### Quantization and Cohomology (Week 20)

#### Posted by John Baez

In this week’s class on Quantization and Cohomology, we introduce Chen’s "smooth spaces" which generalize smooth manifolds and provide a more convenient context for differential geometry. These will allow us to define "smooth categories" and study the principal of least action starting with any smooth category $C$ equipped with a smooth functor $S: C \to \mathbb{R}$ describing the ‘action’.

- Week 20 (Apr. 10) - Smooth spaces and smooth categories. The concept of a "category internal to $K$" where $K$ is any category with pullbacks. The category of smooth manifolds does not have pullbacks. Grothendieck’s dictum. Chen’s category of smooth spaces. Examples: the discrete and indiscrete smooth structures on a set. Any convex set or smooth manifold is a smooth space. The product and coproduct of smooth spaces. Any subset of a smooth space becomes a smooth space. Homework: the category of smooth spaces has pullbacks.

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

Posted at April 11, 2007 3:08 AM UTC
## Smooth spaces

I actually like diffeological terminology better than “smooth space”, but regardless of what one calls the notion, I hope your notes increase general awareness of it. It’s another one of those very good ideas (like the Kurzweil-Henstock integral) that hasn’t quite caught on yet.

I wonder if diffeological/smooth spaces fit into Paul Feit’s general scheme of passing from local to global notions:

P. Feit, Axiomization of passage from “local” structure to “global” object,

Mem. Amer. Math. Soc.101(1993), no. 485.