(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
A 2-module bundle / 2-vector bundle is a fiber ∞-bundle whose typical fiber is a 2-module/2-vector space.
Let $R$ be a commutative ring, or more generally an E-∞ ring. By the discussion at 2-vector space consider the 2-category
equivalent to that whose objects are associative algebras (or generally algebras) $A$ over $R$, (being placeholders for the 2-vector space $A Mod$ which is the category of modules over $A$) whose 1-morphisms are bimodules between these algebras (inducing linear functors between the corresponding 2-vector spaces = categories of modules) and whose 2-morphisms are homomorphisms between those.
Under Isbell duality and by the discussion at Modules – as generalized vector bundles we may think of this 2-category as being that of (generalized) 2-vector bundles over a space called $Spec R$.
See at BDR 2-vector bundle.
The notion of 2-vector bundles based on regarding 2-vector spaces as algebras with bimodules between them (here) is first discussed in
and much further developed in
Last revised on August 23, 2022 at 18:31:24. See the history of this page for a list of all contributions to it.