Classical vs Quantum Computation (Week 11)
Posted by John Baez
Today in our course on Classical vs Quantum Computation we covered lots of examples of 2categories, to show how widespread these gadgets are:

Week 11 (Jan. 25)  Examples of 2categories. The 2category of categories. The fundamental 2groupoid of a topological space. The 2category of topological spaces, maps, and homotopies between maps. The 2category of topological spaces, maps, and homotopies between maps. The 2category implicit in extended topological quantum field theories, due to Jeffrey Morton. The 2category implicit in string theory, due to Stolz and Teichner. Monoidal categories as oneobject 2categories. The 2category of rings,
bimodules and bimodule homomorphisms. Monoidal categories as oneobject 2categories. The 2category of rings, bimodules and bimodule homomorphisms.
Supplementary reading: Jeffrey Morton, A double bicategory of cobordisms with corners.
 Stefan Stolz and Peter Teichner, What is an elliptic object? Section 4.2: the bicategory of conformal 0, 1 and 2manifolds.
Last week’s notes are here; next week’s notes are here.
The newest of these examples deserve a bit of advertisement.
Not enough people seem to have realized that Jeffrey Morton’s paper solves a tough problem in an elegant way. For work on extended topological quantum field theories, we really want a (weak) 2category with
 compact $(n2)$manifolds as objects,
 $(n1)$dimensional cobordisms between these as morphisms,
 and $n$dimensional cobordisms between those as 2morphisms.
It’s an intuitive idea, but actually getting your hands on this 2category seems annoyingly tricky. Jeffrey does it in an elegant by first building a bigger structure — a ‘double bicategory’ in the sense of Dominic Verity — and then finding the desired 2category inside there. In fact, the double bicategory may ultimately be more useful than the 2category, as I sketched in week242.
(In fact, there’s an important picture in this week’s notes that doesn’t really live in the 2category I was describing — only in the double bicategory. As an exercise, try to spot it!)
In string theory we’d like some sort of 2category like the above one for $n = 2$, where the 2morphisms are ‘string worldsheets’. But, to get this to work we really need the string worldsheets to be equipped with complexanalytic structures. The closest thing I know to a full treatment is section 4.2 in the paper by Stolz and Teichner cited above. Does anyone know a reference with more details? There are a lot annoyingly tricky issues.
Re: Classical vs Quantum Computation (Week 11)
We did talk about this way back here. Is there progress on understanding this better?