Lie groupoids as generalized atlases (original) (raw)
Related papers
On the linearization theorem for proper Lie groupoids
2011
We revisit the linearization theorems for proper Lie groupoids around general orbits (statements and proofs). In the the fixed point case (known as Zung's theorem) we give a shorter and more geometric proof, based on a Moser deformation argument. The passing to general orbits (Weinstein) is given a more conceptual interpretation: as a manifestation of Morita invariance. We also clarify the precise conditions needed for the theorem to hold (which often have been misstated in the literature).
Lie groupoids of mappings taking values in a Lie groupoid
Archivum Mathematicum, 2020
Endowing differentiable functions from a compact manifold to a Lie group with the pointwise group operations one obtains the so-called current groups and, as a special case, loop groups. These are prime examples of infinite-dimensional Lie groups modelled on locally convex spaces. In the present paper, we generalise this construction and show that differentiable mappings on a compact manifold (possibly with boundary) with values in a Lie groupoid form infinite-dimensional Lie groupoids which we call current groupoids. We then study basic differential geometry and Lie theory for these Lie groupoids of mappings. In particular, we show that certain Lie groupoid properties, like being a properétale Lie groupoid, are inherited by the current groupoid. Furthermore, we identify the Lie algebroid of a current groupoid as a current algebroid (analogous to the current Lie algebra associated to a current Lie group). To establish these results, we study superposition operators C ℓ (K, f) : C ℓ (K, M) → C ℓ (K, N), γ → f • γ between manifolds of C ℓ-functions. Under natural hypotheses, C ℓ (K, f) turns out to be a submersion (an immersion, an embedding, proper, resp., a local diffeomorphism) if so is the underlying map f : M → N. These results are new in their generality and of independent interest.
Nonassociative analogs of Lie groupoids
2021
We introduce nonassociative geometric objects generalising naturally Lie groupoids, called (smooth) quasiloopoids and loopoids. We prove that the tangent bundles of smooth loopoids is canonically a smooth loopoid (which is nontrivial in case of loopoids) and this is untrue for the cotangent bundles. After providing a few natural constructions, we show how the Lie-like functor associates with these objects skew-algebroids and almost Lie algebroids, respectively, and how discrete mechanics on Lie groupoids can be reformulated in the nonassociative case.
In Ehresmann's footsteps: from Group Geometries to Groupoid Geometries
arXiv: Differential Geometry, 2007
The geometric understanding of Cartan connections led Charles Ehresmann from the Erlangen program of (abstract) transformation groups to the enlarged program of Lie groupoid actions, via the basic concept of structural groupoid acting through the fibres of a (smooth) principal fibre bundle or of its associated bundles, and the basic examples stemming from the manifold of jets (fibred by its source or target projections). We show that the remarkable relation arising between the actions of the structural group and the structural groupoid (which are mutually determined by one another and commuting) may be viewed as a very special (unsymmetrical!) instance of a general fully symmetric notion of "conjugation between principal actions" and between "associated actions", encapsulated in a nice "butterfly diagram". In this prospect, the role of the local triviality looks more incidental, and may be withdrawn, allowing to encompass and bring together much more general situations. We describe various examples illustrating the ubiquity of this concept in Differential Geometry, and the way it unifies miscellaneous aspects of fibre bundles and foliations.
Turkish Journal of Mathematics
In this work we constitute the category of coverings of the Lie fundamental groupoid associated with a connected smooth manifold. We show that this category is equivalent to the category of universal coverings of a connected smooth manifold. In addition, we prove the equivalence of the category of coverings of a Lie groupoid and the category of actions of this Lie groupoid on a connected smooth manifold. Also we present two side results related to actions of Lie groupoids on the manifolds and coverings of Lie groupoids.
Coverings and Actions of Structured Lie Groupoids I
2009
In this work we deal with coverings and actions of Lie group- groupoids being a sort of the structured Lie groupoids. Firstly, we define an action of a Lie group-groupoid on some Lie group and the smooth coverings of Lie group-groupoids. Later, we show the equivalence of the category of smooth actions of Lie group-groupoids on Lie groups and the category of smooth cov- erings of Lie group-groupoids. Further, we prove a theorem which denotes how is obtained a covering Lie group-groupoid and a smooth covering morphism of Lie group-groupoids from a Lie group-groupoid.
Differentiable stratified groupoids and a de Rham theorem for inertia spaces
Journal of Geometry and Physics
We introduce the notions of a differentiable groupoid and a differentiable stratified groupoid, generalizations of Lie groupoids in which the spaces of objects and arrows have the structures of differentiable spaces, respectively differentiable stratified spaces, compatible with the groupoid structure. After studying basic properties of these groupoids including Morita equivalence, we prove a de Rham theorem for locally contractible differentiable stratified groupoids. We then focus on the study of the inertia groupoid associated to a proper Lie groupoid. We show that the loop and the inertia space of a proper Lie groupoid can be endowed with a natural Whitney B stratification, which we call the orbit Cartan type stratification. Endowed with this stratification, the inertia groupoid of a proper Lie groupoid becomes a locally contractible differentiable stratified groupoid. Contents 1. Introduction 2. Fundamentals 2.1. Topological groupoids 2.2. Differentiable groupoids 2.3. Differentiable stratified groupoids 2.4. Morita equivalence 3. Examples of differentiable stratified groupoids 4. The algebroid of a differentiable stratified groupoid 5. A de Rham theorem 6. The inertia groupoid of a proper Lie groupoid as a differentiable stratified groupoid 6.1. The inertia groupoid of a proper Lie groupoid 6.2. The stratification of the loop space 6.3.
Groupoids: unifying internal and external symmetry
Eprint Arxiv Math 9602220, 1996
The aim of this paper is to explain, mostly through examples, what groupoids are and how they describe symmetry. We will begin with elementary examples, with discrete symmetry, and end with examples in the differentiable setting which involve Lie groupoids and their corresponding infinitesimal objects, Lie algebroids.
A differentiable monoid of smooth maps on Lie groupoids
arXiv (Cornell University), 2017
In this article we investigate a monoid of smooth mappings on the space of arrows of a Lie groupoid and its group of units. The group of units turns out to be an infinite-dimensional Lie group which is regular in the sense of Milnor. Furthermore, this group is closely connected to the group of bisections and the geometry of the Lie groupoid. Under suitable conditions, i.e. if the source map of the Lie groupoid is proper, one also obtains a differentiable structure on the monoid and can identify the bisection group as a Lie subgroup of its group of units. Finally, relations between the (sub-)groupoids associated to the underlying Lie groupoid and subgroups of the monoid are obtained. The key tool driving the investigation is a generalisation of a result by A. Stacey. In the present article, we establish this so called Stacey-Roberts Lemma. It asserts that pushforwards of submersions are submersions between the infinite-dimensional manifolds of mappings. The Stacey-Roberts Lemma is of independent interest as it provides tools to study submanifolds of and geometry on manifolds of mappings.