We approach Mackenzie's LA-groupoids from a supergeometric point of view by introducing Q-groupoids, which are groupoid objects in the category of Q-manifolds. There is a faithful functor from the category of LA-groupoids to the category of Q-groupoids. We associate to every Q-groupoid a double complex that provides a …
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In this note a functorial approach to the integration problem of an LA-groupoid to a double Lie groupoid is discussed. To do that, we study the notions of fibred products in the categories of Lie groupoids and Lie algebroids, giving necessary and sufficient conditions for the existence of such. In particular, it turns …
Unified framework for Morita invariant cohomology of Lie groupoids.
problem Proving Morita invariance of cohomology theories for Lie groupoids.
method Viewing cohomology as sheaves of modules on the nerve of the groupoid and establishing criteria for Morita invariance.
result Established criteria for Morita invariant cohomology theories.
The article proves Lie's third theorem for Lie algebroids via singular Lie groupoids.
problem Lie's third theorem does not hold for Lie groupoids and Lie algebroids.
method Introducing a subcategory of diffeological spaces called quasi-etale, constructing a functor mapping singular Lie groupoids to Lie algebroids.
result Lie's third theorem is valid for Lie algebroids within the context of singular Lie groupoids.
New approach to principal groupoid bundles with connections using dg-Lie groupoids.
problem Developing a new perspective on principal bundles with connections.
method Using dg-Lie groupoids and additional adjustment data for Lie groupoids.
result Adjusted connections provide a global formulation of curved Yang-Mills-Higgs theories.
We address the question of duality for the dynamical Poisson groupoids of Etingof and Varchenko over a contractible base. We also give an explicit description for the coboundary case associated with the solutions of the classical dynamical Yang-Baxter equation on simple Lie algebras as classified by the same authors. O…
In this paper we study the Lie groupoids which appear in foliation theory. A foliation groupoid is a Lie groupoid which integrates a foliation, or, equivalently, whose anchor map is injective. The first theorem shows that, for a Lie groupoid G, the following are equivalent: - G is a foliation groupoid, - G has discrete…
In this thesis, we employ simplicial methods to study actions, principal bundles, and bibundles of higher groupoids. Roughly, we use Kan fibrations to model actions of higher groupoids, we use pairs of a Kan fibration and a special acyclic fibration to model principal bundles of higher groupoids, we use inner Kan fibra…
The algebraic part of approach to groupoids started by S. Zakrzewski is presented.
Researchers extend a groupoid approach to calculate Wodzicki residue and Kontsevich-Vishik trace.
problem Calculating Wodzicki residue and Kontsevich-Vishik trace for pseudo-differential operators of any order.
method Groupoid approach to pseudo-differential operators.
result Extension of van Erp and Yuncken's work to operators of any order.
Integrates Manin pairs to simplify Poisson and symplectic groupoid constructions.
problem Integrating Lie algebroids and bialgebroids to groupoids.
method Generalizes results using Manin pairs .
result Simplified geometric approach to integration results.
Given a Poisson (or more generally Dirac) manifold , there are two approaches to its geometric quantization: one involves a circle bundle over endowed with a Jacobi (or Jacobi-Dirac) structure; the other one involves a circle bundle with a (pre-) contact groupoid structure over the (pre-) symplectic groupoid…
Groupoids in Diffeologymath.DG
The paper explains the importance of diffeological groupoids in modern geometry and physics.
problem None explicitly stated, focuses on the concept and applications.
method Expository review of the development and applications of diffeological groupoids.
result The diffeological groupoid serves as a powerful tool in various areas of modern geometry and physics.
New proof of Lie algebroid action equivalence and integrability.
problem Equivalence between Lie algebroid action and integrability.
method Double Lie groupoids and multiplicative foliations.
result Simple characterization of double Lie groupoids inducing Lie groupoid structures.
The purpose of this article is to study Ezra Getzler's approach to the Atiyah-Singer index theorem from the perspective of Alain Connes' tangent groupoid. We shall construct a "rescaled" spinor bundle on the tangent groupoid, define a convolution operation on its smooth, compactly supported sections, and explain how th…
New Hausdorff integrations for Lie algebroids and symplectic groupoids.
problem Integrating Lie algebroids and symplectic groupoids.
method Hausdorff versions of Lie Integration Theorems 1 and 2, Lie equivalences, and algebraic approach to holonomy.
result Generalization of integration of subalgebroids to non-wide cases and detailed exploration of foliation groupoids.
New algebraic approach for approximating Hamiltonian dynamics.
problem Approximating Hamilton-Jacobi solutions on symplectic groupoids.
method Introducing a pre-Lie algebra and Butcher trees for symplectic groupoids.
result New class of Poisson integrators for Hamiltonian dynamics.
Cosymplectic groupoidsmath.SG
Study on cosymplectic groupoids with structural results.
problem Understanding cosymplectic groupoids and their properties.
method Structural analysis of cosymplectic groupoids, comparison with other groupoids.
result Several structural results for cosymplectic groupoids.
Using Fedosov's approach we give a geometric construction of a formal symplectic groupoid over any Poisson manifold endowed with a torsion-free Poisson contravariant connection. In the case of Kaehler-Poisson manifolds this construction provides, in particular, the formal symplectic groupoids with separation of variabl…
Defines Wodzicki residue using groupoids and fibered distributions.
problem Defining and understanding the Wodzicki residue in noncommutative geometry.
method Using groupoid language and filtered manifolds, defining the residue and showing its properties.
result The groupoidal residue is a trace on pseudodifferential operators and matches the usual residue in certain cases.
Groupoids help define Riemann sums on manifolds.
problem Defining Riemann sums on compact manifolds.
method Using groupoids and the van Est map.
result Riemann sums converge to the usual integral.
Solves the invariant linearization problem for Lie groupoids.
problem Understanding invariant linearization for Lie groupoids.
method Introduced a counter-example and a sufficient criterion using compatible complete metrics and covers of proper group actions.
result Proved a sufficient criterion for invariant linearization of Lie groupoids.
PB-groupoids vs VB-groupoidsmath.DG
Paper defines PB-groupoids and their relation to VB-groupoids.
problem Establishing PB-groupoids as a Lie 2-groupoid counterpart.
method Defining PB-groupoids as Lie 2-groupoid actions on diagrams of Lie groupoids and principal bundles.
result PB-groupoids generalize VB-groupoids and provide a new correspondence between them and principal bundles.
Defines Lie and Courant algebroids over Lie groupoids using homological vector fields.
problem Provides a Morita invariant definition of Lie and Courant algebroids over Lie groupoids.
method Views vector fields as Maurer-Cartan elements in a differential graded Lie algebra and as functors and natural transformations.
result Obtains a unifying conceptual framework for studying various algebraic structures.
The paper generalizes bundle gerbes over groupoids and their correspondence with PB groupoids.
problem Generalizing bundle gerbes over groupoids and their properties.
method Developed a functorial correspondence between PB groupoids and bundle gerbes over groupoids.
result Built a correspondence between PB groupoids and bundle gerbes over groupoids, including partial quotients.
We apply the bar construction to the nerve of a double Lie groupoid to obtain a local Lie 2-groupoid. As an application, we recover Haefliger's fundamental groupoid from the fundamental double groupoid of a Lie groupoid. In the case of a symplectic double groupoid, we study the induced closed 2-form on the associated l…
In these lectures notes I discuss the Linearization Theorem for Lie groupoids, and its relation to the various classical linearization theorems for submersions, foliations and group actions. In particular, I explain in some detail the recent metric approach to this problem.
In this paper we construct two groupoids from morphisms of groupoids, with one from a categorical viewpoint and the other from a geometric viewpoint. We show that for each pair of groupoids, the two kinds of groupoids of morphisms are equivalent. Then we study the automorphism groupoid of a groupoid.
A groupoid is a small category in which each morphism has an inverse. A topological groupoid is a groupoid in which both sets of objects and morphisms have topologies such that all groupoid structure maps are continuous. The notion of monodromy groupoid of a topological groupoid generalises those of fundamental groupoi…
This research reinterprets Lie and Cartan's work on geometric structures using Lie groupoids.
problem Revisiting Lie and Cartan's geometric structures from a modern perspective.
method Encoding geometric structures into principal -bundles with a transversally parallelisable foliation.
result Developed a notion of flatness for Lie groupoids encompassing various geometric structures.
We outline the construction of the holonomy groupoid of a locally Lie groupoid and the monodromy groupoid of a Lie groupoid. These specialise to the well known holonomy and monodromy groupoids of a foliation, when the groupoid is just an equivalence relation.
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 b…
New star-product defined on Poisson manifolds using Toeplitz operators.
problem Defining star-products on Poisson manifolds induced by symplectic Lie algebroids.
method Using Toeplitz operators on groupoids with Heisenberg group structure.
result Generalization of Guillemin and Melrose's symplectic approach.
Integrates transitive Lie algebroids to Lie groupoids, explaining obstructions.
problem Integrating transitive Lie algebroids to Lie groupoids.
method Geometric explanation and explicit construction of integration, with obstructions considered.
result Obstructions explained and integration constructed when they vanish.
Lie algebroids are by no means natural as an infinitesimal counterpart of groupoids. In this paper we propose a functorial construction called Nishimura algebroids for an infinitesimal counterpart of groupoids. Nishimura algebroids, intended for differential geometry, are of the same vein as Lawvere's functorial notion…
Review and generalize Haefliger's differentiable cohomology for diffeomorphisms and flat Cartan groupoids.
problem Define and investigate differentiable cohomology for diffeomorphisms and flat Cartan groupoids.
method Define Haefliger's differentiable cohomology for diffeomorphisms, investigate its structure, and generalize to flat Cartan groupoids.
result Define characteristic maps for geometric structures on manifolds associated to flat Cartan groupoids.
We discuss two sorts of generalization of Lie groupoids. One is Lie -groupoids defined as simplicial manifolds with trivial . The other is the stacky Lie groupoid $\cG\rra M$ with $\cG$ a differentiable stack. We build 1-1 correspondence between Lie 2-groupoids and stacky Lie groupoids up to a certain…
We complete the construction of the double Lie algebroid of a double Lie groupoid begun in the first paper of this title. We show that the Lie algebroid structure of an LA--groupoid may be prolonged to the Lie algebroid of its Lie groupoid structure; in the case of a double groupoid this prolonged structure for either …
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-grou…
Glanon groupoidsmath.DG
We introduce the notion of Glanon groupoids, which are Lie groupoids equipped with multiplicative generalized complex structures. It combines symplectic groupoids, holomorphic Lie groupoids and holomorphic Poisson groupoids into a unified framework. Their infinitesimal, Glanon Lie algebroids are studied. We prove that …
Solves open problem on Lie groupoids equivalence.
problem Whether Lie groupoids Morita equivalent are diffeologically Morita equivalent.
method Localisation of 2-categories, anafunctors, Lie groupoids, diffeological groupoids.
result Two Lie groupoids diffeologically Morita equivalent are Morita equivalent in the Lie sense.
Locally conformal symplectic (l.c.s.) groupoids are introduced as a generalization of symplectic groupoids. We obtain some examples and we prove that l.c.s. groupoids are examples of Jacobi groupoids in the sense of \cite{IM}. Finally, we describe the Lie algebroid of a l.c.s. groupoid.
This paper explains the fundamental relation between Jacobi structures and the classical Spencer operator coming from the theory of PDEs so as to provide a direct and geometric approach to the integrability of Jacobi structures. It uses recent results on the integrability of Spencer operators and multliplicative forms …
Characterizes algebraic integrability and minimality of Lie equations for non-commutative pseudogroups.
problem Understanding algebraic integrability and minimality of Lie equations for non-commutative pseudogroups.
method Algebraic characterization and differential Galois theory of rational connections.
result Equivalence of algebraic integrability to the triviality of the differential Galois group and demonstration of minimality under certain conditions.
Integrates singular subalgebroids using diffeological groupoids.
problem Integration of singular subalgebroids.
method Definition of integration via diffeological groupoids with specific properties.
result Holonomy groupoids correspond to singular subalgebroids with submersive property.
We discuss two generalizations of Lie groupoids. One consists of Lie -groupoids defined as simplicial manifolds with trivial . The other consists of stacky Lie groupoids $\cG\rra M$ with $\cG$ a differentiable stack. We build a 1-1 correspondence between Lie 2-groupoids and stacky Lie groupoids up to …
The paper studies deformations of symplectic groupoids using cohomology and spectral sequences.
problem Deformations of symplectic groupoids and their cohomology.
method Deformation cohomology, Moser path methods, Lie groupoids, multiplicative forms, de Rham models, spectral sequences.
result Computations and constructions of deformation cohomology for various types of symplectic groupoids.
Lie groupoids and their orbit spaces are linked through equivalence classes.
problem Understanding the relationship between Lie groupoids and their orbit spaces.
method Introducing lift-complete Lie groupoids and showing equivalence between categories.
result Morita equivalence class of a lift-complete Lie groupoid is determined by its orbit space.