(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
For $\mathbb{G}_m$ the multiplicative group, a $\mathbf{B}^{n-1}\mathbb{G}_m$ principal infinity-bundle, modulated by a cocycle $X\to \mathbf{B}^n \mathbb{G}_m$, classified by a cohomology class in in $H^n(X,\mathbb{G}_m)$.
moduli spaces of line n-bundles with connection on $n$-dimensional $X$