(see also Chern-Weil theory, parameterized homotopy theory)
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 $\mathbf{H}$ 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.
For let $Type$ be the object classifier. Then any bundle $E \to X$ is classified by a morphism
On the other hand, since the pullback to the bundle on some $U$ is trivializable, that bundle over $U$ is classsified by a map that factors through the point which is the name of the fiber $V$
The 1-image-of this point inclusion is the delooping of the automorphism ∞-group of $V$ :
Therefore the fact that $E$ is trivialized over $U$ means that there the classifying maps fit into a commuting diagram of the form
By assumption the left morphism is a 1-epimorphism and by the above construction the right morphism is a 1-monomorphism. Therefore by the (n-connected, n-truncated) factorization system this diagram has an essentially unique lift
This diagonal lift classifies an $\mathbf{Aut}(V)$-principal ∞-bundle and the commutativity of the bottom right triangle exhibits the original bundle $E \to X$ as the associated ∞-bundle to that.
See the references at associated ∞-bundle.
The explicit general definition appears as def. 4.1 in part I of