vector bundle, 2-vector bundle, (∞,1)-vector bundle
real, complex/holomorphic, quaternionic
group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
The canonical line bundle on the product of an abelian group (abelian variety) and its dual Picard scheme. Alternatively its sheaf of sections: the Poincaré sheaf.
The idea is briefly summarized in
Lecture notes:
Monographs:
Discussion in differential geometry and with an eye towards of (topological and geometric) T-duality:
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