The scope of 'categorification'

I mentioned here that there were rival conceptions of noncommutative geometry. This set me wondering whether there might be rival conceptions of categorification. The originator of the term appears to be Louis Crane, who was looking to form 4d topological quantum field theories by categorifying constructions used in 3d theories. Back at the beginning of 1993, John Baez refers (item 4) to several papers written or co-written by Crane, including a draft paper 'Categorification and the construction of topological quantum field theory', written with Igor Frenkel. But one can trace the motivation back earlier.

Baez and Dolan's paper Categorification lays out the scope of their sense of the term:

It is clear, therefore, that the set-based mathematics we know and love is just the tip of an immense iceberg of n-categorical, and ultimately ω-categorical, mathematics. The prospect of exploring this huge body of new mathematics is both exhilariting and daunting. (p. 46)

One can try to categorify anything and everything. The continuation of the discussion about categorying Klein's Erlangen Program, I recorded here (in post and comment), went as follows:

DC:
> I list a bunch of nice 2-groups in HDA5, and if I were
> going to categorify Klein geometry I would I would look
> at a bunch in parallel. The fun of course is seeing the effect and
> significance of the 2-morphisms (sorry, now I'm thinking
> of a 2-group as a 1-object 2-groupoid).

I misread this when I first read it, but even the misreading raised a question. I thought you were talking about 2-morphisms BETWEEN 2-groups (should be 3-morphisms I suppose). Back on the level of Kleinian 1-geometry, what role could group homomorphisms play in the Erlangen program? I guess what's already treated is group inclusions, e.g., projective transformations within affine transformations within Euclidean transformations. But is there scope for more interesting homomorphisms?

JB: Good point. Sure! Each group determines, or we could say "is", a Klein geometry, but groups form a category - in fact a 2-category, since groups are categories - so we get a 2-category of Klein geometries.

Ignoring the 2-morphisms for a moment, though they're interesting and important, let's think about the morphisms: group homomorphisms, viewed as morphisms between Klein geometries.

The examples you mention are already very interesting, because they show how geometry is a unified subject, not just a bunch of isolated "geometries". They show that including a little group in a big one can be seen as making a geometry "more flexible", by adding new transformations.

But, there's another way inclusions of groups show up: in Klein geometry, a "figure" is more or less given by the subgroup that stabilizes it. But, a subgroup is an inclusion of groups! So, the examples you listed of group inclusions can also be seen as "figures"! The most famous example is getting affine geometry by taking projective geometry and restricting to the transformations that stabilize a "point at infinity", or "line at infinity", or...

I hadn't noticed this dual viewpoint, for some reason.

Anyway, some other examples come from outer automorphisms of groups: the outer automorphism of SL(n) is the "duality" that switches points and lines, and for Spin(8) one has 3! acting as outer automorphisms, called "triality". For simple Lie groups one can read these off from the Dynkin diagramn
symmetries.

There are also inner automorphisms, which are just "changes of reference frame". The difference between inner and outer automorphisms is nicely handled by the 2-categorical structure of Grp. (Ever figured out what a natural transformation between a functor betwen groups is?)

This leaves the epimorphisms of groups: quotient maps. When you mod out the Euclidean group by the translation subgroup, you get the rotation group. What does this short exact sequence mean for the corresponding 3 Klein geometries?

This is a lot of fun, and it will be even more fun to categorify.

END

In this sense of the word 'categorify' you can try it on just about anything. Now, someone else who is very much involved with categorification is Dror Bar-Natan, a former doctoral student of Edward Witten. From here and here, we glean the following:

categorification (a bold suggestion of I. Frenkel, that much of math is the Euler characteristic of some "higher math", much like much of algebra is q-algebra at q=1)

Conjecture: (I. Frenkel, though he may disown this version)
1. Every object in mathematics is the Euler characteristic of a complex.
2. Every operation in mathematics lifts to an operation between complexes.
3. Every identity in mathematics is true up to homotopy at complex−level.

Now, it seems a shame to have this position confined to a couple of hand-outs, expressed in a few lines, and attributed to someone else who might not agree with it. I would like to find out whether there are any differences of conception either in terms of scope or emphasis. No doubt this would become clear if I could attend this conference on categorification.

A lot of work going by the name categorification has centred on categorifying the Jones polynomial, in the understanding of which Witten played such an important part. Bar-Natan's Khovanov's Homology for Tangles and Cobordisms describes this work. Here is an extract from the paper which contains an enormous amount for a philosopher to ponder:

1.2. The plan. A traditional math paper sets out many formal definitions, states theorems and moves on to proving them, hoping that a “picture” will emerge in the reader’s mind as (s)he struggles to interpret the formal definitions. In our case the “picture” can be summarized by a rather fine picture that can be uploaded to one’s mind even without the formalities, and, in fact, the formalities won’t necessarily make the upload any smoother. Hence we start our article with the picture, Figure 1 on page 5, and follow it in Section 2 by a narrative description thereof, without yet assigning any meaning to it and without describing the “frame” in which it lives — the category in which it is an object. We fix that in Sections 3 and 4: in the former we describe a certain category of complexes where our picture resides, and in the latter we show that within that category our picture is a homotopy invariant. The nearly tautological Section 5 discusses the good behaviour of our invariant under arbitrary tangle compositions. In Section 6 we refine the picture a bit by introducing gradings, and in Section 7 we explain that by applying an appropriate functor F (a 1+1-dimensional TQFT) we can get a computable homology theory which yields honest knot/link invariants.

While not the technical heart of this paper, Sections 8–10 are its raison d’être. In Section 8 we explain how our machinery allows for a simple and conceptual explanation of the functoriality of the Khovanov homology under tangle cobordisms. In Section 9 we further discuss the “frame” of Section 3 finding that in the case of closed tangles (i.e., knots and links) and over rings that contain 1/2 it frames very little beyond the original Khovanov homology while if 2 is not invertible our frame appears richer than the original. In Section 10 we introduce a generalized notion of Euler characteristic which allows us to “localize” the assertion “The Euler characteristic of Khovanov Homology is the Jones polynomial”.

The final Section 11 contains some further “odds and ends”.

If you would like to see the picture in glorious technicolor, then look here.