nLab Lie 2-group

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

• Every ordinary Lie group $G$ is in particular a Lie 2-group with $G$ as its space of objects and only identity morphisms.

• Every discrete 2-group is a Lie 2-group equipped with discrete smooth structure;

• For $A$ an abelian Lie group there is a Lie 2-group denoted $\mathbf{B}A$ or $(A \to 1)$, which has a single object and $A$ as its space of morphism.

  For $A = U(1)$ the circle group, the Lie 2-group $\mathbf{B}U(1)$ is the circle 2-group .

• Every crossed module of Lie groups $(H \to G )$, $G \to Aut(H)$ gives an example of a strict Lie 2-group, using the general relation between crossed modules and strict 2-groups.

  For instance the crossed module $Spin(n) \to O(n)$. Or $(U(1) \to 1)$. Etc.

• The string 2-group $String(G)$ has various different but equivalent incarnations as a Lie 2-group. One is given by the crossed module $\hat \Omega_* G \to P_* G$ internal to Frechet Lie groups.

• For $X$ a Lie groupoid, its automorphism infinity-group is a smooth 2-group.

Constructions and Applications

Delooping 2-groupoids

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

Principal 2-bundles

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

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

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

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

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

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

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

Related concepts

• group, Lie group, Lie groupoid

• 2-group, Lie 2-group, Lie 2-groupoid

  • Poisson Lie 2-group

• ∞-group, smooth ∞-group, smooth ∞-groupoid

References

A 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

• John Baez, Alissa Crans, Urs Schreiber, Danny Stevenson, From Loop Groups to 2-Groups (arXiv:math/0504123)

Discussion in a more comprehensive context is in

• Urs Schreiber, differential cohomology in a cohesive topos

with an introduction in section 1.3.1 and a general abstract discussion in 3.3.2.

On the cohomology of Lie 2-groups:

• Grégory Ginot, Ping Xu, Cohomology of Lie 2-groups (pdf)

• Christoph Wockel, Categorified central extensions, étale Lie 2-groups and Lieʼs Third Theorem for locally exponential Lie algebras (web)

• Chris Schommer-Pries, Central Extensions of Smooth 2-Groups and a Finite-Dimensional String 2-Group (arXiv)