Bicategories of Action Groupoids

TL;DR

This paper establishes the equivalence of three different bicategory constructions of action Lie groupoids through localization techniques, generalizing known orbifold cases and analyzing weak equivalences.

Contribution

It proves the equivalence of three bicategory models of action Lie groupoids via localization, extending results from orbifold groupoids and characterizing weak equivalences.

Findings

The three localization methods produce equivalent bicategories.

Any weak equivalence decomposes into two specific equivariant weak equivalences.

Generalization of orbifold groupoid results to action Lie groupoids.

Abstract

We prove that the 2-category of action Lie groupoids localised in the following three different ways yield equivalent bicategories: localising at equivariant weak equivalences \`a la Pronk, localising using surjective submersive equivariant weak equivalences and anafunctors \`a la Roberts, and localising at all weak equivalences. These constructions generalise the known case of representable orbifold groupoids. We also show that any weak equivalence between action Lie groupoids is isomorphic to the composition of two particularly nice forms of equivariant weak equivalences.