The n-Category Café

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

there is a Lie $n$-algebra $g_\mu$ for every Lie $k$-algebra with an Lie $(n+1)$-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 $n$-algebras up to $n$-groups and looking at their representation theory relates to the categorification that Aaron is after.

(Every such Lie $n$-algebra can be integrated to an $\omega$-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.

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.

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 $n$-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 $(\infty,n)$-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.

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.

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 $\mathbb{F}_q$. 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.

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 .