Quantization and Cohomology (Week 18)
Posted by John Baez
Today was the last class on Quantization and Cohomology for the winter quarter! We wrapped up with a summary of what we’d done this quarter, and a sketch of the big picture:

Week 18 (Mar. 13) 
The big picture. Building a Hilbert space from
a finite category C equipped with an "amplitude" functor
A: C → U(1). Example: discretized version of the free particle
on the line. Generalizing from particles to strings by categorifying everything in sight.
Building a 2Hilbert space from a finite 2category C
equipped with a 2functor A: C → U(1)Tor, where
U(1)Tor is the 2group of U(1)torsors.
Supplementary reading:
Last week’s notes are here; next week’s notes are here.
Next quarter we’ll dig more deeply into the secret theme of this course: the way cohomology shows up naturally when we repeatedly categorify our work so far. Classes resume at the beginning of April. Ciao!
Re: Quantization and Cohomology (Week 18)
Concerning the issue mentioned around the Good question on slide 5051: you know I have been thinking about this quite a bit lately.
My impression is that it is of great help to write down the theory for $n=1$ completely in terms of arrows. This answers a lot of subtle questions when it comes to categorification.
For instance, I believe it does answer the question what the “space of functions” on configuration space categorifies to.
Also, the choice of 2category of 2vector spaces then answers automatically the question what these “2functions” should be valued in.
Or so I think. I might of course be wrong. I would love to discuss this stuff in more detail. There are not too many people in the world thinking about this stuff from this point of view.
From your reaction to my last comments I gained the impression that you would maybe rather not spread out everything in public, with your student Alex Hoffnung being supposed to work on this. If that’s so, I would just as well enjoy taking this to private email and keeping top secrecy about any further development.
Given my latest rate of publications, my activity here should hardly be a threat to anyone else’s career. So it would be a pity if we could not exchange more ideas on this.
Maybe I should just email Alex about this… (?)