Quasifold groupoids and diffeological quasifolds

TL;DR

This paper develops a categorical framework for quasifolds and their groupoids, showing an equivalence between effective quasifold groupoids and their diffeological orbit spaces, extending previous work.

Contribution

It introduces a bicategory of quasifold groupoids and proves an equivalence with diffeological quasifolds under certain conditions, extending earlier results.

Findings

Effective quasifold groupoids are categorically equivalent to their diffeological orbit spaces.

The bicategory of quasifold groupoids embeds into the bicategory of Lie groupoids.

The results extend previous work on the structure of quasifolds and their groupoids.

Abstract

Quasifolds are spaces that are locally modelled by quotients of $\mathbb{R}^n$ by countable affine group actions. These spaces first appeared in Elisa Prato's generalization of the Delzant construction, and special cases include leaf spaces of irrational linear flows on the torus, and orbifolds. We consider the category of diffeological quasifolds, which embeds in the category of diffeological spaces, and the bicategory of quasifold groupoids, which embeds in the bicategory of Lie groupoids, bibundles, and bibundle morphisms. We prove that, restricting to those morphisms that are locally invertible, and to quasifold groupoids that are effective, the functor taking a quasifold groupoid to its diffeological orbit space is an equivalence of the underlying categories. These results complete and extend earlier work with Masrour Zoghi.