What is curvature, in terms of holonomy functors?

It is well known and understood that linear connections, as holonomy functors, are composition-preserving mappings from the path groupoid to a structure group $G$. This extends the idea of a 1-form line integral to a non-Abelian setting.

It is known, although still a work in progress (http://ncatlab.org/nlab/show/connection+on+a+2-bundle), that using higher category theory we can do the very same for 2-forms, 3-forms, etcetera, raising the degree of the groups.

The geometrical picture is very nice. I was wondering, though, if there were such a picture for curvature on a good old bundle (vector or principal) of degree 1. That is: can we see curvature as a mapping between 2-dimensional surfaces and a group $G$ (the holonomy group)? How can we make this mapping not Abelian, to prevent what happens in the classic second homotopy group?

I have some personal ideas about how to do such a thing, but I'd first rather knowing if it has been already done nicely elsewhere.

Thanks!

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This question is tightly linked to this very interesting one: math.stackexchange.com/questions/49583/…. – geodude Jul 26 '13 at 9:35