group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
Let $\mathbf{H}$ be an ambient (∞,1)-topos. Let $V, X$ be two objects of $\mathbf{H}$. Then a $V$-fiber bundle over $X$ in $\marthbf{H}$ is a morphism $E \to X$ such that there is an effective epimorphism $U \to X$ and an (∞,1)-pullback square of the form
Externally this is a $V$-fiber $\infty$-bundle.
See at associated ∞-bundle for more.
A fiber $\infty$-bundle whose typical fiber $V$ is a pointed connected object, hence a delooping $\mathbf{B}G$ of an ∞-group $G$
is a $G$-∞-gerbe.
Every $V$-fiber $\infty$-bundle is the associated ∞-bundle to an automorphism ∞-group-principal ∞-bundle.
See the references at associated ∞-bundle.
The explicit general definition appears as def. 4.1 in part I of