group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
An -vector bundle is an fiber ∞-bundle whose fibers are n-vector spaces of sorts.
For the circle 2-group and the category of 2-vector spaces (objects are categories equivalence to Mod for some associative algebra or algebroid , morphisms are bimodules) there is a canonical 1-dimensional ∞-representation
on the 1-dimensional 2-vector space .
For a cocycle for a circle 2-bundle, the composite
is the corresponding classifying map for the “associated line 2-bundle”.
A section of is a twisted vector bundle with twist given by .
principal bundle / torsor / groupoid principal bundle / associated bundle
(∞,1)-vector bundle / -vector bundle