Lie Groupoids and Differentiable Stacks

Abstract

This is a concise introduction to the theory of Lie groupoids, with emphasis in their role as models for stacks. After some preliminaries, we review the foundations on Lie groupoids, and we carefully study equivalences and proper groupoids. Differentiable stacks are geometric objects which have manifolds and orbifolds as special instances, and can be presented as the transverse geometry of a Lie groupoid. Two Lie groupoids are equivalent if they are presenting the same stack, and proper groupoids are presentations of separated stacks, which by the linearization theorem are locally modeled by linear actions of compact groups. We discuss all these notions in detail. Our treatment diverges from the expositions already in the literature, looking for a complementary insight over this rich theory that is still in development.

39 pages. Essentially the same as previous version. We have just replaced the vague terms 'orbispace' and 'weak equivalence' by the more specific ones 'differentiable stack' and 'Morita map'

References in corpus (3)

• Topological and Smooth Stacks
• Homotopy Theory of Orbispaces
• Stability and invariants of Hilsum-Skandalis maps

Cited by in corpus (20)

• Vector bundles over Lie groupoids and algebroids
• Riemannian metrics on Lie groupoids
• Riemannian metrics on differentiable stacks
• Orbispaces as differentiable stratified spaces
• Morita equivalences of vector bundles
• Differential forms with values in VB-groupoids
• Faithful Semitoric Systems
• On the coadjoint Virasoro action
• Principal actions of stacky Lie groupoids
• Categorical approach to the Baum-Connes conjecture for étale groupoids
• Multiplicative Connections and Their Lie Theory
• On Hausdorff integrations of Lie algebroids
• On invariant linearization of Lie groupoids
• Geodesics on Riemannian stacks
• The general linear 2-groupoid
• Poly-Poisson Sigma models and their relational poly-symplectic groupoids
• Shifted coisotropic structures for differentiable stacks
• Stratification of the transverse momentum map
• Diffeological Morita Equivalence
• Higher Vector Bundles