group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
For $\mathcal{X}$ an (∞,1)-topos, a 2-gerbe $P$ in $\mathcal{X}$ is an object which is
The first condition says that it is an (∞,1)-sheaf with values in 2-groupoids. The second says that $P \to *$ is an effective epimorphism and that the 0-th homotopy sheaf is the terminal sheaf. In the literature this is often stated as saying that $P$ is a) locally connected and b) locally non-empty .
For $\mathcal{X}$ an (∞,1)-topos, an abelian 2-gerbe $P$ in $\mathcal{X}$ is an object which is
principal 3-bundle / 2-gerbe / bundle 2-gerbe
A comprehensive discussion of nonabelian 2-gerbes is in
A more expository discussion is in
Abelian 2-gerbes are a special case (see ∞-gerbe) of the discussion in section 7.2.2 of
See also