nLab bundle 2-gerbe

Contents

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

infinitesimal cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular 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& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\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

A model for the supergravity C-field in terms of nonabelian bundle 2-gerbes:

Last revised on February 8, 2024 at 13:45:07. See the history of this page for a list of all contributions to it.