Orbispaces as differentiable stratified spaces

Abstract

We present some features of the smooth structure, and of the canonical stratification on the orbit space of a proper Lie groupoid. One of the main features is that of Morita invariance of these structures - it allows us to talk about the canonical structure of differentiable stratified space on the orbispace (an object analogous to a separated stack in algebraic geometry) presented by the proper Lie groupoid. The canonical smooth structure on an orbispace is studied mainly via Spallek's framework of differentiable spaces, and two alternative frameworks are then presented. For the canonical stratification on an orbispace, we extend the similar theory coming from proper Lie group actions. We make no claim to originality. The goal of these notes is simply to give a complementary exposition to those available, and to clarify some subtle points where the literature can sometimes be confusing, even in the classical case of proper Lie group actions.

arXiv comment49 pages. Final version to appear in Letters in Mathematical Physics. Changed title; corrected example 4.16

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• Singular symplectic cotangent bundle reduction of gauge field theory
• Orbifold Euler characteristics of non-orbifold groupoids
• Stratification of the transverse momentum map
• On Fredholm boundary conditions on manifolds with corners I: Global corner's cycles obstructions
• Commutative Control Data for Smoothly Locally Trivial Stratified Spaces