### Breaking Out of the Box

#### Posted by David Corfield

While the Café’s gone a little quiet of late – and with two of its owners tripping off to Delphi soon while the other’s still on holiday, things can only get quieter – there are some interesting things happening abroad. In fact, walking in this morning, I was thinking up something to say about a vague sense I had from reading about canopolises, but when I reached the office and tuned into the blogosphere, there’s Noah Snyder clearly articulating the thought I’d had that things could be taken further.

The larger question is why did we ever restrict ourselves to ends of boxes when we could be letting the string ends of our $n$-categories wander about on the surfaces of spheres?

Snyder gives us a new table of shapes becoming more circle-like as one moves to the right. Cubes become cylinders (where canopolises live) become spheres, and he adds:

The traditional n-category theory perspective is to look at the left-most column as fundamental, and think of moving further right as adding more duals. This is how Lauda-Pfeiffer think of Khovanov homology, for example. Those of us infused with the Jones seminar propaganda tend to think of the rightmost column as the most important since it most clearly exhibits all the structure present.

Now we have to think about integrating this story with what we know and love here. Will $n$-groupoids feel they’re being left out? What happens to those fundamental $n$-categories with duals of stratified spaces we talked about so intensely?

And are there other shapes awaiting us? Tori sitting in a torus?

## Re: Breaking Out of the Box

Quite a while back Barrett and Westbury showed that spherical categories give rise to 3d topological quantum field theories. Spherical categories are monoidal categories where the relevant string diagrams live on a sphere instead of a square.

Later, Marco Mackaay went up a dimension and showed that spherical 2-categories give rise to 4d topological quantum field theories.

My only worry about spherical categories is that someone will be reminded of a famous physics joke and start snickering if I say “Consider a spherical category…”.