(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
An $n$-vector bundle is an fiber ∞-bundle whose fibers are n-vector spaces of sorts.
For $\mathbf{B}U(1)$ the circle 2-group and $2 Vect$ the category of 2-vector spaces (objects are categories equivalence to $A$ Mod for some associative algebra or algebroid $A$, morphisms are bimodules) there is a canonical 1-dimensional ∞-representation
on the 1-dimensional 2-vector space $Vect_{\mathcal{C}}$.
For $g : X \to \mathbf{B}^2 U(1)$ a cocycle for a circle 2-bundle, the composite
is the corresponding classifying map for the “associated line 2-bundle”.
A section of $\rho(g)$ is a twisted vector bundle with twist given by $\rho$.
principal bundle / torsor / groupoid principal bundle / associated bundle
(∞,1)-vector bundle / $(\infty,n)$-vector bundle