group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
A bundle 2-gerbe is a special presentation of the total space of a $\mathbf{B}^2 U(1)$-principal 3-bundle, where $\mathbf{B}^2 U(1)$ is the circle Lie 3-group.
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 $X$ is
a surjective submersion $Y \to X$;
on the fiber product $Y \times_X Y$ a bundle gerbe $\mathcal{L} \to Y\times_X Y$;
a morphims of bundle gerbes $\pi_1^* \mathcal{L} \otimes\pi_2^* \mathcal{L} \to \pi_1^* \mathcal{L}$;
which is associative up to a choice of coherent 2-morphisms.
principal 3-bundle / 2-gerbe / bundle 2-gerbe
Bundle 2-gerbes were briefly introduced in
and further developed in
drawing on ideas from Stevenson’s PhD thesis (arXiv:math/0004117).
A general picture of bundle $n$-gerbes (with connection) as circle (n+1)-bundles with connection classified by Deligne cohomology is in