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June 1, 2007

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 GG-bundle over MM are smooth functors hol:PMGhol: P M \to G; gauge transformations are smooth natural transformations between these. Connections on a fixed principal G-bundle PMP \to M are smooth functors hol:PMTrans(P)hol: P M \to Trans(P); gauge transformations are smooth natural transformations between these. Connections on an arbitrary, or variable principal GG-bundle over MM are smooth anafunctors hol:PMGhol: 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 “π\pi-local ii-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.

Posted at June 1, 2007 6:14 PM UTC

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3 Comments & 1 Trackback

Read the post Quantization and Cohomology (Week 25)
Weblog: The n-Category Café
Excerpt: How describing bundles in terms of Cech cohomology secretly amounts to describing them in terms of smooth anafunctors.
Tracked: June 1, 2007 7:09 PM

Re: Quantization and Cohomology (Week 24)

notion of “π\pi-local ii-trivialization” for a functor, which generalizes the concept of smooth anafunctor

Thanks for mentioning this!

The way I think about a π\pi-local ii-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 choiceofπlocalitrivialization\stackrel{\mathrm{choice}\;\mathrm{of}\;\pi-\mathrm{local}\; i-\mathrm{trivialization}}{\rightarrow} descent data canon.\stackrel{\mathrm{canon}.}{\sim} anafunctor.

So I am not sure if I would say that the notion ofπ\pi-local ii-trivialization generalizes that of anafunctor. Rather, I think of it as something connecting anafunctors with “global functors”.

Posted by: urs on June 3, 2007 6:13 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 24)

transport is supposed to mean: admits some smooth local trivialization

We all know the danger of identifying things that are only equivalent, but in this case I
don’t see they are even equivalent???

Posted by: jim stasheff on June 4, 2007 12:44 AM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 24)

in this case I don’t see they are even equivalent

I am not exactly sure which two supposedly equivalent things are you asking about here.

The equivalence of transport functors with their descent data is the main result in my paper with Konrad. That every such descent data gives an anafunctor is described in the section on anafunctors. That also every anafunctor gives rise to the coresponding descent data (cocycle data) is what John is talking about in the lecture here. In terms of the language used in our paper, the proof is here.

Posted by: urs on June 4, 2007 10:36 AM | Permalink | Reply to this

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