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 Gbundle $P \to M$ 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: 2bundles. Section 2.2.2, on "2maps",
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.

Toby Bartels, Higher
gauge theory I: 2bundles. Section 2.2.2, on "2maps",
describes smooth anafunctors between smooth categories. Section 2.2.3
describes what I’m calling smooth ananatural transformations.
Last week’s notes are here; next week’s notes are here.
Posted at June 1, 2007 6:14 PM UTC
Re: Quantization and Cohomology (Week 24)
Thanks for mentioning this!
The way I think about a $\pi$local $i$trivialization is as the bridge between the anafunctor (= “descent data” = “transition data” = “cocycle data”) and the global object defined by it.
Given one globally defined transport functor (where “transport” is supposed to mean: “admits some smooth local trivialization”) we may choose a fixed smooth local trivialization. From this choice we then obtain a descent datum, which is canonically equivalent to an anafunctor.
global functor $\stackrel{\mathrm{choice}\;\mathrm{of}\;\pi\mathrm{local}\; i\mathrm{trivialization}}{\rightarrow}$ descent data $\stackrel{\mathrm{canon}.}{\sim}$ anafunctor.
So I am not sure if I would say that the notion of$\pi$local $i$trivialization generalizes that of anafunctor. Rather, I think of it as something connecting anafunctors with “global functors”.