Lie Groupoids and Differentiable Stacks

TL;DR

This paper introduces the theory of Lie groupoids and differentiable stacks, emphasizing their roles in modeling geometric objects like manifolds and orbifolds, and explores their equivalences and properties.

Contribution

It provides a detailed, alternative perspective on Lie groupoids and differentiable stacks, focusing on their equivalences, proper groupoids, and local models, expanding understanding beyond existing literature.

Findings

Proper groupoids present separated stacks.

Linearization theorem models proper groupoids locally by linear actions.

Lie groupoids serve as models for differentiable stacks.

Abstract

This is a concise introduction to the theory of Lie groupoids, with emphasis in their role as models for stacks. After some preliminaries, we review the foundations on Lie groupoids, and we carefully study equivalences and proper groupoids. Differentiable stacks 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 stack, and proper groupoids are presentations of separated stacks, 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.