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

  • (shape modality \dashv flat modality \dashv sharp modality)

    (ʃ)(ʃ \dashv \flat \dashv \sharp )

  • dR-shape modality\dashv dR-flat modality

    ʃ dR dRʃ_{dR} \dashv \flat_{dR}

  • tangent cohesion

    • differential cohomology diagram
    • differential cohesion

      • (reduction modality \dashv infinitesimal shape modality \dashv infinitesimal flat modality)

        (&)(\Re \dashv \Im \dashv \&)

      • graded differential cohesion

        • fermionic modality\dashv bosonic modality \dashv rheonomy modality

          (Rh)(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)

        • 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 }

          </semantics></math></div>

          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)

          Last revised on September 15, 2011 at 12:40:40. See the history of this page for a list of all contributions to it.