nLab
3-groupoid of Lie 3-algebra valued forms

Context

\infty-Lie theory

∞-Lie theory

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

\infty-Chern-Weil theory

Contents

Idea

Given a smooth manifold UU and a Lie 3-algebra 𝔤\mathfrak{g}, the 3-groupoid of Lie 3-algebra valued forms over UU has as objects ∞-Lie algebroid valued differential forms with values in 𝔤\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 GG-principal 3-bundles over UU with nontrivial connection, for GG the 3-Lie group related to 𝔤\mathfrak{g} by Lie integration.

Regarded as a presheaf of 3-groupoids over all suitable manifolds UU, 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)