nLab
n-vector bundle

Definition

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

Examples

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

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

  on the 1-dimensional 2-vector space ${\mathrm{Vect}}_{𝒞}$.

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

  $$\rho (g):X\to B^{2}U(1)\stackrel{\rho }{\to }2\mathrm{Vect}$$\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 $\rho (g)$ is a twisted vector bundle with twist given by $\rho$.

Related concepts

• principal bundle / torsor / groupoid principal bundle / associated bundle

• principal 2-bundle / gerbe / bundle gerbe

• principal 3-bundle / bundle 2-gerbe

• principal ∞-bundle / associated ∞-bundle

• vector bundle

• 2-vector bundle

• (∞,1)-vector bundle / $(\mathrm{\infty },n)$-vector bundle