Lie groupoids and their orbispaces

This is a concise introduction to the theory of Lie groupoids, with emphasis in their role as models for orbispaces. After some preliminaries, we review the foundations on Lie groupoids, and we carefully study equivalences and proper groupoids. Orbispaces are geometric objects which have manifolds and orbifolds as special instances, and can be presented as the transverse geometry of a Lie groupoid. Two Lie groupoids are equivalent if they are presenting the same orbispace, and proper groupoids are presentations of separated orbispaces, which by the linearization theorem are locally modeled by linear actions of compact groups. We discuss all these notions in detail. Our treatment diverges from the expositions already in the literature, looking for a complementary insight over this rich theory that is still in development.