nLab 2-vector bundle

Contents

Context

Bundles

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Idea

A 2-module bundle / 2-vector bundle is a fiber ∞-bundle whose typical fiber is a 2-module/2-vector space.

Definition

Let RR be a commutative ring, or more generally an E-∞ ring. By the discussion at 2-vector space consider the 2-category

2Vect RAlg R 2 Vect_R \simeq Alg_R

equivalent to that whose objects are associative algebras (or generally algebras) AA over RR, (being placeholders for the 2-vector space AModA Mod which is the category of modules over AA) 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 SpecRSpec R.

Examples

References

Via completing nn-tunles of vector bundles

See at BDR 2-vector bundle.

As algebra bundles with bimodule bundles between them

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.