nLab connection on a 3-bundle

Context

Differential cohomology

differential cohomology

Contents

Definition

For $G$ a Lie 3-group, a connection on a $G$-principal 3-bundle coming from a cocycle $g:X\to BG$ is a lift of the cocycle to the 3-groupoid of Lie 3-algebra valued forms $B{G}_{\mathrm{conn}}$

$\begin{array}{ccc}& & B{G}_{\mathrm{conn}}\\ & {}^{\nabla }↗& ↓\\ X& \stackrel{g}{\to }& BG\end{array}$\array{ && \mathbf{B}G_{conn} \\ & {}^{\mathllap{\nabla}}\nearrow & \downarrow \\ X &\stackrel{g}{\to}& \mathbf{B}G }

References

The higher parallel transport of local 3-connections is considered in

Examples of 3-connections obtained from fibrations of Courant algebroids are discussed in

• Olivier Brahic, On the infinitesimal Gauge Symmetries of closed forms (arXiv)

A discussion of fully general local 3-connections is in

and the globalization is in

For a discussion of all this in a more comprehensive context see section xy of