Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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 \left\[G\right]$, where $\left\[G\right]$ 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 \left\[G\right]$, 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 \left\[TG\right]$ (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?

share|improve this question

1 Answer 1

up vote 3 down vote accepted

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.

share|improve this answer
    
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

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.