The category of reduced orbifolds in local charts

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.