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.

中文翻译：

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作为可微分层空间的轨道空间

我们提出了一个适当的李群群在轨道空间上的光滑结构和典型分层的一些特征。主要特征之一是这些结构的 Morita 不变性——它允许我们讨论由适当李群表示的轨道空间（类似于代数几何中的分离堆栈的对象）上可微分层空间的规范结构。轨道空间上的规范光滑结构主要通过 Spallek 的可微空间框架进行研究，然后提出了两种替代框架。对于轨道空间上的规范分层，我们扩展了来自适当李群作用的类似理论。我们不主张原创性。