tightly in their own little boxes; communication is hampered, and new approaches are frustrated.

There. I think Jim’s exclamation of “jeunesse!” was directed at me, but it seems that I am becoming what in this country we call a Grumpy Old Man.

Posted by: Tom Leinster on March 31, 2008 3:31 PM | Permalink | Reply to this

Re: In defence of the MSC

I had a recent paper reclassified from algebraic topology to geometric topology. But I don’t use any geometric techniques, and I do talk in passing about how my techniques come up when using algebraic topology tools like the fundamental group of a knot complement.

But where the hell does knot theory go in the arXiv? For that matter, where do tangles go in the MSC? Invariants of knots? That’s the best I can come up with.

Posted by: John Armstrong on March 31, 2008 5:02 PM | Permalink | Reply to this

Re: In defence of the MSC

Tom L wrote:

There. I think Jim’s exclamation of ‘jeunesse’ was directed at me, but it seems that I am becoming what in this country we call a Grumpy Old Man.

jeunesse refereed to not knowing how to use the paper version - or was that tongue well located in cheek?

your complaint about reclassification is well taken and will be discussed with the powers that be

Posted by: jim stasheff on March 31, 2008 10:54 PM | Permalink | Reply to this

Re: In defence of the MSC

Thanks. Sorry to rant, but the frustration and anger are real, and were certainly felt by my friend X.

It’s really true that I’ve never used the paper version and didn’t (still don’t?) understand how to. I started my PhD in 1996, and by the time I needed Mathematical Reviews, it was available online.

Posted by: Tom Leinster on March 31, 2008 11:04 PM | Permalink | Reply to this

Re: In defence of the MSC

I started my PhD only a couple of years before Tom and spent many a happy hour (well maybe not quite) in the library reading the paper copy. I would flip it open to the 57M pages, or whatever, and see what interesting papers were around and what people had to say about them.

I hardly ever use MathSciNet now.

The maths library at Edinburgh was actually classified according to MSC and it was fantastic. It was pretty easy to find stuff and to browse related things. Then we got a new Head Librarian who thought it was too much hassle to check what the MSC was when a new book was bought, and so he decided, unilaterally, to switch to the Library of Congress system. Apart from the chaos of reclassification over a period of several years, having the two systems badly intermingled, Library of Congress classification just didn’t seem as well adapted at the MSC (all of the yellow Springer GTMs classified together, for instance.)

Posted by: Simon Willerton on April 1, 2008 1:10 AM | Permalink | Reply to this

Re: In defence of the MSC

But any classification is problematic if applied unthinkingly, MSC included. For example, it leads to this kind of complaint from Arnold (final page):

To facilitate the search of mathematical information, Russian mathematicians have tried to cover most of the present day mathematics in the more than one hundred volumes of the Encyclopaedia of mathematical sciences, several dozens of which have already been translated into English. The idea of this collection was to represent the living mathematics as an experimental science, as a part of physics rather than the systematic study of corollaries of the arbitrary sets of axioms, as Hilbert and Bourbaki proposed. I hope that this Encyclopaedia is useful as the source describing the real origins of mathematical ideas and methods (see, for in­ stance, my paper on the catastrophe theory in vol. 5). Unfortunately, in the Library of Congress, and hence in all the USA libraries, the volumes of the Encyclopaedia of Mathematical Sciences are scattered according to the author/subject alphabetical order, which makes its use as an encyclopaedia extremely difficult. I have seen, however, all the collection arranged on one shelf in some European universities, for example in France.

Posted by: David Corfield on April 1, 2008 9:13 AM | Permalink | Reply to this

Re: In defence of the MSC

Tom Leinster wrote, regarding Mathematical Reviews:

It’s really true that I’ve never used the paper version and didn’t (still don’t?) understand how to.

It’s probably a bit late for a lesson now, but if I remember correctly, each volume had an index of authors and was also classified by subject, and one did ones best to use those to find stuff. It seems unworkable in retrospect — but at the time it seemed great, just like those stone hand axes.

Actually the main thing I miss about those days is sitting in the library with a stack of Mathematical Reviews, going through year by year, and reading reviews of all the papers in a given subject class. I’d mainly be looking for papers that would help me with some specific problem, but I had a lot of fun bumping into other stuff, and learning the whole landscape of the subject.

I don’t use the library much anymore. As a student, I went there often, browsing through books on diverse subjects until my brain hurt, my self-confidence shrunk to a speck, and I wanted to run out screaming. I remember seeing a big fat book entitled K-Theory, and having no idea what K-theory was — and not being sure I wanted to. It was intimidating and somehow depressing. But, being young, I’d bounce back fast, and come back to the library for more…

Now I can sit at home and download more information than I can understand in a twinkling of an eye. So, I guess it’s mainly the sense of place — the library as sanctum of knowledge — that’s diminished. It’s still there; it’s still great. But, I don’t feel I have time to go there often.

Posted by: John Baez on April 1, 2008 6:40 PM | Permalink | Reply to this

Re: In defence of the MSC

Off-topic, but what you say John reminds me of some of the (all but lost) joys of my childhood, leafing through hardcover encyclopedias (like the World Books I grew up with). You pull down say volume ‘M’, intending to read about mathematics, but then somewhere close by there’s Madagascar, and so there are so many hours of fun just browsing around with no fixed purpose in mind… Wikipedia with its hypertext is a whole other experience, but there’s nothing in it that quite compares to curling up in a big chair and letting one’s mind wander indiscriminately over the pages of a traditional encyclopedia.

Posted by: Todd Trimble on April 1, 2008 8:54 PM | Permalink | Reply to this

Re: In defence of the MSC

Herb Wilf points out that even laptops or hand helds haven’t reached the 3 C criteria for reading a book in htat chair or even in bed
I forget the other 2 but one of his C’s is Cuddly

Posted by: jim stasheff on April 3, 2008 2:38 PM | Permalink | Reply to this

Re: In defence of the MSC

I think my iPhone is quite “cuddly”, actually. Could be a little bigger, but definitely cuddly.

Posted by: John Armstrong on April 4, 2008 7:06 PM | Permalink | Reply to this

Re: In defence of the MSC

Todd wrote:

Wikipedia with its hypertext is a whole other experience, but there’s nothing in it that quite compares to curling up in a big chair and letting one’s mind wander indiscriminately over the pages of a traditional encyclopedia.

I used to do that too.

Of course the Wikipedia is just a subset of the web, whose links allow for all sorts of marvelous explorations, and I spend lots of time on such explorations — I guess it’s my main hobby. A bunch of what I bump into eventually finds its way into This Week’s Finds, or my diary…

I could do this curled up in a big chair if I wanted to. So, what element of romance has been lost, apart from the special patina attached to childhood memories?

Maybe it’s the randomness of topic built into the alphabetical organization of an old-fashioned encyclopedia. You’re not likely to blunder from Mathematics to Madagascar on the web!

People also complain that the web encourages the formation of communities of like-minded people who don’t bump into drastically different sorts of people. Like, umm, us here.

(By the way, don’t forget Borges’ encyclopedia.)

Posted by: John Baez on April 2, 2008 2:30 AM | Permalink | Reply to this

Re: In defence of the MSC

Like, umm, us here

who just made it to the editorial of the Notices AMS thanks to our ‘discussion’ about the DARPA 23

Posted by: jim stasheff on April 3, 2008 2:41 PM | Permalink | Reply to this

Re: In defence of the MSC

I pinioned:

It is far more refined than the arXiv system (now there’s a system that I don’t use as a reader and just find annoying as an author).

Jim rejoined:

I hope you mean you don’t use the arXiv classifying system (but then how do you get daily updates as to what has been posted?)

as opposed to not using the arXiv

To reassure you, I meant that I don’t use the arXiv classifying system when I browse the arXiv. I most certainly do use the arXiv, both as a researcher and an author.

The volume on the arXiv is still (but maybe not for that much longer) small enough for it to be reasonable to look at all the new postings for maths each day. It gets a bit annoying when I’ve been away for a week or so, though. I don’t subscribe to the mailing list as I find the web interface a little easier to work with. So I don’t bother with the classification when looking at new postings.

As for searching, the classification is too broad to allow much refinement and the algorithm seems fast enough that I’ve yet to have to restrict matters that much.

Actually, to be honest, I don’t search the arXiv that much. I “bookmark” papers that I think might be interesting and have my own search engine for those papers. Being local and on a smaller data set, it works much faster. I just need to adapt the system to use the new arXiv API (the “bookmark” feature downloads the record from the arXiv so with the old system I had to be careful not to run afoul of the rampaging robot defence).

In terms of search order, I usually use:

  1. MathSciNet
  2. Wikipedia (and maybe Planet Math)
  3. Google

Of course, it depends a little on what I’m looking for. My goal is to find a few MathSciNetWhacks but I haven’t looked all that hard yet. Actually, “Sonic hedgehog” is a MathSciNetWhack.

Sorry. End of the day, got a bit silly there.

Andrew

PS I once did trigger the rampaging robot defence on the arXiv and have no wish to do so again.

Posted by: Andrew Stacey on March 31, 2008 3:34 PM | Permalink | Reply to this

Re: Categorified Quantum Groups

ah, jeunesse!

Posted by: jim stasheff on March 31, 2008 2:47 PM | Permalink | Reply to this

Re: Categorified Quantum Groups

David wrote:

Did the Geometric Representation Theory Seminar reach the point of having categorified quantum groups?

Not quite — we ran into an obstacle. We only groupoidified ‘half’ of the quantum group associated to a Dynkin diagram of type ADE. Jim began this in lecture 22, and we continued working on it through the rest of the seminar.

(By the way, I got too busy to do blog entries for the seminar after lecture 25. Since I don’t need to teach this spring, I’ll resume blogging soon, and take us on up to the last lecture.)

Not that I’m after differences of approach, of course.

There’s definitely a difference of approach: Jim and I are doing ‘groupoidification’, based on groups defined over 𝔽 q\mathbb{F}_q. The other folks are mainly using the homology of groups defined over \mathbb{C}. But these are related by the Weil conjectures.

(Just so nobody gets the wrong idea: the Weil conjectures are theorems, and these theorems are incredibly easy to check in the very special cases I’m talking about.)

What I objected to was the idea of erecting walls between different thoughts on categorification, when what we need are better roads.

Posted by: John Baez on March 28, 2008 8:22 PM | Permalink | Reply to this

Re: Categorified Quantum Groups

It’s probably still worth articulating differences to make clearer what would need to be done to build better roads, while not erecting walls.

Posted by: David Corfield on March 30, 2008 12:52 PM | Permalink | Reply to this

Re: Categorified Quantum Groups

Hey you mathematicians ! Don’t forget that it’s Alexander Grothendieck’s 80th birthday! Listen to my instrumental psychedelic tribute to him “Gothendieck” at http://www.myspace.com/russvr .

Posted by: Russ Van Rooy on March 29, 2008 4:51 PM | Permalink | Reply to this

Re: Categorified Quantum Groups

Cool! Here is a link to the song “Grothendieck” by Russell van Rooy. Wasn’t his birthday yesterday though?

Posted by: Bruce Bartlett on March 29, 2008 5:05 PM | Permalink | Reply to this

Re: Categorified Quantum Groups

I bet he’s a regular lurker here. He wouldn’t be happy if we got his birthday wrong.

Posted by: Jamie Vicary on March 29, 2008 5:18 PM | Permalink | Reply to this

Re: Categorified Quantum Groups

I never got the impression that he was ever particularly happy.

Posted by: John Armstrong on March 29, 2008 8:30 PM | Permalink | Reply to this

Re: Categorified Quantum Groups

Perhaps nobody ever gets his birthday right.

Posted by: Jamie Vicary on March 29, 2008 9:42 PM | Permalink | Reply to this

Re: Categorified Quantum Groups

Vous êtes tous si enfantin.

Posted by: Alexander Grothendieck on March 30, 2008 7:16 AM | Permalink | Reply to this

Re: Categorified Quantum Groups

Do you know where one finds a good review of the “Longue Marche”?

Posted by: Thomas Riepe on April 5, 2008 7:58 AM | Permalink | Reply to this

Gothendieck

Russ van Rooy:

Listen to my instrumental psychedelic tribute to him “Gothendieck” at http://www.myspace.com/russvr.

I was really hoping this wasn’t a typo, but apparently it is. It reminded me of Sam Nelson, who used to be a practicing goth as well as a knot theorist and a nice, not actually scary guy.

Posted by: John Baez on April 1, 2008 6:50 PM | Permalink | Reply to this

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