Morphisms between spaces of leaves viewed as fractions (original) (raw)
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On Morphic Actions and Integrability of LA-Groupoids
Lie theory for the integration of Lie algebroids to Lie groupoids, on the one hand, and of Poisson manifolds to symplectic groupoids, on the other, has undergone tremendous developements in the last decade, thanks to the work of Mackenzie-Xu, Moerdijk-Mrčun, Cattaneo-Felder and Crainic-Fernandes, among others. In this thesis we study-part of-the categorified version of this story, namely the integrability of LA-groupoids (groupoid objects in the category of Lie algebroids), to double Lie groupoids (groupoid objects in the category of Lie groupoids) providing a first set of sufficient conditions for the integration to be possible. Mackenzie's double Lie structures arise naturally from lifting processes, such as the cotangent lift or the path prolongation, on ordinary Lie theoretic and Poisson geometric objects and we use them to study the integrability of quotient Poisson bivector fields, the relation between "local" and "global" duality of Poisson groupoids and Lie theory for Lie bialgebroids and Poisson groupoids. In the first Chapter we prove suitable versions of Lie's 1-st and 2-nd theorem for Lie bialgebroids, that is, the integrability of subobjects (coisotropic subalgebroids) and morphisms, extending earlier results by Cattaneo and Xu, obtained using different techniques. We develop our functorial approach to the integration of LA-groupoids [65] in the second Chapter, where we also obtain partial results, within the program, proposed by Weinstein, for the integration of Poisson groupoids to symplectic double groupoids. The task of integrating quotients of Poisson manifolds with respect to Poisson groupoid actions motivates the study we undertake in third Chapter of what we refer to as morphic actions, i.e. groupoid actions in the categories of Lie algebroids and Lie groupoids, where we obtain general reduction and integrability results. In fact, applying suitable proceduresà la Marsden-Weinstein zero level reduction to "moment morphisms", respectively of Lie bialgebroids or Poisson groupoids, canonically associated to a Poisson G-space, we derive two approches to the integration of the quotient Poisson bivector fields. The first, a kind of integration via symplectic double groupoids, is not always effective but reproduces the "symplectization functor" approch to Poisson actions of Lie groups, very recently developed by Fernandes-Ortega-Ratiu, from quite a different perspective. We earlier implemented this approach successfully in the special case of complete Poisson groups [64]. The second approach, relying both on a cotangent lift of the Poisson G-space and on a prolongation of the original action to an action on suitable spaces of Lie algebroid homotopies, produces necessary and sufficient integrability conditions for the integration and gives a positive answer to the integrability problem under the most natural assumptions. Könnte jeder brave Mann solche Glöcken finden, seine Feinde würden dann ohne Mühe schwinden 1 [ ? ] 2
Lie groupoids as generalized atlases
Central European Journal of Mathematics, 2004
Starting with some motivating examples (classical atlases for a manifold, space of leaves of a foliation, group orbits), we propose to view a Lie groupoid as a generalized atlas for the “virtual structure” of its orbit space, the equivalence between atlases being here the smooth Morita equivalence. This “structure” keeps memory of the isotropy groups and of the smoothness as well. To take the smoothness into account, we claim that we can go very far by retaining just a few formal properties of embeddings and surmersions, yielding a very polymorphous unifying theory. We suggest further developments.
On the Lie 2-algebra of sections of an LA-groupoid
Journal of Geometry and Physics, 2019
In this work we introduce the category of multiplicative sections of an LA-groupoid. We prove that these categories carry natural strict Lie 2-algebra structures, which are Morita invariant. As applications, we study the algebraic structure underlying multiplicative vector fields on a Lie groupoid and in particular vector fields on differentiable stacks. We also introduce the notion of geometric vector field on the quotient stack of a Lie groupoid, showing that the space of such vector fields is a Lie algebra. We describe the Lie algebra of geometric vector fields in several cases, including classifying stacks, quotient stacks of regular Lie groupoids and in particular orbifolds, and foliation groupoids.
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.
On (Co)morphisms of Lie Pseudoalgebras and Groupoids
Journal of Algebra, 2007
We give a unified description of morphisms and comorphisms of Lie pseudoalgebras, showing that the both types of morphisms can be regarded as subalgebras of a Lie pseudoalgebra, called the psi\psipsi-sum. We also provide similar descriptions for morphisms and comorphisms of Lie algebroids and groupoids.
The monodromy groupoid of a Lie groupoid
1995
The monodromy groupoid of a Lie groupoid Cahiers de topologie et géométrie différentielle catégoriques, tome 36, n o 4 (1995), p. 345-369. http://www.numdam.org/item?id=CTGDC\_1995\_\_36\_4\_345\_0 © Andrée C. Ehresmann et les auteurs, 1995, tous droits réservés. L'accès aux archives de la revue « Cahiers de topologie et géométrie différentielle catégoriques » implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ 345 THE MONODROMY GROUPOID OF A LIE GROUPOID by Ronald BROWN and Osman MUCUK CAHIERS DE TOPOLOGIE ET GEOMETRIE DIFFERENTIELLE CATEGORIQUES Volume XXXVI-4 (1995) Resume: Dans cet article, on montre que, sous des conditions générales, Punion disjointe des recouvrements universels des étoiles d'un groupoide de Lie a la structure d'un groupoide de Lie dans lequel la projection possède une propriete de monodromie pour les extensions des morphismes locaux reguliers. Ceci complete un rapport 46ta,iU6 de r6sultats announces par J. Pradines. of his PhD wor k at Bangor (Nlucuk [18]), and to the examiner s (Kirill Mackenzie and Mark Lawson), for helpful comments.
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.
Fundamentals of Lie categories
arXiv (Cornell University), 2023
We introduce the basic notions and present examples and results on Lie categories-categories internal to the category of smooth manifolds. Demonstrating how the units of a Lie category C dictate the behavior of its invertible morphisms G(C), we develop sufficient conditions for G(C) to form a Lie groupoid. We show that the construction of Lie algebroids from the theory of Lie groupoids carries through, and ask when the Lie algebroid of G(C) is recovered. We reveal that the lack of invertibility assumption on morphisms leads to a natural generalization of rank from linear algebra, develop its general properties, and show how the existence of an extension C → G of a Lie category to a Lie groupoid affects the ranks of morphisms and the algebroids of C. Furthermore, certain completeness results for invariant vector fields on Lie monoids and Lie categories with well-behaved boundaries are obtained. Interpreting the developed framework in the context of physical processes, we yield a rigorous approach to the theory of statistical thermodynamics by observing that entropy change, associated to a physical process, is a functor.
2000
Our definition of correspondence between groupoids (which generalizes the notion of homomorphism) is obtained by weakening the conditions in the definition of equivalence of groupoids in [5]. We prove that such a correspondence induces another one between the associated C*-algebras, and in some cases besides a Kasparov element. We wish to apply the results obtained in the particular case of K-oriented maps of leaf spaces in the sense of [3].
Spaces with local equivalence relations, and their monodromy
Topology and its Applications, 1996
We elaborate a suggestion of Grothendieck, and study the invariant sheaves for a local equivalence relation on a space (e.g., a foliation). One of our purposes is to compare this to the standard model for the leaf-(quotient-)space of a foliation, given by the holonomy groupoid. To this end, we prove that, under suitable connectedness assumptions, Grothendieck's invariant sheaves can be described in terms of a closely related, but different, "monodromy" groupoid. Our second purpose is to prove that every Ctale groupoid arises this way.