### Quantization and Cohomology (Week 23)

#### Posted by John Baez

This week in our seminar on Quantization and Cohomology, we tackled connections on bundles from a modern viewpoint:

- Week 23 (May 7) - Principal bundles. The transport groupoid $Trans(P)$ of a principal $G$-bundle $P$ over a smooth space $M$. Connections as smooth functors $hol: P M \to Trans(P)$ where $P M$ is the path groupoid of $M$. Proof that connections are described locally by smooth functors $hol: P U \to G$ where $U$ is a neighborhood in $M$. Theorem: smooth functors $hol: P U \to G$ are in 1-1 correspondence with $Lie(G)$-valued 1-forms on $U$.

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

To some extent we were making things up as we went along. Here’s a technical improvement that James Dolan pointed out, which didn’t make it into the notes.

We saw that not any smooth functor

$hol: P M \to Trans(P)$

gives a connection on the principal bundle $P$ over $M$. Rather, we only want smooth functors with

$hol(x) = P_x$

for each point $x \in M$. But, how can we say this in a nice arrow-theoretic way?

At first I thought we could to equip $P M$ and $Trans(P)$ with functors to $Disc(M)$, the discrete category corresponding to $M$ — that is, the category with points of $M$ as objects and only identity morphisms. I wanted to express the equation $hol(x) = P_x$ by demanding that

$hol: P M \to Trans(P)$

make the resulting triangle commute.

But, this makes no sense: there *are* no functors from $P M$ or $Trans(P)$ to the discrete category corresponding to $M$! I wriggled out of this problem by equipping $P M$ and $Trans(P)$ with functors *from* the discrete category corresponding to $M$, and demanding that $hol: P M \to Trans(P)$ make the resulting triangle commute.

In fancy lingo: instead of working with ‘categories over $Disc(M)$’, I realized I could work with ‘categories under $Disc(M)$’. This is what appears in the notes.

However, this feels funny — after all, we’re talking about a bundle *over* $M$. James pointed out the right solution near the end of class. We should work with categories over $Codisc(M)$, the codiscrete category corresponding to $M$. This has the points of $M$ as objects and exactly one morphism from any object to any other.

Both $P M$ and $Trans(P)$ are categories over $Codisc(M)$, and if we demand that

$hol: P M \to Trans(P)$

make the resulting triangle commute, we get

$hol(x) = P_x$

Moreover, $Codisc(M)$ also solves another problem for us — see the end of the notes.

So, treating path groupoids and transport groupoids as lying over $Codisc(M)$ is a good idea.

(By the way: I could explain this better if I knew how to draw nice commuting triangles in this environment!)

## Re: Quantization and Cohomology (Week 23)

Shouldn’t Codisc be better named Indisc? Or is that too… risqué?