### News on Measures on Groupoids?

#### Posted by Urs Schreiber

Over at his Theoretical Atlas blog, Jeffrey Morton reports from Quantum Gravity and Quantum Geometry 2008, briefly indicating the content of a couple of talks. This one here especially caught my attention:

Benjamin Bahr gave another talk dealing with categorical issues - namely, how to get measures on certain groupoids, such as, indeed, the groupoid of connections on a manifold. In fact, he treated various cases under the same framework: flat and non-flat connections, on manifolds and on graphs - and others.

I have posted the following comment, which however is still “awaiting moderation”. While it awaits, I thought I’d share this here:

Hi,

thanks for the report!

I am thrilled by this bit:

Benjamin Bahr gave another talk dealing with categorical issues - namely, how to get measures on certain groupoids, such as, indeed, the groupoid of connections on a manifold.

I had not heard of Bahr’s work before. A quick search on the web didn’t yield anything (but maybe I didn’t search carefully enough). But I am very interested in this general question. Have thought a bit about it myself.

Do you happen to have any further details on this? Is there anything written available in any form?

Do you know if his groupoid measures are at all related to Tom Leinster’s weightings on categories I, II, III, IV?

Does Bahr go as far as saying something about path integrals for gauge theory? Does he mention connections to BRST-BV methods which are the standard tool for handling integration over things like the groupoid of gauge connections?

For the case of finite groups and hence necessarily flat connections, the groupoids they live in are, naturally, well understood. More strikingly, it is known in these cases that the right “path integral” (just a sum in this case) measure on these groupoids – which appears in Dijkgraaf-Witten theory but also in its higher version, the Yetter-Martins-Porter model that you mention above – is precisely the Baez-Dolan-Leinster groupoid measure for Dijkgraaf-Witten, and its higher version for the Cat-enriched case for the Martins-Porter model. (I recall this for DW and prove it for 2DW in section 1.4 of On $\Sigma$-models and nonabelian differential cohomology).

Does Bahr’s approach say anything about this? Does he reproduce the measures for DW and higher DW theory? Do you happen to know?

I’d be grateful for whatever information you might have.

## Re: News on Measures on Groupoids?

His abstract is a start.