Two papers dealing with the categorification of knot invariants:
Sergei Gukov, Amer Iqbal, Can Kozcaz, Cumrun Vafa, Link Homologies and the Refined Topological Vertex.
R. Dijkgraaf, D. Orlando, S. Reffert, Dimer Models, Free Fermions and Super Quantum Mechanics.
Guess what the latter’s suggestion is for an “easy to read introduction to the concept of categorification” (p. 18)?
Posted at May 16, 2007 9:50 AM UTCTrackBack URL for this Entry: https://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/1276
Some physicists are expressly interested in learning more about categories, but after looking for literature many come away with the impression that it is a difficult and esoteric subject.
Just recently expressed so here over on the blog Rantings of an angry physicist.
(Not sure why this one is angry, but I sure do not want to get into discussing why he might be ;-)
That’s a pity, because categories make many intricate-looking things much simpler than they usually appear. I think what often causes the confusion is that powered by the simplifying power of (-)categories people can go to places neither reachable nor possibly even visible by other means – which then causes the impression that category theory itself is a highly impenetrable subject.
We should post an entry here listing the best introductions into categories for physicists. John Baez’s and Bob Coecke’s stuff, etc, with some helpful hints. I posted brief comment on that to the angry guy’s blog already. When I find the time I might post a more detailed entry here.
If category theory is the “abstract nonsense” what is the “concrete nonsense” then? Ah?
Here I’ve found a talk by Israel Gelfand “Mathematics as an adequate language. A few remarks” from his 90th birthday conference.
What Gelfand says:
…Maybe, instead of categories one should study structures with the “Heredity Principle”…
is he against categories or what?
What is the “Heredity Principle”?
I am not sure how to abstract a general principle from Gelfand’s example of quasideterminants.
If I had to guess at a general principle, it would be based on the following. Quasideterminants allow you to take a “determinant”-style operation on block matrices which gives a general matrix. One could generalize to “block matrices OF block matrices”, with a “general quasideterminant” giving a block matrix, with its own quasideterminant giving a matrix. A “heredity principle” seems to suggest that there is a whole sequence of generations of structures, each of which inherits something from the previous - possibly some reduction operation, such as, in this example, by a quasideterminant. But I don’t see any obvious candidate principle either.
Why this would seem like a good substitute for categories, I don’t understand, but it does seem to have the feature of adding extra layers of structure, which may be good to do. The quasideterminant as an analog of decategorification may be a bit farfetched, but I conjecture he had some such notion in mind. If so, this doesn’t seem a very fruitful train of thought (if that metaphor isn’t too mixed). But I imagine he chased that train farther than I did, so I can’t be sure.
I see. Sounds vaguely like iterated internalization. Could it be that we are talking about determinants on endomorphisms of -vector spaces in some sense here?
I mean, a matrix whose entries are matrices is something which I may regard as a 2-morphism of Kapranov-Voevodsky 2-vector spaces: these are matrices whose entries are linear maps!
But that’s just an observation. I haven’t tried to think about if this has any relevance for quasideterminants.
It’s great to see “categorification” beginning to appear more and more in the physics literature. I’m especially pleased that the paper by Dijkgraaf, Orlando and Reffert actually cites John Baez and James Dolan’s Categorification paper “for an easy to read introduction to the concept of categorification”. I feel we’re becoming mainstream now :-)
(Talking about Robbert Dijkgraaf, he’s actually an alumni of Utrecht, where I did a masters in Theoretical Physics. He wrote his Phd thesis on “A Geometrical Approach to Two Dimensional Conformal Field Theory” in 1989. It’s pretty cool, lots of pictures (he was an artist before a physicist!), and it also contains quite a lot about TQFT’s. For instance, the proof (physics-style) that 2d TQFT’s are the same as Frobenius algebras. Also, he introduced the Dijkgraaf-Witten finite group model there as well.)
Anyhow, it’s nice to see categorical ideas entering the consciousness of theoretical physics. I remember when I gave my Master’s talk at Utrecht on the theme of “categorical aspects of TQFT’s” in 2005. It didn’t go down too well 
. I drew lots of pictures and tried to make it as relevant as possible. But when I got to the point of trying to convince the audience that a connection on a principal bundle (i.e. gauge theory) can be thought of as a functor which (basically) assigns group elements to paths (à la John’s QG notes)… I was met with some dubious stares and one or two not-entirely-pleasant questions. I got a similar response when I tried to argue that a TQFT was a functor from nCob into Vect 
.
Heh, I don’t blame them though. Sometimes one gets the same response when one tries to explain those things to mathematicians ! 
For those who don’t already know, it’s worth pointing out that Dijkgraaf is a real leader in the mathematical side of string theory. People listen to him. So, it’s possible that we’ll soon see more research on the interface between physics and categorification.
Re: Two Physics Papers Involving Categorification
I am not in the least bit surprised. As a newcomer committed to obtaining a better understanding of the interface between modern physics and mathematics, I find the dedication of the founders of The n-Category Cafe to achieving clear communication to be extremely unusual and valuable. Watching you three experts think out loud as you struggle towards clarity is a privilege that I truly cherish. Also, I think that John’s “This Weeks Finds” is unique as a reference resource, a teaching tool, a source of inspiration, an example of how to express rather than impress, and a cool way to have fun too. The n-Category Cafe is ascending to this stature as well. Solving difficult technical problems is a major challenge requiring the best brains our planet has to offer – but it is simple compared to the almost impossible task of communicating with each other clearly. You guys know that. Many people never learn it. Thank you.