∞-Lie theory (higher geometry)
Formal Lie groupoids
For a Lie algebra, the groupoid of -valued forms is the groupoid whose objects are differential 1-forms with values on , and whose morphisms are gauge transformations between these.
This carries the structure of a generalized Lie groupoid , which is a differential refinement of the delooping Lie groupoid of the Lie group corresponding to :
its -parameterized smooth families of objects are Lie algebra valued differential forms on . Its -parameterized families of morphisms are gauge transformations of these forms by -valued smooth functions on .
A cocycle with coefficients in is a connection on a bundle.
For more discussion of this see ∞-Lie groupoid – Lie groups.
For a Lie group the groupoid of -valued differential forms is as a groupoid internal to smooth spaces, the sheaf of groupoids
that to a smooth test space assigns the functor category of smooth functors (functors internal to smooth spaces) from the path groupoid of to the one-object delooping groupoid .
The groupoid is canonically equivalent to the smooth groupoid where
- objects are smooth -valued 1-forms ;
- morphisms are given by smooth -valued functions such that
Here is the right invariant Maurer-Cartan form on . A common way to write this is .
A proof is in SchrWalI.
Differential nonabelian cohomology
The cohomology with coefficients in classifies -principal bundles connection on a bundle with connection.
More is true: there is a natural canonical equivalence of groupoids
For these differential cocycles model the Yang-Mills field in physics.
Details are in
The definition in terms of differential forms is def 4.6 there. The equivalence to is proposition 4.7.
See also ∞-Chern-Weil theory introduction