group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
A principal 3-bundle is a $G$-principal ∞-bundle over a 3-group $G$.
For $G = \mathbf{B}^2 U(1)$ the circle Lie 3-group, the total space of a $\mathbf{B}^2 U(1)$-principal 3-bundle is modeled by a bundle 2-gerbe.
See connection on a 3-bundle .