# nLab 3-groupoid of Lie 3-algebra valued forms

∞-Lie theory

## Examples

### $\infty$-Lie algebras

#### $\infty$-Chern-Weil theory

∞-Chern-Weil theory

∞-Chern-Simons theory

∞-Wess-Zumino-Witten theory

# Contents

## Idea

Given a smooth manifold $U$ and a Lie 3-algebra $\mathfrak{g}$, the 3-groupoid of Lie 3-algebra valued forms over $U$ has as objects ∞-Lie algebroid valued differential forms with values in $\mathfrak{g}$, as morphisms gauge transformations of these, as 2-morphisms 2-gauge transformations and so on.

This can be understood as the 3-groupoid of trivial $G$-principal 3-bundles over $U$ with nontrivial connection, for $G$ the 3-Lie group related to $\mathfrak{g}$ by Lie integration.

Regarded as a presheaf of 3-groupoids over all suitable manifolds $U$, this is a non-concrete 3-Lie groupoid.

A cocycle with coefficients in this 3-groupoid is a connection on a 3-bundle.

## References

For Lie 3-algebras coming from differential 2-crossed modules, at least parts of this data have been discussed in

Revised on August 24, 2011 02:45:12 by Urs Schreiber (131.211.239.234)