[Paper Review] Lie Groupoids and their Orbispaces

del Hoyo, L Matías|arXiv (Cornell University)|Dec 30, 2012

Homotopy and Cohomology in Algebraic Topology14 references4 citations

TL;DR

This paper presents a comprehensive introduction to Lie groupoids as foundational tools for modeling orbispaces—geometric objects that generalize manifolds and orbifolds. It establishes equivalences between Lie groupoids as presentations of the same orbispace and shows that proper groupoids present separated orbispaces, locally modeled by linear actions of compact groups, offering a novel, complementary perspective on this evolving theory.

ABSTRACT

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.

Motivation & Objective

• To provide a clear, self-contained introduction to Lie groupoids and their role in modeling orbispaces.
• To clarify the notion of equivalence between Lie groupoids as presenting the same orbispace.
• To characterize proper Lie groupoids as presentations of separated orbispaces.
• To establish the connection between proper groupoids and local models via linear actions of compact groups.
• To offer a complementary perspective to existing literature, enriching the understanding of an ongoing development in differential geometry.

Proposed method

• Reviewing foundational concepts of Lie groupoids, including source and target maps, and the smooth structure of the space of arrows.
• Defining and analyzing groupoid equivalences through Morita equivalence, ensuring they present the same transverse geometry.
• Introducing proper Lie groupoids via the properness of the diagonal map, ensuring good geometric behavior.
• Applying the linearization theorem to show that proper groupoids are locally modeled on linear actions of compact Lie groups.
• Using transverse geometry to interpret orbispaces as quotients of Lie groupoids, generalizing the notion of orbifolds.
• Emphasizing geometric intuition and structural clarity over technical formalism, diverging from standard expositions to offer new insight.

Experimental results

Research questions

• RQ1 How do Lie groupoids serve as models for orbispaces, and what conditions ensure they present the same geometric object?
• RQ2 What characterizes an equivalence between two Lie groupoids in the context of orbispace presentation?
• RQ3 In what sense do proper Lie groupoids represent separated orbispaces, and how does this relate to local structure?
• RQ4 How does the linearization theorem connect proper groupoids to linear actions of compact groups?
• RQ5 What new geometric and categorical insights does this approach offer compared to existing treatments of orbifolds and orbispaces?

Key findings

• Equivalence of Lie groupoids corresponds precisely to presenting the same orbispace, establishing a Morita equivalence framework.
• Proper Lie groupoids are shown to present separated orbispaces, ensuring well-behaved quotient structures.
• The linearization theorem implies that proper groupoids are locally modeled on linear actions of compact Lie groups, providing a local classification.
• Orbispaces generalize both manifolds and orbifolds, with Lie groupoids serving as their natural geometric models.
• The paper's approach diverges from standard literature by emphasizing structural clarity and geometric intuition, offering a complementary perspective on an active research area.
• Transverse geometry of a Lie groupoid captures the essential features of the associated orbispace, making it a central tool in the theory.