The n-Category Café

Quantization and Cohomology (Week 19)

Posted by John Baez

The spring quarter has begun here at U. C. Riverside! Our seminar on Quantization and Cohomology resumed today. This time we’ll try to bring cohomology more explicitly into the picture — and we’ll start by seeing how it arises in a modern approach to classical mechanics:

• Week 19 (Apr. 3) - Finding critical points of an action functor $S: C \to \mathbb{R}$. For this, $C$ should be a ‘smooth category’ and $S$ should be something like a ‘smooth functor’. How can we make these concepts precise? The example where $C$ is the smooth path groupoid of a manifold equipped with a 1-form (for example, a cotangent bundle equipped with its canonical 1-form). The definition of ‘smooth category’ - that is, a category internal to some category of ‘smooth spaces’.

Last week’s notes are here; next week’s notes are here.

Of course, we’ll eventually see that $C$ should be a category internal to some nice category of smooth spaces, as explained in this paper:

• John Baez and Urs Schreiber, Higher gauge theory II: 2-connections (draft version).

and $S$ should be a smooth anafunctor or ‘2-map’ in the sense of Bartels:

• Toby Bartels, Higher gauge theory I: 2-bundles.