Quotients of singular foliations and Lie 2-group actions

Abstract

Androulidakis–Skandalis (2009) showed that every singular foliation has an associated topological groupoid, called holonomy groupoid. In this note, we exhibit some functorial properties of this assignment: if a foliated manifold $(M,{\mathcal{F}}_M)$ is the quotient of a foliated manifold $(P,{\mathcal{F}}_P)$ along a surjective submersion with connected fibers, then the same is true for the corresponding holonomy groupoids. For quotients by a Lie group action, an analogue statement holds under suitable assumptions, yielding a Lie 2-group action on the holonomy groupoid.