# Orbispaces as differentiable stratified spaces

## TL;DR

This paper explores the smooth and stratified structures of orbispaces derived from proper Lie groupoids, emphasizing Morita invariance and clarifying existing theories with alternative frameworks.

## Contribution

It provides a detailed exposition of the differentiable stratified space structure on orbispaces, extending existing theories and clarifying subtle points in the literature.

## Key findings

- Morita invariance of stratification structures
- Extension of stratification theory from Lie group actions to orbispaces
- Comparison of multiple frameworks for differentiable structures

## 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.