nLab
n-vector bundle

Context

Bundles

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Definition

An nn-vector bundle is an fiber ∞-bundle whose fibers are n-vector spaces of sorts.

Examples

  • For BU(1)\mathbf{B}U(1) the circle 2-group and 2Vect2 Vect the category of 2-vector spaces (objects are categories equivalence to AA Mod for some associative algebra or algebroid AA, morphisms are bimodules) there is a canonical 1-dimensional ∞-representation

    ρ:BBU(1)2Vect \rho : \mathbf{B} \mathbf{B} U(1) \to 2 Vect

    on the 1-dimensional 2-vector space Vect 𝒞Vect_{\mathcal{C}}.

    For g:XB 2U(1)g : X \to \mathbf{B}^2 U(1) a cocycle for a circle 2-bundle, the composite

    ρ(g):XB 2U(1)ρ2Vect \rho(g) : X \to \mathbf{B}^2 U(1) \stackrel{\rho}{\to} 2 Vect

    is the corresponding classifying map for the “associated line 2-bundle”.

    A section of ρ(g)\rho(g) is a twisted vector bundle with twist given by ρ\rho.

Revised on December 12, 2012 17:08:43 by Urs Schreiber (71.195.68.239)