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June 1, 2007

Quantization and Cohomology (Week 25)

Posted by John Baez

In this week’s seminar on Quantization and Cohomology, we looked at a simplified version of a claim made last week:

  • Week 25 (May 22) - Bundles, connections, cohomology and anafunctors. A simplified version of the claim made last week: principal GG-bundles over MM correspond to smooth anafunctors hol:Disc(M)Ghol: Disc(M) \to G, where Disc(M)Disc(M) is the smooth category with points of MM as objects and only identity morphisms. Bundle isomorphisms correspond to smooth ananatural transformations between these. To prove this, use Cech 1-cocycles to describe principal GG-bundles, and Cech 0-cochains to describe isomorphisms between these. Claim: the first Cech cohomology consists of smooth anafunctors modulo smooth ananatural transformations.

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

Last week we claimed that:

  • Principal GG-bundles with connection over MM correspond to smooth anafunctors hol:PMGhol: P M \to G, where PMP M is the path groupoid of MM.
  • Gauge transformations between such GG-bundles with connection correspond to smooth ananatural transformations between such anafunctors.
This time we left out the connections and began sketching how to prove this:
  • Principal GG-bundles over MM correspond to smooth anafunctors hol:Disc(M)Ghol: Disc(M) \to G, where Disc(M)Disc(M) is the smooth category with MM as the space of objects, and only identity morphisms.
  • Gauge transformations between principal GG-bundles over MM correspond to smooth ananatural transformations between such anafunctors.

This got us into Cech cohomology!

Posted at June 1, 2007 6:45 PM UTC

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Read the post Quantization and Cohomology (Week 26)
Weblog: The n-Category Café
Excerpt: Cech cohomology in terms of anafunctors and ananatural transformations.
Tracked: June 1, 2007 7:42 PM

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