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A connection on a bundle is given locally by a Lie algebra-valued 1-form. Gauge transformations act in the usual way on the forms, and form a groupoid.

A connection on a 2-bundle is given locally by a pair of Lie algebra-valued forms (a 1-form, and a 2-form), which form a so-called Lie 2-algebra-valued form. A pair given by a gauge transformation and a 1-form acts on our original pair, and forms a 2-groupoid.

Is this correct?

The real question is, anyway:

What happens for 3-bundles? How do we write a Lie 3-algebra valued form in terms of traditional forms?

Do we have again a pair with a 3-form and a 2-form, or a triplet? If they are they objects of some 3-groupoid, what are the morphisms, and how are they composed? (For example, are they a triplet with a gauge transformation, a 1-form, and a 2-form?)

Sorry for being so uninformed. The article on nLab is still a work in progress and not very explicit, and the references cited there are complete, but still too advanced for me (I am looking for a general idea).

Apart from the answer, any easy reference that does not stop at order 2 would be very welcome.

Thank you.

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  • $\begingroup$ I wish I could say this was an "answer", but my educated guess is that you allow for 1,2, and 3-holonomy to all live in different structure groups. That means you need a 3-form a 2-form and a 1-form on an open set to give a connection. Now you need to transform 3-holonomy along an intersection so you'd need a 2-form (acting on the "faces" of the 3-track) in addition to the other information from the 2-connection story. In short: A triplet $C_i, B_i, A_i$ and a triplet $c_{ij}, a_{ij}, g_{ij}$. $\endgroup$
    – cheyne
    Commented Feb 17, 2016 at 17:39
  • $\begingroup$ I know this is a bit late, but I'm pretty sure your answer is here: arxiv.org/pdf/0907.2566.pdf $\endgroup$
    – cheyne
    Commented Aug 17, 2017 at 22:57

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