nLab
Lie 2-group

Context

Group Theory

\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

Contents

Idea

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.

Examples

Constructions and Applications

Delooping 2-groupoids

By the discussion at looping and delooping, every Lie 2-group GG induces a delooping Lie 2-groupoid BG\mathbf{B}G: this has a single object, the space of morphisms is G 0G_0, the space of 2-morphisms is G 1G_1 and the horizontal composition is given by the group product.

Principal 2-bundles

For XX a smooth manifold (or itself a Lie groupoid such as an orbifold, or generally any smooth ∞-groupoid), morphisms

g:XBG g : X \to \mathbf{B}G

of smooth ∞-groupoids from XX to the delooping Lie 2-groupoid BG\mathbf{B}G classify smooth GG-principal 2-bundles over XX.

If G=AUT(H)G = AUT(H) is the automorphism 2-group of a Lie group HH then these are equivalently smooth HH-gerbes over XX.

Notice that a morphism of smooth \infty-groupoids XBGX \to \mathbf{B}G is presented by an 2-anafunctor of 2-groupoid valued presheaves, given by a span

C(U i) g BG X, \array{ C(U_i) &\stackrel{g}{\to}& \mathbf{B}G \\ \downarrow^{\simeq} \\ X } \,,

where C(U i)C(U_i) is the Cech nerve 2-groupoid of some covering. The top morphism here encodes degree-1 nonabelian Cech hypercohomology with coefficients in GG.

References

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:

Revised on February 6, 2013 23:53:21 by Urs Schreiber (82.113.106.234)