A connection on a bundle 2-gerbe is a special cocycle representative for circle n-bundles with connection, hence for degree 4 Deligne cohomology, hence for degree 4 Cheeger-Simons differential characters.
The definition is built by iteration on the definition of bundle gerbe:
a bundle 2-gerbe over a manifold is
a surjective submersion ;
a morphims of bundle gerbes ;
which is associative up to a choice of coherent 2-morphisms.
Bundle 2-gerbes were briefly introduced in
and further developed in
drawing on ideas from Stevenson’s PhD thesis (arXiv:math/0004117).