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 maybe 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.