  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.

Differential Geometry