# Quotients of singular foliations and Lie 2-group actions

## TL;DR

This paper explores how the holonomy groupoid of a singular foliation behaves under quotients, showing functorial properties and the emergence of Lie 2-group actions in certain cases.

## Contribution

It establishes functorial relationships for holonomy groupoids under quotients and introduces Lie 2-group actions in the context of foliated quotients.

## Key findings

- Holonomy groupoids are functorial under certain quotients.
- Quotients by Lie group actions induce Lie 2-group actions on holonomy groupoids.
- The results generalize the understanding of foliation quotients and their symmetry structures.

## Abstract

Every singular foliation has an associated topological groupoid, called holonomy groupoid (see arXiv:math/0612370). 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 analog statement holds under suitable assumptions, yielding a Lie 2-group action on the holonomy groupoid.