3-groupoid of Lie 3-algebra valued forms
∞-Lie theory (higher geometry)
Formal Lie groupoids
Given a smooth manifold and a Lie 3-algebra , the 3-groupoid of Lie 3-algebra valued forms over has as objects ∞-Lie algebroid valued differential forms with values in , 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 -principal 3-bundles over with nontrivial connection, for the 3-Lie group related to by Lie integration.
Regarded as a presheaf of 3-groupoids over all suitable manifolds , this is a non-concrete 3-Lie groupoid.
A cocycle with coefficients in this 3-groupoid is a connection on a 3-bundle.
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