The n-Category Café

Quantization and Cohomology (Week 24)

Posted by John Baez

In this week’s seminar on Quantization and Cohomology, we drew a big chart comparing three approaches to connections and gauge transformations. The most sophisticated uses the idea of “smooth anafunctor” developed by Toby Bartels. A smooth anafunctor is something that looks locally, but perhaps not globally like a smooth functor!

• Week 24 (May 15) - Connections and smooth anafunctors: review and prospectus. Connections on the trivial principal $G$-bundle over $M$ are smooth functors $hol: P M \to G$; gauge transformations are smooth natural transformations between these. Connections on a fixed principal G-bundle $P \to M$ are smooth functors $hol: P M \to Trans(P)$; gauge transformations are smooth natural transformations between these. Connections on an arbitrary, or variable principal $G$-bundle over $M$ are smooth anafunctors $hol: P M \to G$; gauge transformations are smooth ananatural transformations between these. The definition of smooth anafunctor.
  
  Supplementary reading:
  
  
  • Toby Bartels, Higher gauge theory I: 2-bundles. Section 2.2.2, on "2-maps", describes smooth anafunctors between smooth categories. Section 2.2.3 describes what I’m calling smooth ananatural transformations.
    
    
  • Urs Schreiber and Konrad Waldorf, Parallel transport and functors. This develops some closely related ideas, including a more flexible notion of “$\pi$-local $i$-trivialization” for a functor, which generalizes the concept of smooth anafunctor in a useful way.

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