David Roberts
2-bundle

Idea

2-bundles are a categorification of the usual notion of bundle. Most often principal 2-bundles are treated, but there is interest in 2-vector bundles for the purposes of interpreting elliptic cohomology.

Papers

The seminal work on 2-bundles is Toby Bartel’s PhD thesis:

• Higher Gauge Theory I: 2-bundles (webpage)

which covers principal 2-bundles, internal category theory, anafunctors, associated 2-bundles, descent data for 2-bundles. Toby introduced internal anafunctors and my thesis draws heavily on that material (covering it in my own way).

There are other papers that deal with 2-bundles, or related objects, such as Igor Bakovic’s PhD thesis (principal bigroupoid bundles), and various papers referenced at the nLab page on principal 2-bundles.

For stuff on 2-vector bundles see work by Nils Baas and coworkers: Two-vector bundles and forms of elliptic cohomology, Two-Categorical Bundles and Their Classifying Spaces.

(rough) Definition

A 2-bundle on a topological space $X$ is a topological groupoid $E\to X$ over $X$ such that there is an open cover $U\to X$ and an equivalence $U{\times }_{X}E\to U\times F$ in the bicategory $\mathrm{Ana}/U$ of anafunctors over $U$, where $F$ is some topological groupoid.