nLab
3-groupoid of Lie 3-algebra valued forms

Idea

Given a smooth manifold $U$ and a Lie 3-algebra $𝔤$, the 3-groupoid of Lie 3-algebra valued forms over $U$ has as objects ∞-Lie algebroid valued differential forms with values in $𝔤$, 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 $𝔤$ 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.

Related concepts

• groupoid of Lie-algebra valued forms

• 2-groupoid of Lie 2-algebra valued forms

• 3-groupoid of Lie 3-algebra valued forms

• ∞-groupoid of ∞-Lie-algebra valued forms

References

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

• João Faria Martins, Roger Picken, The fundamental Gray 3-groupoid of a smooth manifold and local 3-dimensional holonomy based on a 2-crossed module (arXiv:0907.2566)