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October 20, 2008

Categorification in New Scientist

Posted by John Baez

Here’s an article on knot theory that mentions categorification:

  • Richard Elwes, Fundamental secrets are tied up in knots, New Scientist, October 15, 2008.

Richard Elwes is a mathematician and reporter based in Leeds, UK.

Most of the story is about the Jones polynomial and other quantum invariants of knots. I’ll just quote a bit at the end:

Then, in 1999, an exciting breakthrough came — once again from an entirely unexpected quarter. A radically different technique not only spawned a new generation of knot invariants, but also suggested that the mathematics behind knots might have a more profound significance than anyone suspected. The technique’s name was categorification.

Categorification turns the normal guiding logic of mathematics — the abstraction and simplification of the real world — on its head. Abstraction and simplification are all very well, but the results are often too simple to describe as much as we might want. This can be illustrated by the difficulties small children wrestle with when learning to count. Why do three apples and three oranges both reduce to the same “3”, when apples and oranges are entirely different things?

The answer is that the mathematical structure that “3” inhabits - the system of ordinary numbers - is an abstraction shorn of any information about the things it represents. In this case, categorification involves replacing this system with a richer structure, a “category”, in which the rigid equations that define the number system - “1 + 2 = 3”, for example - are replaced with weaker statements comparing the sizes of different sets of objects. This category offers a real-world flexibility that numbers on their own do not: its sets can be different even if they have the same size. In this structure, the original number system appears as the category’s “shadow”, obtained by collapsing all sets of size 3 down to just one representative: the number 3.

Might what works for the number system also work for other mathematical objects? This was the philosophy adopted by Mikhail Khovanov, a mathematician at the University of California, Davis, when he revisited Jones’s invariant in 1999. Instead of breaking it down into smaller finite-type expressions, he looked for some grander structure of which it was just the most visible shadow.

His search was a spectacular success. The overarching category that he found is conceptually difficult, but mathematically it is far less awkward than Kontsevich’s integral, and it’s a more reliable description of a knot than Jones’s formula. Even better, thanks to an ingenious computer program written in 2006 by Dror Bar-Natan of the University of Toronto, Canada, it can now be computed efficiently for any knot, potentially opening up its use to researchers in other areas.

Khovanov’s category is not perfect: there are still instances of knots that share the same category. So work continued and, in 2005, together with Lev Rozansky of the University of North Carolina, Chapel Hill, Khovanov unveiled a new invariant operating at an even higher level. Not only does this categorify many quantum invariants beyond Jones’s, it also subsumes Khovanov’s original category, as well as several other knot invariants discovered in the meantime.

The power of the Khovanov–Rozansky category is getting us close to the perfect knot description, though early indications are that we are not there yet. A still broader family of quantum invariants may yet need to be incorporated before we can say we have the loose ends of the knot problem tied up. Nevertheless, we seem to be edging towards the ultimate mathematical solution.

With early experiments in categorification yielding such riches, physicists and mathematicians are waking up to the idea that the approach might apply to more than just knots. Recalling the deep connection of knots with quantum theory that inspired Jones in the first place, some researchers think they have spotted a tantalising hint that whole chunks of mathematical physics are just shadows of larger categorical structures.

Striking analogies have already been found between the categorical descriptions of quantum mechanics and Einstein’s relativity - the twin theoretical pillars of modern physics that have so far seemed fundamentally incompatible. And John Baez, Alexander Hoffnung and Christopher Rogers of the University of California, Riverside, have recently argued that string theory — a favoured starting point for a theory of everything — can be viewed as a categorification of particle physics.

If a categorification embracing both relativity and quantum theory can be found, then the inconsistencies that physicists see between these two theories might yet prove to be an illusion. Such a tying together of all of physics would indeed be a categorical triumph for the humble knot.

This is the first time I’ve seen a popular science magazine try to explain categorification. Knot theory is a good context for explaining this. A picture of knots that are isomorphic but not equal might help, since anyone can see they’re different but still ‘the same in a way’ — and untying a knot is just proving that it’s isomorphic to the trivial knot.

Posted at October 20, 2008 11:28 AM UTC

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11 Comments & 0 Trackbacks

Re: Categorification in New Scientist

Very interesting. Are there any books on categorification that you would recommend?

Posted by: Rajiv Das on October 20, 2008 3:35 PM | Permalink | Reply to this

Re: Categorification in New Scientist

Not yet. For something not very technical, I’d recommend this.

Posted by: John Baez on October 20, 2008 5:03 PM | Permalink | Reply to this

Re: Categorification in New Scientist

People may be interested to learn a bit about Richard Elwes. He seems to be pretty serious about explaining math to a broad audience… but he seems to have a sense of humor, too:

That’s his photo from an article he wrote on the classification of finite simple groups, over at Plus magazine.

Posted by: John Baez on October 20, 2008 5:49 PM | Permalink | Reply to this

Re: Categorification in New Scientist

Isn’t that Weird Al Yankovic?

pic

Posted by: Bruce Bartlett on October 20, 2008 6:28 PM | Permalink | Reply to this

New Pseudoscientist redeemed; Re: Categorification in New Scientist

Years ago I subscribed to New Scientist for my son. I have kept renewing the subscription. He got his double Math and CS degree at age 18, but still reads them when he comes home about every other week from law school.

Although, as the John Baez-Greg Egan dual entity has explained, NS is often drifting into Physics crackpottery, it is strong in other areas, and fun to read. I even xerox and distribute some articles from it as part of homework assignments to the Bio, Anatomy and Physiology classes that I teach.

I was delighted by the Knot Theory/Category Theory article. It is a gem, and the Baez quote is icing on the cake, to mix a metaphor.

Posted by: Jonathan Vos Post on October 21, 2008 12:59 AM | Permalink | Reply to this

Re: New Pseudoscientist redeemed; Re: Categorification in New Scientist

Physics crackpottery
cf psychoceramics

Posted by: jim stasheff on October 21, 2008 1:26 AM | Permalink | Reply to this

Lou Kauffman and the Arrow Polynomial; Re: Categorification in New Scientist

How does this fit the story’s big picture? Louis H. Kauffman is an amazing thinker and writer!

Title: Virtual Crossing Number and the Arrow Polynomial
Authors: H. A. Dye, Louis H. Kauffman
Comments: 21 pages, 28 figures
Subjects: Geometric Topology (math.GT)

(Submitted on 21 Oct 2008)

We introduce a new polynomial invariant of virtual knots and links and use this invariant to compute a lower bound on the virtual crossing number.

Posted by: Jonathan Vos Post on October 22, 2008 6:24 AM | Permalink | Reply to this

Re: Lou Kauffman and the Arrow Polynomial; Re: Categorification in New Scientist

Cf:

  • arXiv:math/0701333
    Title: Virtual crossings, convolutions and a categorification of the SO(2N) Kauffman polynomial
    Authors: Mikhail Khovanov, Lev Rozansky

but as I read this paper, it is NOT a categorification but rather a lifting/resolution of the polynomial by a chain complex

Or is this commonly called ‘categorification’? I would have expected a category rather than a chain complex??

Posted by: jim stasheff on October 22, 2008 2:27 PM | Permalink | Reply to this

Re: Lou Kauffman and the Arrow Polynomial; Re: Categorification in New Scientist

This is a bit late, considering the new post on categorification, but this was my reply by email at the time, which I was asked to make public:

Hi Jim,

here are some of my own thoughts - they may not be fully informed.

there are two apparently different meanings to categorification these days: Baez-Dolan and one as used by knot theorists. I suspect they are not all that different, but have a different flavour.

Knot theorists say categorification when they replace the usual knot invariants (polynomials) with collections of vector spaces, much in the vein of when E. Noether replaced betti numbers with abelian groups. Then the polynomial is the Poincare series of some appropriate graded space.

Baez-Dolan categorification is the one which most often appears at the Cafe - replacing things with categories.

Best,

David

I was I think a bit misleading - I didn’t mean to imply there are two different sorts of categorification, but that they are only apparently different. See John’s new post linked to above for details.

Posted by: David Roberts on October 24, 2008 2:32 AM | Permalink | Reply to this

Re: Lou Kauffman and the Arrow Polynomial; Re: Categorification in New Scientist

Thanks for posting this comment publicly. You knew or suspected that that the difference between ‘Khovanov’ and ‘Baez–Dolan’ categorification is only apparent — but a lot of people don’t, especially young folks working on knot homology who haven’t had time to step back and consider how widespread categorification actually is. So, I used your comment as an excuse for a quick post.

There should really be a bigger Wikipedia article on this subject: as ‘categorification’ sweeps the mathematical landscape, more and more people will wonder what it means.

Posted by: John Baez on October 24, 2008 3:47 AM | Permalink | Reply to this

Re: Lou Kauffman and the Arrow Polynomial; Re: Categorification in New Scientist

Jonathan wrote, speaking of Kauffman’s latest paper on virtual knots:

How does this fit the story’s big picture? Louis H. Kauffman is an amazing thinker and writer!

I met Kauffman in Lisbon this summer, and over dinner attempted to extract a category-theoretic account of the ‘deep inner meaning’ of virtual knot theory. He helped me understand it much better… but Kauffman is not a category theorist at heart, so what follows is my own version of the ‘deep inner meaning’, for which he is not to blame.

(I wrote a bit about this in my diary. I was planning to discuss it in This Week’s Finds, but haven’t gotten around to it yet. So, here goes.)

Abstractly, virtual knot theory is the study of the free symmetric monoidal category with duals on one object equipped X with a morphism R:XXXXR: X \otimes X \to X \otimes X that satisfies the Yang–Baxter equation. (In physics jargon, we call such a morphism an ‘R-matrix’.) This gives our category a braiding in addition to the original symmetric braiding. We draw the new braiding as a crossing, and the original symmetric one as a ‘virtual crossing’, marked by a circle below:



These are the Reidemeister moves in virtual knot theory. The moves at left are the usual Reidemeister moves, while those at right concern the ‘virtual crossing’, and those at the bottom involve both kinds of crossing.

This sort of structure arises a lot in the theory of quantum groups, where besides the usual symmetric braiding in the category of vector spaces we also get another nonsymmetric braiding.

There’s also a fascinating interpretation of virtual knot theory in terms of knots drawn on compact 2-manifolds, where the virtual crossing is a crossing that involves one strand going over a ‘handle’ — or more picturesquely, ducking into a ‘wormhole’. Greg Kuperberg has proved a nice theorem about this interpretation.

I would like someday to understand his theorem in terms of the ‘free symmetric monoidal category with duals on a solution of the Yang–Baxter equation’. Why does this category know about compact 2-manifolds?

Posted by: John Baez on October 24, 2008 4:18 AM | Permalink | Reply to this

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