nLab
bundle 2-gerbe

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Contents

Idea

A bundle 2-gerbe is a special presentation of the total space of a B 2U(1)\mathbf{B}^2 U(1)-principal 3-bundle, where B 2U(1)\mathbf{B}^2 U(1) is the circle Lie 3-group.

A connection on a bundle 2-gerbe is a special cocycle representative for circle n-bundles with connection, hence for degree 4 Deligne cohomology, hence for degree 4 Cheeger-Simons differential characters.

The definition is built by iteration on the definition of bundle gerbe:

a bundle 2-gerbe over a manifold XX is

  • a surjective submersion YXY \to X;

  • on the fiber product Y× XYY \times_X Y a bundle gerbe Y× XY\mathcal{L} \to Y\times_X Y;

  • a morphims of bundle gerbes π 1 *π 2 *π 1 *\pi_1^* \mathcal{L} \otimes\pi_2^* \mathcal{L} \to \pi_1^* \mathcal{L};

  • which is associative up to a choice of coherent 2-morphisms.

Examples

References

Bundle 2-gerbes were briefly introduced in

and further developed in

drawing on ideas from Stevenson’s PhD thesis (arXiv:math/0004117).

A general picture of bundle nn-gerbes (with connection) as circle (n+1)-bundles with connection classified by Deligne cohomology is in

  • Pawel Gajer, Geometry of Deligne cohomology Invent. Math., 127(1):155–207 (1997) (arXiv)

Revised on September 15, 2011 12:40:40 by Urs Schreiber (131.211.239.169)