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.
| Original language | English |
|---|---|
| Pages (from-to) | 805-859 |
| Number of pages | 55 |
| Journal | Letters in Mathematical Physics |
| Volume | 108 |
| Issue number | 3 |
| DOIs | |
| Publication status | Published - 1 Mar 2018 |
Funding
The authors would like to thank Rui Loja Fernandes, David Martinez Torres and Hessel Posthuma for useful discussions related to this paper. The first author was supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek—Vici Grant No. 639.033.312. The second author was supported by Fundação para a Ciência e Tecnologia grant SFRH/BD/71257/2010 under the POPH/FSE programmes and by the Centre for Mathematics of the University of Coimbra—UID/MAT/00324/2013, funded by the Portuguese Government through FCT/MEC and co-funded by the European Regional Development Fund through the Partnership Agreement PT2020. Open access funding provided by Max Planck Society.
Keywords
- Lie groupoids
- Morita types
- Orbispaces
- Stacks