Cohomology and Extensions
Formal Lie groupoids
The notion of Lie 2-group is the generalization of the notion of Lie group as groups are generalized to 2-groups:
it is a smooth 2-group that happens to have a model given by a Lie groupoid equipped with the structure of a group object (in general only up to homotopy).
One general way to make the notion precise is as a special case of an smooth ∞-groupoid, namely a 1-truncated ∞-group object in ∞-stacks over the site CartSp/SmthMfd, possibly with some representability condition:
these are stacks on the site of smooth manifolds (representable by Lie groupoids and) equipped with group structure: “group stacks” or “gr-stacks”.
Special cases of this have simpler definitions. For instance a crossed module internal to Diff is a model for a strict and comparatively tame Lie 2-group.
Analogous to how the infinitesimal version of a Lie group is a Lie algebra, the infinitesimal version of a Lie 2-group is a Lie 2-algebra.
Constructions and Applications
By the discussion at looping and delooping, every Lie 2-group induces a delooping Lie 2-groupoid : this has a single object, the space of morphisms is , the space of 2-morphisms is and the horizontal composition is given by the group product.
For a smooth manifold (or itself a Lie groupoid such as an orbifold, or generally any smooth ∞-groupoid), morphisms
of smooth ∞-groupoids from to the delooping Lie 2-groupoid classify smooth -principal 2-bundles over .
If is the automorphism 2-group of a Lie group then these are equivalently smooth -gerbes over .
Notice that a morphism of smooth -groupoids is presented by an 2-anafunctor of 2-groupoid valued presheaves, given by a span
where is the Cech nerve 2-groupoid of some covering. The top morphism here encodes degree-1 nonabelian Cech hypercohomology with coefficients in .
An first exposition is in the lecture notes
- Alissa Crans, A survey of higher Lie theory (pdf)
A general review of Lie 2-groups, as well as a discussion of the example of the string 2-group is in
Discussion in a more comprehensive context is in
with an introduction in section 1.3.1 and a general abstract discussion in 3.3.2.
On the cohomology of Lie 2-groups: