# nLab bundle 2-gerbe

Contents

cohomology

### Theorems

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

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

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

infinitesimal cohesion

tangent cohesion

differential cohesion

singular cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

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 $\mathbf{B}^2 U(1)$-principal 3-bundle, where $\mathbf{B}^2 U(1)$ 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 $X$ is

• a surjective submersion $Y \to X$;

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

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

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

## 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 $n$-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.