The n-Category Café

The bicategory that Chris considers is similar to, but different from, the week double category in

Jeffrey Morton
A Double Bicategory of Cobordisms With Corners
arXiv:math/0611930

which John discussed in week 242.

Chris defines a 2-category whose

- objects are disjoint unions of points

- morphisms are 1-dimensional cobordisms between these

- 2-morphisms go between cobordisms with the same boundaries and are (that’s my formulation of the pictures he drew) surfaces with a height function $\Sigma \to [0,1]$ such that the preimage of 0 is the incoming 1-dimensional morphism, the preimage of 1 the outgoing one and each source-source point runs to a corresponding target-source point.

Chris proves a big theorem which says that this bicategory is equivalent to one obtained from a handful of generators modulo a bunch of relations.

The generators he presented include precisely those in the pictures on p. 27 of Baez, Dolan Higher-dimensional Algebra and Topological Quantum Field Theory, but also some more. I suppose that, hence, Chris is talking about the 2-category that places like this would be called $2Tang$ or the like. But I can’t tell for sure.

In any case, Chris invokes Morse theory, singularity theory and Cerf theory to prove that his generators and relations are sufficient. The general strategy here is quite simialr to the proof that Aaron Lauda and Hendryk Pfeiffer give for similar but different situation

Aaron D. Lauda, Hendryk Pfeiffer
Open-closed strings: Two-dimensional extended TQFTs and Frobenius algebras
arXiv:math/0510664 (blog)

In that article the sufficiency of the generators and relations is proven by showing that they allow to reduce every surface to one of a certain “normal form”. Chris Schommer-Pries emphasized that this only works because we have a good classification of 2-manifolds, whereas his proof immediately generalizes to higher dimensions.

In fact, he is working with Chris Douglas already on the $d=3$ case. The singularity theory they use is under control up to $d=7$ and hence they can already see how to generalize everything up to that dimension, the main problem being on the algebraic side, where one has to come to grips with “symmetric monoidal weak 7-categories”.

In fact, it seems that the number 7 appearing here which was attributed by Mike Hopkins to joint work by Arthur Bartels, Chris Douglas, and André Henriques is really the 7 in Chris Schommer-Pries’s work.

The nice thing about having a presentation in terms of generators and relations is that it allows to classify functors out of this 2-category $Bord^2$ into some other bicategory.

This way Chris obtains the following nice corollary:

2-Functors from these extended 2-cobordisms to the bicategory of algebras, bimodules and bimodule homomorphisms $$Bord^2 \to Bimod(\mathbb{R})$$ are given, up to equivalence, by trace-Morita classes of separable symmetric Frobenius algebras.

These algebras are of course what the 2-functor assigns to a single point.

Notice that, while refined to a 2-category, Chris’s $Bord^2$ is a “localized” version of “just” closed string worldsheets. He told me about a cool idea how to generalize his work to the open-closed case. That idea was very close to my heart, but I am not sure if I am allowed to talk about it here in public.