Every singular foliation has an associated topological groupoid, called holonomy groupoid [1]. In this note we exhibit some functorial properties of this assignment: if a foliated manifold (M, F M ) is the quotient of a foliated manifold (P, 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.
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