### Kock on 1-Transport

#### Posted by Urs Schreiber

A few months ago, I had a very inspiring e-mail conversation with Anders Kock on the concepts of connection, parallel transport and holonomy.

I had learned of Anders Kock’s work through the central role it played in Breen & Messing’s work on connections in nonabelian gerbes in terms of *combinatorial differential forms*.

Since John and I were thinking about another approach # describing such 2-transport, I was interested in understanding which use we could make of synthetic reasoning.

From our point of view, a principal $n$-connection should be a suitably well-behaved $n$-functor with domain some $n$-category of “$n$-paths in the base” and target some $n$-groupoid (of morphisms between fibers of an $n$-bundle, for instance).

At that time we were mainly concentrating on modelling $n$-paths as thin-homotopy classes of smooth maps from $n$-cubes into the base space. For $n=1$ it is an old theorem that, with a suitable notion of smoothness and target the transport groupoid of a principle bundle, such functors are in bijection with connections on that bundle.

We were essentially looking for a categorified version of this statement. But it was rather obvious that working with these classes of $n$-paths up to thin homotopy was rather cumbersome.

So, I was thinking, it would be nice if there were a description of smooth $n$-transport which was closer to the way physicists handle such things intuitively anyway, namely in terms of “small infinitesimal straight pieces” of $n$-paths. It seemed to me that the formalism used in particular by Anders Kock might be a way to make that approach precise.

Apparently partly in reaction to this motivation, Anders Kock has now written up a paper which is intended as a first step in this program:

Anders Kock
*Connections and path connections in groupoids*

preprint.

As a preparation for underanding the case $n \gt 1$, this work formalizes the notion of a connections in a fiber bundle, and the notion of holonomy of a connection, in synthetic differtial geometric terms.

The main concept that Anders Kock introduces and uses is that of a connection as a morphism of *graphs*.

The idea is (in my heuristic paraphrasing) to pass from smooth transport functors on categories to their differential version, which are mere morphisms on graphs. Heuristically, by passing to infinitesimals, we forget how to compose morphisms, since such composition is a first step towards “integrating” many infinitesimal paths to one composite finite path.

Please take notice that this is not how Anders Kock describes it. But I do think it gives the right idea of the spirit of his definitions.

A graph map *need not* respect composition (when applied to the underlying graph of a category), but it may. If it does, we say it is *flat*.

This is best understood my noting that what played the role of the groupoid of thin-homotopy classes of paths in my above description, is now, roughly, the graph of “infinitesimally close points” in the base space. A connection should associate to each such pair the infinitesimal parallel transport along a small path between the two points.

While such graph morphisms describing connections usually are not flat, they all become flat (under sufficiently nice conditions) when pulled back to finite paths.

Namely, we may imagine mapping the interval into our base space and pulling all the structure we had over there back to our interval. Any bundle will thus become trivial (trvially so). Its connection necessarily becomes flat, meaning that the pulled-back connection graph morphism now in fact is an honest functor.

Anders Kock calls this a *partial integration* of the connection along the given path. It gives rise to the notion of holonomy or parallel transport along finite paths. Thus one makes contact with the description of connections that I described at the beginning.

I am very much delighted to see these ideas in (electronic) print. I am sure there is a nice generalization of these concepts that allows one to treat smooth $n$-transport for $n \gt 1$ in the same fashion

In fact, I have thought about this issue, too, since then.

A while ago I made a remark on how the results of my work with John ($n=2$) should look like in synthetic language.

There are meanwhile also more detailed notes on this here.

A general remark I would like to make in closing is this:

I believe that an equivalent way to think of morphisms of graphs, with domain the (infinitesimal) pair groupoids, with target a given groupoid, is to think of *pseudo*functors to a 2-groupoid extending the original groupoid.

I have more details on that idea here.

## Re: Kock on 1-Transport

I think you know what I am thinking, but I will spare the electrons :)