[Paper Review] Convenient categories of reduced orbifolds

Anke Pohl|arXiv (Cornell University)|Jan 5, 2010

Homotopy and Cohomology in Algebraic Topology3 references3 citations

TL;DR

This paper introduces a new category of reduced smooth orbifolds using marked proper effective étale Lie groupoids, achieving an isomorphism between the category of orbifolds and the groupoid category—offering a stricter equivalence than previous categorical equivalences. The construction ensures precise correspondence via explicit marking of groupoid structures.

ABSTRACT

It is well-known that reduced smooth orbifolds and proper effective foliation Lie groupoids form equivalent categories. However, for certain recent lines of research, equivalence of categories is not sufficient. We propose a notion of maps between reduced smooth orbifolds and a definition of a category in terms of marked proper effective \'etale Lie groupoids such that the arising category of orbifolds is isomorphic (not only equivalent) to this groupoid category.

Motivation & Objective

• To address the limitation of categorical equivalence in recent orbifold research, where isomorphism is required for structural precision.
• To define a new category of reduced smooth orbifolds that is isomorphic, not just equivalent, to the category of marked proper effective étale Lie groupoids.
• To provide a framework where orbifold maps are rigorously encoded through groupoid morphisms with explicit marking conditions.
• To ensure that the category of orbifolds captures the full structure of the underlying groupoids without loss or ambiguity.

Proposed method

• Introduces the concept of 'marked' proper effective étale Lie groupoids to encode orbifold maps precisely.
• Defines a category of reduced smooth orbifolds using these marked groupoids as objects and compatible morphisms as arrows.
• Establishes a strict isomorphism between the category of orbifolds and the category of marked groupoids.
• Uses the étale and proper conditions on Lie groupoids to ensure smoothness and properness of the orbifold structure.
• Applies the notion of 'reduced' orbifolds to eliminate redundant automorphisms and simplify the groupoid representation.
• Ensures that the groupoid morphisms respect the marking, which encodes the orbifold map data faithfully.

Experimental results

Research questions

• RQ1 How can a category of reduced smooth orbifolds be constructed such that it is isomorphic to a category of groupoids, rather than merely equivalent?
• RQ2 What marking conditions on proper effective étale Lie groupoids are necessary and sufficient to represent orbifold maps faithfully?
• RQ3 In what way does the use of marked groupoids improve the categorical description of orbifolds compared to standard equivalence?
• RQ4 How does the isomorphism between the orbifold category and the marked groupoid category resolve structural ambiguities in existing formulations?
• RQ5 What conditions ensure that the groupoid-based construction fully captures the geometric and differentiable structure of reduced orbifolds?

Key findings

• The category of reduced smooth orbifolds is isomorphic to the category of marked proper effective étale Lie groupoids, providing a strict categorical correspondence.
• The marking condition on groupoid morphisms ensures that orbifold maps are represented without ambiguity or redundancy.
• The construction resolves the issue of categorical equivalence by achieving isomorphism, which is essential for certain modern applications in orbifold geometry.
• The use of étale and proper groupoids preserves the smooth and proper structure of the orbifolds, ensuring geometric consistency.
• The framework allows for a precise, structure-preserving description of orbifold maps through groupoid morphisms with explicit data.
• The reduced condition eliminates non-trivial automorphisms that do not contribute to the orbifold's geometric data, simplifying the category.