∞-Lie theory (higher geometry)
Formal Lie groupoids
For a Lie 2-algebra the 2-groupoid of -valued forms is the 2-groupoid whose objects are differential forms with values in , whose morphisms are gauge transformations between these, and whose 2-morphisms are higher order gauge transformations of those.
This naturally refines to a non-concrete Lie 2-groupoid is the 2-truncated ∞-Lie groupoid whose -parameterized smooth families of objects are smooth differential forms with values in a Lie 2-algebra, and whose morphisms are gauge transformations of these.
This is the higher category generalization of the groupoid of Lie-algebra valued forms.
A cocycle with coefficients in this 2-groupoid is a connection on a 2-bundle.
For strict Lie 2-algebras
Consider a Lie strict 2-group corresponding to a Lie crossed module with action . Write for the corresponding delooping 2-groupoid, the one coming from the crossed complex
Write for the corresponding differential crossed module with action
The 2-groupoid of Lie 2-algebra valued forms is defined to be the 2-stack
which assigns to the following 2-groupoid:
An object is a pair
A 1-morphism is a pair
The composite of two 1-morphisms
is given by the pair
a 2-morphism is a function
and composition is defined as follows: vertical composition is given by pointwise multiplication (DR: the order still needs sorting out!) and horizontal composition is given as horizontal composition in the one-object 2-groupoid .
For general Lie 2-algebras
We consider now a general Lie 2-algebra.
Let and be the two vector spaces involved and let
be a dual basis, respectively. The structure of a Lie 2-algebra is conveniently determined by writing out the most general Chevalley-Eilenberg algebra
with these generators.
We thus have
for collections of structure constants (the bracket on ) and (the differential ) and (the action of on ) and (the “Jacobiator” for the bracket on ).
These constants are subject to constraints (the weak Jacobi identity and its higher coherence laws) which are precisely equivalent to the condition
Over a test space a -valued form datum is a morphism
from the Weil algebra .
This is given by a 1-form
and a 2-form
The curvature of this is , where the 2-form component (“fake curvature”) is
and whose 3-form component is
(flat Lie 2-algebra valued forms)
The full sub-2-groupoid on flat Lie 2-algebra valued forms, i.e. those pairs for which the 2-form curvature
and the 3-form curvature
vanishes is a resolution of the underlying discrete Lie 2-groupoid of the Lie 2-groupoid .
This is discussed at ∞-Lie groupoid in the section strict Lie 2-groups – differential coefficients.
Let be the smooth 2-fundamental groupoid functor and let be the path 2-groupoid functor, taking values in the 2-catgeory of 2-groupoids internalization to diffeological spaces. Then
the 2-groupoid of Lie 2-algebra valued forms for which both 2- and 3-form curvature vanish is canonically equivalent to
the 2-groupoid of Lie 2-algebra valued forms for which the 2-form curvature vanishes is canonically equivalent to
The equivalence is given by 2-dimensional parallel transport. A proof is in SchrWalII.
The following proposition asserts that the Lie 2-groupoid of Lie 2-algebra valued forms is the coefficient object for for differential nonabelian cohomology in degree 2, namely for connections on principal 2-bundles and in particular on gerbes.
This is discussed and proven in SchrWalII for the case where the 2-form curvature is restricted to vanish. In this case the above can be written as
where is a resolution of the path 2-groupoid of .
The 2-groupoid of Lie 2-algebra valued forms described in definition 2.11 of
- Schreiber, Waldorf, Smooth functors versus differential forms (web).
There are many possible conventions. The one reproduced above is supposed to describe the bidual opposite 2-category of the 2-groupoid as defined in that article, with the direction of 1- and 2-morphisms reversed.
differential cohomology in an (∞,1)-topos – survey - connections on 2-bundles.