Categorified central extensions, étale Lie 2-groups and Lieʼs Third Theorem for locally exponential Lie algebras

Wockel, Christoph

doi:10.1016/j.aim.2011.07.003

Cited by 12 publications

(12 citation statements)

References 46 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The above notion of central extension of smooth 2-group is more general then the notion introduced by Wockel [66]. In particular it is invariant under equivalence of smooth 2-group, and includes the following examples not covered by Wockel's treatment.…”

mentioning

confidence: 91%

Central extensions of smooth 2–groups and a finite-dimensional string 2–group

Schommer-Pries

2011

Geom. Topol.

126

View full text Add to dashboard Cite

We provide a model of the String group as a central extension of finite-dimensional 2-groups in the bicategory of Lie groupoids, left-principal bibundles, and bibundle maps. This bicategory is a geometric incarnation of the bicategory of smooth stacks and generalizes the more naive 2-category of Lie groupoids, smooth functors and smooth natural transformations. In particular this notion of smooth 2-group subsumes the notion of Lie 2-group introduced by Baez and Lauda [5]. More precisely we classify a large family of these central extensions in terms of the topological group cohomology introduced by Segal [56], and our String 2-group is a special case of such extensions. There is a nerve construction which can be applied to these 2-groups to obtain a simplicial manifold, allowing comparison with the model of Henriques [23]. The geometric realization is an A 1 -space, and in the case of our model, has the correct homotopy type of String.n/. Unlike all previous models [58; 60; 33; 23; 7] our construction takes place entirely within the framework of finitedimensional manifolds and Lie groupoids. Moreover within this context our model is characterized by a strong uniqueness result. It is a canonical central extension of Spin.n/. 57T10, 22A22, 53C08; 18D10

show abstract

mentioning

confidence: 91%

Central extensions of smooth 2–groups and a finite-dimensional string 2–group

Schommer-Pries

2011

Geom. Topol.

126

View full text Add to dashboard Cite

show abstract

“…quotients of smooth manifolds by almost free actions of non-compact Lie groups, and leaf spaces of foliated manifolds.Étale differentiable stacks have been studied by various authors, c.f. [20,23,31,14,13,32,30,5,7]. Higherétale differentiable stacks are higher categorical analogues ofétale differentiable stacks allowing not only points to have automorphisms, but also the automorphisms of points to have automorphisms themselves, and so on.…”

Section: Introductionmentioning

confidence: 99%

On the homotopy type of higher orbifolds and Haefliger classifying spaces

Carchedi¹

2016

Advances in Mathematics

View full text Add to dashboard Cite

We describe various equivalent ways of associating to an orbifold, or more generally a higherétale differentiable stack, a weak homotopy type. Some of these ways extend to arbitrary higher stacks on the site of smooth manifolds, and we show that for a differentiable stack X arising from a Lie groupoid G, the weak homotopy type of X agrees with that of BG. Using this machinery, we are able to find new presentations for the weak homotopy type of certain classifying spaces. In particular, we give a new presentation for the Borel construction M × G EG of an almost free action of a Lie group G on a smooth manifold M as the classifying space of a category whose objects consists of smooth maps R n → M which are transverse to all the G-orbits, where n = dim M − dim G. We also prove a generalization of Segal's theorem, which presents the weak homotopy type of Haefliger's groupoid Γ q as the classifying space of the monoid of self-embeddings of R q , B (Emb (R q )) , and our generalization gives analogous presentations for the weak homotopy type of the Lie groupoids Γ Sp 2q and RΓ q which are related to the classification of foliations with transverse symplectic forms and transverse metrics respectively.

show abstract

“…Note that this concept of "locally smooth" Lie 2-group is only known to be equivalent to the one mentioned above in very special cases [WW13] that do not govern the situation we have in this paper. The value of the current article is that it extends the result of [Woc11a] to a global one. One can do this because we weaken (in a certain sense) the category of "smooth 2-spaces" from [Woc11a], which is nothing but the category of Lie groupoids with strict morphisms, to the bicategory of smooth stacks which is equivalent to the one of Lie groupoids with generalized morphisms and 2-morphisms.…”

mentioning

confidence: 95%

“…We obtain a version of Lie's Third Theorem asserting that each locally exponential Lie algebra (see Definition A.4) with topologically split center integrates to anétale Lie 2-group. The same question was studied in [Woc11a] by a completely different and less powerful concept of Lie 2-group, since it only admits a notion of smoothness "near the identity". Note that this concept of "locally smooth" Lie 2-group is only known to be equivalent to the one mentioned above in very special cases [WW13] that do not govern the situation we have in this paper.…”

mentioning

confidence: 99%

“…One can do this because we weaken (in a certain sense) the category of "smooth 2-spaces" from [Woc11a], which is nothing but the category of Lie groupoids with strict morphisms, to the bicategory of smooth stacks which is equivalent to the one of Lie groupoids with generalized morphisms and 2-morphisms. To obtain our result we then have enhanced the approach from [Woc11a] significantly because generalized morphisms are more involved than the strict ones. This effort is eventually rewarded by obtaining a globally smooth object integrating an infinite-dimensional Lie algebra.…”

mentioning

confidence: 99%

See 1 more Smart Citation

Integrating central extensions of Lie algebras via Lie 2-groups

Wockel¹,

Zhu²

2016

J. Eur. Math. Soc.

Self Cite

View full text Add to dashboard Cite

The purpose of this paper is to show how central extensions of (possibly infinitedimensional) Lie algebras integrate to central extensions ofétale Lie 2-groups in the sense of [Get09,Hen08]. In finite dimensions, central extensions of Lie algebras integrate to central extensions of Lie groups, a fact which is due to the vanishing of π 2 for each finite-dimensional Lie group. This fact was used by Cartan (in a slightly other guise) to construct the simply connected Lie group associated to each finitedimensional Lie algebra.In infinite dimensions, there is an obstruction for a central extension of Lie algebras to integrate to a central extension of Lie groups. This obstruction comes from non-trivial π 2 for general Lie groups. We show that this obstruction may be overcome by integrating central extensions of Lie algebras not to Lie groups but to central extensions ofétale Lie 2-groups. As an application, we obtain a generalization of Lie's Third Theorem to infinite-dimensional Lie algebras.

show abstract

Categorified central extensions, étale Lie 2-groups and Lieʼs Third Theorem for locally exponential Lie algebras

Cited by 12 publications

References 46 publications

Central extensions of smooth 2–groups and a finite-dimensional string 2–group

Central extensions of smooth 2–groups and a finite-dimensional string 2–group

On the homotopy type of higher orbifolds and Haefliger classifying spaces

Integrating central extensions of Lie algebras via Lie 2-groups

Contact Info

Product

Resources

About