nLab
nonabelian bundle gerbe

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Idea

A nonabelian bundle gerbe (as studied by Aschieri-Cantini-Jurco) is a model for the Lie groupoid which is the total space of a smooth AUT(H)AUT(H)-principal 2-bundle for AUT(H)AUT(H) the Lie 2-group that is the automorphism 2-group of a Lie group HH.

Specifically, a nonabelian bundle gerbe on a smooth manifold XX is given by a surjective submersion YXY \to X and an HH-bibundle PY× XYP \to Y\times_X Y together with a morphism of HH-bibundles

μ:π 0 *Pπ 2 *Pπ 1 *P \mu : \pi_0^* P \otimes \pi_2^* P \to \pi_1^* P

that is associative in the evident sense. This construction serves to model pullbacks of Lie 2-groupoids of the form

P˜ EAUT(H) C(Y) g BAUT(H) X, \array{ \tilde P &\to& \mathbf{E}AUT(H) \\ \downarrow && \downarrow \\ C(Y) &\stackrel{g}{\to}& \mathbf{B}AUT(H) \\ \downarrow^{\mathrlap{\simeq}} \\ X } \,,

where on the rght we have the universal principal 2-bundle.

The resulting Lie groupoid P˜\tilde P is an extension of the Cech groupoid C(Y)C(Y) by AUT(H)AUT(H). This generalizes the case of ordinary bundle gerbes, which are models for BU(1)\mathbf{B}U(1)-principal 2-bundles, for BU(1)\mathbf{B}U(1) the circle 2-group.

This can all be extended to topological groupoids, and to structure 2-groups given by more general crossed modules than HAut(H)H\to Aut(H).

References

Revised on September 9, 2010 07:43:30 by David Roberts (203.24.207.216)