Descent for n-Bundles

TL;DR

This paper extends the classical theory of principal bundles to local n-bundles for general n, providing finite-dimensional models for Lie 2-groups like String(n).

Contribution

It introduces a construction for local n-bundles for general n and establishes analogues for simplicial Lie groups, linking strict Lie n-groups to local n-bundles.

Findings

Existence of local n-bundles analogous to principal G-bundles.

Analogues for simplicial Lie groups of Moore's results.

Finite-dimensional models for Lie 2-groups such as String(n).

Abstract

Given a Lie group G, one constructs a principal G-bundle on a manifold X by taking a cover U of X, specifying a transition cocycle on the cover, and descending the trivialized bundle along the cover. We demonstrate the existence of an analogous construction for local n-bundles for general n. We establish analogues for simplicial Lie groups of Moore's results on simplicial groups; these imply that bundles for strict Lie n-groups arise from local n-bundles. Our construction leads to simple finite dimensional models of Lie 2-groups such as String(n).