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 maps of groupoid structure are continuous. The notion of monodromy groupoid of a topological groupoid generalizes those of fundamental groupoid and universal cover. It was earlier proved that the monodromy groupoid o...

We introduce singular subalgebroids of an integrable Lie algebroid, extending the notion subalgebroid by dropping constant rank requirement. lay bases a theory for subalgebroids: we construct associated holonomy groupoids, adapting procedure Androulidakis-Skandalis foliations, in way that keeps track choice groupoid integrating ambient algebroid. The groupoids are topological and suitable nonco...

In this work we deal with actions of vector groupoid which is a new concept in the literature‎. ‎After we give the definition of the action of a vector groupoid on a vector space‎, ‎we obtain some results related to actions of vector groupoids‎. ‎We also apply some characterizations of the category and groupoid theory to vector groupoids‎. ‎As the second part of the work‎, ‎we define the notion...

We give a new construction of the holonomy groupoid regular foliation in terms partial connection on diffeological principal bundle germs transverse parametrisations, which may be viewed as systematisation Winkelnkemper’s original using ideas from gauge theory. extend these to construct novel for any foliated bundle, we prove sits at top hierarchy jet groupoids associated with bundle. This show...

In the present paper we study some geometric objects and constructions related to groupoids of matrix subalgebras in a fixed matrix algebra. 1. The definition of groupoids In this paper we will freely use the notation from [3]. First recall some definitions and constructions [3]. 1.1. Groupoids Gk, l. Let Mkl(C) be the complex matrix algebra. Unital ∗-subalgebras isomorphic Mk(C) in some unital...

Let $\widetilde{X}$ be a smooth Riemannian manifold equipped with proper, free, isometric and cocompact action of discrete group $\Gamma$. In this paper we prove that the analytic surgery exact sequence Higson-Roe for is isomorphic to associated adiabatic deformation Lie groupoid $\widetilde{X}\times_\Gamma\widetilde{X}$. We then generalize result context smoothly stratified manifolds. Finally,...

Abstract Let $G$ be a connected complex semi-simple Lie group and ${\mathcal {B}}$ its flag variety. For every positive integer $n$, we introduce Poisson groupoid over ${{\mathcal {B}}}^n$, called the $n$th total configuration of flags $G$, which contains family sub-groupoids whose spaces are generalized double Bruhat cells bases Schubert in {B}}^n$. Certain symplectic leaves these then shown t...

We develop the theory of multiplicative Ehresmann connections for Lie groupoid submersions covering identity, as well their infinitesimal counterparts. construct obstructions to existence such connections, and we prove several interesting classes groupoids algebroids, including all proper groupoids. show that many notions from principal bundle have analogues in this general setup, 1-forms, curv...

Abstract We define a new notion of fiberwise linear differential operator on the total space vector bundle E . Our main result is that operators are equivalent to (polynomial) derivations an appropriate line over E ? {E^{\ast}} believe this might represent first step towards definition m...

In this paper, a notion of principal $2$-bundle over Lie groupoid has been introduced. For such $2$-bundles, we produced short exact sequence VB-groupoids, namely, the Atiyah sequence. Two notions connection structures viz. strict connections and semi-strict on arising respectively, from retraction up to natural isomorphism have We constructed class $\mathbb{G}=[G_1\rightrightarrows G_0]$-bundl...