# Connections on principal bundles via stacks?

Let G be a Lie group and M a smooth manifold. Suppose that P is a principal G-bundle over M. Then by Yoneda, this corresponds to a smooth map $p:M \to [G]$, where $[G]$ is the differentiable stack associated to G. If P is equipped with a connection, how does this fit in this picture?

I was thinking of using the fact that we can obtain $P$ as the weak pullback of the classifying map p along the universal principal G-bundle $* \to [G]$, and then using that the tangent functor preserves finite weak limits. Then, if we are given a connection in terms of a G-invariant subbundle of $TP$, we could express its inclusion map $H \to TP$ as a map $H \to TM$ together with a certain 2-cell between the composite of the canonical map $H \to [TG]$ (the universal TG-bundle composed with the unique map to the terminal object) and T(p) composed with $H \to TM$. But then of course, we need to put conditions on our map $H \to TM$ to make sure that the induced map $H \to TP$ is an inclusion of vector bundles that moreover defines a G-invariant Ehresmann connection. Maybe this is not the way to go...

Does anyone know of a nice way of encoding a connection in this stacky language?

-

The only thing which has to be replaced in the representation of a principal bundle as a suitable class of a maps $M\to [*/G]$ in order to introduce a flat connection is to replace $M$ by its fundamental 1-groupoid $\Pi_1(M)$, or, if the connection is not flat, by the thin homotopy version $P_1(M)$ of it, cf. nlab:path groupoid. The same way it works for higher categorical generalizations, see Schreiber:differential nonabelian cohomology and for details also Sec. 7.4 (from page 27 on in version 1) in arxiv/1004.2472.

-
Is there a way to avoid this thin homotopy business? – babubba Apr 20 '10 at 15:38
Look at 7.4.3 in the arXiv paper for such an alternative: one can consider the curvature as an obstruction, and similarly to the problem of extensions of groups, one can look at the cohomology which measures the obstruction. This leads to curvature-twisted cocycles on the usual fundamental 1-groupoid (no thin classes). – Zoran Skoda Apr 20 '10 at 16:17
So, this $P_1(M)$ guy gives us stack on Diff which admits an epimorphism from a diffeological space (in particular a sheaf on Diff), and maps from its associated stack to $\left[G\right]$ classify principal bundles on G with connection? (and then a connection is a choice of factorization of the map $M \to \left[G\right]$ through $M -> \left[P_1(M\right)\right]$?) – David Carchedi Apr 20 '10 at 17:40
You do not need associated stacks, but using weak functors of presheaves of higher categories; they can be represented by anafunctors or any other similar device. – Zoran Skoda Apr 21 '10 at 17:43

In case you have not seen it, the answer given by Chris Schommer-Pries to the following question might be of interest to you:

What is the classifying space of "G-bundles with connections"

If I understand correctly, the only difference with your question is that you want to fix the base manifold $M$ of your principal $G$-bundles, in which case the stack of principal $G$-bundles with connection on $M$ probably is $Bun_{M,G}^{\nabla} = \left[\Omega^1(M\times -;\mathfrak{g})/G\right]$ ?

If in addition you want to fix a particular bundle $P$ on $M$ and look at all possible $G$-connections on it, then the end of your post is reminiscent of Kobayashi's bundle of connections in his PhD thesis work:

S. Kobayashi (1957). "Theory of Connections". Annali di Matematica Pura ed Applicata 43: 119–194. doi:10.1007/bf02411907

which is also referred to on Wikipedia:

https://en.wikipedia.org/wiki/Connection_(principal_bundle)#Bundle_of_principal_connections

The $G$-connections on $P$ are the sections of the fibre bundle $(TP)/G \to TM$, where the map to $TM$ is induced by the differential of $P\to M$ and the action of $G$ on $TP$ is induced by its action on $P$.

-
I now realize that the question is 4 years old so what I have written above is probably not relevant to the OP anymore! But maybe it is not worth deleting either :-) – Oliver Feb 26 '15 at 21:11