Orbifolds and groupoids

TL;DR

This paper establishes a correspondence between complex orbifold atlases and proper étale effective groupoids via a 2-functor, demonstrating a bijection between their equivalence classes.

Contribution

It constructs a 2-functor linking orbifold atlases to groupoids and proves this induces a bijection between their equivalence classes, extending the understanding of orbifold structures.

Findings

The 2-functor F maps orbifold atlases to groupoids.

F induces a bijection between equivalence classes.

The work extends the relation between orbifolds and groupoids.

Abstract

We define a 2-category structure (Pre-Orb) on the category of reduced complex orbifold atlases. We construct a 2-functor F from (Pre-Orb) to the 2-category (Grp) of proper \'etale effective groupoid objects over the complex manifolds. Both on (Pre-Orb) and (Grp) there are natural equivalence relations on objects: (a natural extension of) equivalence of orbifold atlases in (Pre-Orb) and Morita equivalences in (Grp). We prove that F induces a bijection between the equivalence classes of its source and target.