The n-Category Café

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 $G$-bundles over $M$ correspond to smooth anafunctors $hol: Disc(M) \to G$, where $Disc(M)$ is the smooth category with points of $M$ 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 $G$-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 $G$-bundles with connection over $M$ correspond to smooth anafunctors $hol: P M \to G$, where $P M$ is the path groupoid of $M$.
• Gauge transformations between such $G$-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 $G$-bundles over $M$ correspond to smooth anafunctors $hol: Disc(M) \to G$, where $Disc(M)$ is the smooth category with $M$ as the space of objects, and only identity morphisms.
• Gauge transformations between principal $G$-bundles over $M$ correspond to smooth ananatural transformations between such anafunctors.

This got us into Cech cohomology!