We introduce a notion of metric on a Lie groupoid, compatible with multiplication, and we study its properties. We show that many families of Lie groupoids admit such metrics, including the important class of proper Lie groupoids. The exponential map of these metrics allows us to establish a linearization theorem for Riemannian groupoids, obtaining both a simpler proof and a stronger version of...

This paper develops a generalized formulation of Lagrangian mechanics on fibered manifolds, together with a reduction theory for symmetries corresponding to Lie groupoid actions. As special cases, this theory includes not only Lagrangian reduction (including reduction by stages) for Lie group actions, but also classical Routh reduction, which we show is naturally posed in this fibered setting. ...

We propose a notion of 1-homotopy for generalized maps. This notion generalizes those of natural transformation and ordinary homotopy for functors. The 1-homotopy type of a Lie groupoid is shown to be invariant under Morita equivalence. As an application we consider orbifolds as groupoids and study the notion of orbifold 1-homotopy type induced by a 1-homotopy between presentations of the orbif...

We provide a framework for extensions of Lie algebroids, including non-abelian extensions and Lie algebroids over different bases. Our approach involves Ehresmann connections, which allows straight generalizations of classical constructions. We exhibit a filtration in cohomology and explain the associated spectral sequence. We also give a description of the groupoid integrating an extension in ...

We extend the notion of the cardinality of a discrete groupoid (equal to the Euler characteristic of the corresponding discrete orbifold) to the setting of Lie groupoids. Since this quantity is an invariant under equivalence of groupoids, we call it the volume of the associated stack rather than of the groupoid itself. Since there is no natural measure in the smooth case like the counting measu...

The purpose of this paper is to prove two theorems which concern the position of étale groupoids among general smooth (or ”Lie”) groupoids. Our motivation comes from the non-commutative geometry and algebraic topology concerning leaf spaces of foliations. Here, one is concerned with invariants of the holonomy groupoid of a foliation [4, 34], such as the cohomology of its classifying space [14],...

We introduce the notions of differentiable groupoids and stratified groupoids, generalizations Lie in which spaces objects arrows have structures spaces, respectively compatible with groupoid structure. After studying basic properties these including Morita equivalence, we prove a de Rham theorem for proper locally contractible groupoids. then focus on study inertia associated to groupoid. show...

Abstract We show that the 2d Poisson Sigma Model on a groupoid arises as an effective theory of 3d Courant associated with double underlying Lie bialgebroid. This field-theoretic result follows from Lie-theoretic one involving coisotropic reduction odd cotangent bundle by generalized space algebroid paths. also provide several examples, including case symplectic groupoids in which we relate rea...

This note gives a categorical development arising from a theorem of A.A. Klyachko relating the Lie operad to roots of unity. We examine the "substitude" structure on the groupoid C whose homsets are the cyclic groups. The roots of unity representations of the cyclic groups form a Lie algebra for a certain oplax monoidal structure on the category of linear representations of C . Mathematics Subj...

For a finitely presented discrete group $\Gamma$, we introduce two generalizations of the orbifold Euler characteristic and $\Gamma$-orbifold to class proper topological groupoids large enough include all cocompact Lie groupoids. The $\Gamma$-Euler is defined as an integral with respect over orbit space groupoid, $\Gamma$-inertia usual associated groupoid. A key ingredient application o-minimal...