The category of reduced orbifolds in local charts

TL;DR

This paper introduces a new categorical framework for reduced smooth orbifolds using marked proper effective étale Lie groupoids, establishing an isomorphism of categories that enhances the understanding of orbifold morphisms.

Contribution

It defines a new category of reduced smooth orbifolds with maps, showing it is isomorphic to a category of marked proper effective étale Lie groupoids, strengthening categorical equivalences.

Findings

Established a category of orbifolds with morphisms

Proved the isomorphism between orbifold and Lie groupoid categories

Enhanced categorical understanding of reduced orbifolds

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.