The Scope of Categorification II
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.