2-vector bundle in nLab

Contents

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 R≃Alg 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

• line 2-bundle

• stringor bundle

• vector bundle

• principal 2-bundle

• BDR 2-vector bundle

• (∞,1)-vector bundle / (∞,n)-vector bundle

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

• Urs Schreiber, Konrad Waldorf, §4.4 of: Connections on non-abelian Gerbes and their Holonomy, Theory Appl. Categ., 28 17 (2013) 476-540 [arXiv:0808.1923, tac:28-17]

and much further developed in

• Peter Kristel, Matthias Ludewig, Konrad Waldorf, The insidious bicategory of algebra bundles [arXiv:2204.03900]

The example of the stringor bundle:

• Peter Kristel, Matthias Ludewig, Konrad Waldorf, A representation of the string 2-group, [arXiv:2206.09797]

Reviewed in:

• Konrad Waldorf, The stringor bundle, talk at QFT and Cobordism, CQTS (Mar 2023) [web]