The Holonomy Groupoids of Singularly Foliated Bundles

We define a notion of connection in a fibre bundle that is compatible with a singular foliation of the base. Fibre bundles equipped with such connections are in plentiful supply, arising naturally for any Lie groupoid-equivariant bundle, and simultaneously generalising regularly foliated bundles in the sense of Kamber–Tondeur and singular foliations. We define hierarchies of diffeological holonomy groupoids associated to such bundles, which arise from the parallel transport of jet/germinal conservation laws. We show that the groupoids associated in this manner to trivial singularly foliated bundles are quotients of Androulidakis–Skandalis holonomy groupoids, which coincide with Androulidakis–Skandalis holonomy groupoids in the regular case. Finally we prove functoriality of all our constructions under appropriate morphisms.