The Scope of Categorification II
I wrote last month about an apparent difference of opinion in the scope of categorification. Without wishing to give the impression of there being some great ideological divide, we might designate its two wings Frenkelian and Baezian. (How unfair it is that some surnames lend themselves so well to being turned into adjectives. What chance has a Higginbotham to establish a school of thought?). On the ArXiv today we have a paper - Open-closed TQFTs extend Khovanov homology from links to tangles - by former Baez student Aaron Lauda and Hendryk Pfeiffer which penetrates right into the Frenkelian territory of Khovanov homology. The abstract begins
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:
So, I think the issue is why you would want to leave the ground in the first place. A large part of the attraction is a love of discovering common constructions going on behind the scenes. Here's Robin Houston having Fun with Rel:
We use a special kind of 2-dimensional extended Topological Quantum Field Theories (TQFTs), so-called open-closed TQFTs, in order to extend Khovanov homology from links to arbitrary tangles, not necessarily even.Whenever you see that 'extended' in front of TQFT, you know you're heading up the ladder to 2-categories. Extended TQFTs are looking to attach algebraic objects to cobordisms between manifolds with corners, and there seems to be no better way to treat these than with higher-category theoretic tools.
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:
From my perspective, it can't harm mathematics to have a few great visions. Once you've seen that ladder heading up to omega-categorical heaven, it's hard to stop on the first rung.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.
None of us at arsmath are big category theory fans, so we’ll just have to single-handedly restore the balance.
So, I think the issue is why you would want to leave the ground in the first place. A large part of the attraction is a love of discovering common constructions going on behind the scenes. Here's Robin Houston having Fun with Rel:
One of the great joys of category theory is the way you can so often watch familiar structures emerge unexpectedly from general constructions.NB. As they say, Lauda and Pfeiffer are using a 'special kind of 2-dimensional extended Topological Quantum Field Theories' which means they are only working explicitly at the 1-categorical level.
We have chosen to work with open-closed TQFTs mainly because the extension of a 2-dimensional TQFT to an open-closed TQFT is much better understood than the corresponding question for the Temperley–Lieb 2-category. (p. 9)This 2-category is treated by Khovanov in A functor-valued invariant of tangles. They note, however, related work in their earlier paper:
Various extensions of open-closed topological field theories have also been studied. Baas, Cohen, and Ramırez have extended the symmetric monoidal category of open-closed cobordisms to a symmetric monoidal 2-category whose 2-morphisms are certain diffeomorphisms of the open-closed cobordisms. This work extends the work of Tillmann who defined a symmetric monoidal 2-category extending the closed cobordism category. (p. 4)And observe that:
All of the technology outlined above is defined for manifolds with faces of arbitrary dimension. Thus, our work suggests a natural framework for studying extended topological quantum field theories in dimensions three and four. Using 3-manifolds or 4-manifolds with faces, one can imagine defining a category (most likely higher-category) of extended three or four dimensional cobordisms. (p. 45)So, probably, even special extended 3d-TQFTs will require another rung of the ladder.
18 Comments:
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"?
I'm sure you're right to want to keep names out of it. It still might be useful to have a pair of terms to distinguish between a kind of categorification which wants to categorify everything infinitely often, and one which just looks to categorify some things once or twice. I'm tempted by Großcategorification and Kleincategorification after 19th century conceptions of the scope of Germany.
I don't have anything against n-category research; it just doesn't fill me with enthusiasm. It is a curious fact about the internet that category theory is so well represented, while real analysis (for example) is not.
Walt
For a category theoretic approach to analysis, take a look at the book by Moerdijk and Reyes on Smooth Toposes! Or start thinking about complex analysis in terms of conformal geometry...sorry, couldn't help myself!
Ugh. I recognize that some category theorists think that all of mathematics can be reduced to category theory. But since that hasn't actually happened yet, the existence of category-theoretic approaches to something does not obviate my point that category theory has a much bigger internet presence than other fields. Not that this is the fault of category theorists, of course, who have done an admirable job of advocating their point of view.
(I've read the Moerdijk and Reyes book. I thought it was okay.)
Although differences in mathematical aesthetics are very deeply felt, it seems unlikely that there'll ever be mathematical blogs of the Not Even Wrong variety, singling out a specific theory. Perhaps if Jaffe and Quinn had written their 1993 Bulletin AMS article ten years later, it would have generated a lot of blog traffic. But this was a debate about the encroachment of foreign disciplinary methods into mathematics.
Still, I'd love to see a sustained debate about the merits of n-categories, or perhaps non-geometry.
Although differences in mathematical aesthetics are very deeply felt, it seems unlikely that there'll ever be mathematical blogs of the Not Even Wrong variety, singling out a specific theory.
If I understand correctly (it was really before my time), Rene Thom's catastrophe theory seems to have evoked those kind of feelings, see e.g. this:
Catastrophe theory has been subject to criticism due to the fact that, as Thom himself has observed, "the theory did not permit quantitative prediction."
Along with fractals and chaos theory, I think it was the overblown claim that catastrophe theory could take the application of mathematicis into vast new realms that provoked the attacks. Catastrophe theory as a mathematical theory is just (a portion of) singularity theory. It's when you claim it will help you model prison riots and animal attacks that you get attacked. (Perhaps there's a catastrophe theoretic model of such attacks - just claim a little and you're fine if funding is good, but as funding drops, and the claims get exaggerated, expect trouble.)
Can we ever expect to see more internal rows, such as Hilbert vs Brouwer?
David wrote:
Still, I'd love to see a sustained debate about the merits of n-categories, or perhaps non-geometry
Can we ever expect to see more internal rows, such as Hilbert vs Brouwer?
Why do I get the feeling you're eager to watch a fight?
I know, as a philosopher you think it's good for proponents of different views to make the explicity and fight it out... but more and more, I'm not sure it's always worth bothering!
Maybe I've just been reading too many blogs, especially those nasty food fights about the merits and demerits of string theory... but at this point, I feel the last thing I want to do is argue with anyone about the merits of n-category theory - there's just too much incredibly cool n-category theory to be done to waste my time on such baloney! If someone else doesn't like it, fine - they can do something else.
It's a bit like trying to convince skeptics that chocolate tastes better than vanilla. Why bother? If they don't like chocolate, let them leave it to me.
Anyway, I think lots of mathematicians have an attitude somewhat like this. Most of them would prefer to do something most people find dull and unpleasant, like proving logarithmic Sobolev inequalities, rather than spend their time convincing other people that this activity is actually important and fun. The amazing thing is how synergetic this community of individualistic eccentrics can be.
I know, as a philosopher you think it's good for proponents of different views to make the [their case?]explicitly and fight it out... but more and more, I'm not sure it's always worth bothering!
I don't want fighting to break out. But then nor do I want silence. One step up from silence is individuals presenting their positions with no thought of engaging in any dialectic. There's no possibility of pointless fighting here, so we can surely encourage this activity without the threat of nastiness intruding. Clearly, I don't need to preach to you to do this, but others seem to be more reluctant. Presumably we agree that Cartier's 'A Mad Day's Work' was a good thing and that this kind of post is to be encouraged.
Step 2 up the ladder allows for someone whose field has been characterised by someone else to query this characterisation or present their own opinion. Your pessimistic (realistic) view of human nature seems to imply that given the entrenchment of people's opinions, and their lack of generosity in understanding others, this will rapidly degenerate into acrimony.
Add to this the idea that:
It's a bit like trying to convince skeptics that chocolate tastes better than vanilla. Why bother? If they don't like chocolate, let them leave it to me.
and clearly it's not worth taking the second step. But do you really believe this? Do you really think that work on Smarandache theories is only less valuable than work on n-categories in the same way that some people favour a flavour of ice cream that you don't like? Don't you believe that if only humankind could manage not to destroy itself, that the great story of mathematics as told in the year 3000 would find some space for the 21st century episode which saw omega-categories finally nailed, whereas Smarandache would not appear in the index? This is the wager you have made with your career, and it involves you at a very deep level. It is understandable then that debates about the direction of a science are more rancorous than those about ice cream.
But persist we must. Aristotle taught us that the right way for a science to proceed is via dialectic. I'm itching to see a Connesian or an Artinian reply to Le Bruyn's post, just as I was glad to see Connes' response to Cartier's amalgamation of his and Grothendieck's ideas of space. I would have preferred it not to be just a passing comment, however.
The ability to engage with another person's position in a just way requires many virtues. Virtues, as Aristotle taught us, are skills to be learned. You are right to doubt that we sufficiently possess them. But, if we can't even manage to be virtuous enough discussing mathematics with each other, can there be any hope for humanity when it comes to discussing the just allocation of resources? Perhaps your diary answers this question for me.
John wrote:
It's a bit like trying to convince skeptics that chocolate tastes better than vanilla. Why bother? If they don't like chocolate, let them leave it to me.
David wrote:
Do you really think that work on Smarandache theories is only less valuable than work on n-categories in the same way that some people favour a flavour of ice cream that you don't like?
Don't worry, I regard math as even more important than ice cream....
I was getting a bit cynical and flip from a night of reading too many blogs. So, I was trying to joke that convincing people to enjoy n-category theory is just as counterproductive as convincing my friends to like chocolate ice cream. It only leaves less chocolate for me! Let the suckers eat vanilla if they think it's so great.
Luckily, there is plenty of n-category theory to go around, so the above is just a joke. It might not be if I were Andrew Wiles up in my attic. He surely was not going around trying to convince people of the importance of Fermat's Last Theorem!
But less cynically, here's my real opinion. I think n-categories are truly important and mind-bogglingly fun. The best way to let other people share this fun is not to engage in any sort of agonistic debate - it's to show people how much fun n-categories are!
Don't you believe that if only humankind could manage not to destroy itself, that the great story of mathematics as told in the year 3000 would find some space for the 21st century episode which saw omega-categories finally nailed, whereas Smarandache would not appear in the index?
Yup!
Maybe you're right that exposition by individuals and groups of the delights of their work is enough. And since we don't have anywhere near enough at this level, it's hard to tell whether more would contribute anything. And perhaps charging people up to engage in debate would backfire by putting off gentler souls speaking at all. In any case, these personal accounts are already the result of a dialectical process with other people's ideas. The trouble is that even these accounts are discouraged. I mentioned before that Cartier told me that he would never have written A Mad Day's Work had it not been commissioned, imagining it would never have been accepted by a journal.
I wonder though if your programme wasn't going so well. Once Lurie and others nail down elliptic cohomology, and Witten utters the words '2-group' and 'biadjuction' you're made. What though if things were going against you, like the participant of the Minneapolis conference who felt that the use of homotopy theoretic tools such as model categories was all wrong? What if Smarandache theory took a grip on US math and marginalised the math you love? Showing them how much fun it is to do what you do would be as effective as a conservationist showing by example how good it is to cycle and recycle, or a proponent of marriage showing others what fun they're having in their own relationship.
Marginalised seems to be how some of the anti-string theory bloggers feel. Is there nothing of value in their discussions? If we could extract the 2% of commentary which genuinely sought to understand the other's point of view, that might be worth preserving. But then such an extraction would likely be something very close to the thoughtful individual's presentation of their own case.
Perhaps we should be aiming for something along the lines of Aquinas' Summa Theologica, the individual presentation of the outcome of a long-running debate.
David writes:
I wonder though if your programme wasn't going so well. Once Lurie and others nail down elliptic cohomology, and Witten utters the words '2-group' and 'biadjunction' you're made.
I wouldn't consider myself "made" if Witten gets interested in my work - I hope it doesn't happen.
Whenever he embraces a subject, a huge swarm of less creative people come crowding in after him and try to quickly publish papers on it. A lot gets done quickly, but in slapdash sort of way; a bunch of cool new ideas get invented, but the swarm moves on to the next promising territory before clarity is attained, leaving a slagheap of papers where once there had been a wide-open space full of promise....
You can see this happening with the geometric Langlands program now! As you probably know, Witten recently turned his interest to this subject. He wrote a huge paper on it, and now he's writing a book. Unsurprisingly, you can now see see masses of people cramming - trying to rapidly bone up on the geometric Langlands program.
It's not really bad in the long run; a lot of progress gets made, and time will eventually sort out the mess. As we've discussed before, a lot of exquisite math can be found in the brownfields left behind by overly hasty research programs.
But, "following Ed" is not my sort of game. I don't like the feeling of haste and hubbub. I hate it when people get interested in something just because some bigshot is working on it. I think scientists should work hard to develop their own coherent worldview, and base their research on that. I like idiosyncratic individuals, not mobs and bandwagons.
I like to have lots of time to sit in my back yard and think (especially since I have wireless Internet back there). I like to sit and think until things that seemed complicated start seeming simple. I think that working on the simplest things offers me the greatest leverage in the long run. I guess in this way I'm more of a mathematician, or even a logician, than a theoretical physicist.
So, it's no accident that after working on an approach to quantum gravity much less popular than string theory, I switched to something even less popular: n-categories. And if Witten gets the crowd working on n-categories, I may want to switch to something else.
But, I really don't think it'll happen. The Witten-led crowd has a particular core set of skills - mainly quantum field theory and differential geometry - and Witten has a wonderful ability to find new realms in which to apply those skills. The theory of n-categories is sort of different, though of course there's a wonderful overlap.
(One might even say that n-categories are like Feynman path integrals - a not yet completely understood tool which nonetheless serves as a magic key to many rooms in mathematics.)
Marginalised seems to be how some of the anti-string theory bloggers feel. Is there nothing of value in their discussions?
In a sense I'm the prototype of all anti-string theory bloggers. I spent years on sci.physics.research explaining loop quantum gravity and pointing out how string theorists exaggerate the virtues of their work and play down its defects (see this and this, for example). I got tired of this when certain people brought the discussion down to the level of crude ad hominem arguments. Indeed that helped me decide it was time to quite moderating sci.physics.research. Since Peter Woit now has a popular anti-string blog, and a much thicker skin than mine, I don't feel a duty to continue fighting this particular battle. Instead, I have a little lecture I give, in which I castigate all theoretical physicists in an even-handed manner. :-)
I realize I didn't quite answer your questions about the importance of debate in the sciences. I don't really have much in the way of general recommendations for what everyone else should do. But, I hope you see that I take this issue seriously, and I feel my current fairly unconfrontational approach is the right thing for me now.
David wrote:
It still might be useful to have a pair of terms to distinguish between a kind of categorification which wants to categorify everything infinitely often, and one which just looks to categorify some things once or twice. I'm tempted by Großcategorification and Kleincategorification after 19th conceptions of the scope of Germany.
Nice - but we're doing "Kleincategorification" over on the Klein geometry thread!
Whenever he embraces a subject, a huge swarm of less creative people come crowding in after him and try to quickly publish papers on it. A lot gets done quickly, but in slapdash sort of way; a bunch of cool new ideas get invented, but the swarm moves on to the next promising territory before clarity is attained, leaving a slagheap of papers where once there had been a wide-open space full of promise....
This could be taken from the pages of the memorable article by Jaffe and Quinn which sparked off that passionate debate. I spoke about this debate in the context of the proper timing of the introduction of rigour in chapter 7 of my book.
John wrote:
Whenever [Witten] embraces a subject, a huge swarm of less creative people come crowding in after him and try to quickly publish papers on it. A lot gets done quickly, but in slapdash sort of way; a bunch of cool new ideas get invented, but the swarm moves on to the next promising territory before clarity is attained, leaving a slagheap of papers where once there had been a wide-open space full of promise....
David wrote:
This could be taken from the pages of the memorable article by Jaffe and Quinn which sparked off that passionate debate. I spoke about this debate in the context of the proper timing of the introduction of rigour in chapter 7 of my book.
I hope it's clear that I'm not complaining about the lack of rigor. I'm complaining about a swarm of people writing hundreds short papers on the same subject in a short time, each referring to many of the previous ones, nobody taking the time to distill the matter to its essence.
Even if all the papers contained nothing but rigorous theorems, I would still find this annoying. It's fine if you wish to devote yourself to one specialized subject, rapidly master the literature, and compete with the crowd to extract some big nuggets before this vein of ore looks exhausted and it's time to move on. I'm sure this is fun for people with a competitive streak. But there are other people who like to slowly mull over one topic and nurse it to perfection - or like me, mull over lots of topics and gradually form a web of connections until something interesting emerges.
And, you know, it's just possible that some of the people in that Jaffe-Quinn dispute were secretly annoyed about the fast-paced "swarming" style of theoretical physics more than any lack of rigor. I forget if any of them came out and said this.
David, it does appear you're itching to spectate a fight. Among the primary of virtues, understood in the Aristotelian fashion or otherwise, is the ability to combat and wage war. Why is that so and what do you have to gain by it?
I said in an earlier comment that I don't want fighting to break out. On the other hand, I firmly believe that some occasional intense scrutiny of the presuppositions underlying one's work can often be highly beneficial. There are, by and large, two dangers for a discipline. That too many personal schemes or frameworks coexist, excluding the possibility of detailed work. And that a single powerful governing framework dominate the field, preventing any opportunity for fresh thinking. Having overcome the former problem we easily lapse into the latter.
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