Singular Riemannian foliations (SRF) defined by Molino are given by a compatibility condition between the metric and a well-behaved leaf decomposition of the Riemannian manifold, but it does not see the other aspects of singular foliations defined by Androulidakis-Skandalis. We present the notion of module SRFs adapted to this definition, and define the notion of Morita equivalence for SRFs such that the underlying singular foliations are Hausdorff Morita equivalent as recently introduced by Garmendia and Zambon.
Then we introduce the category of I-Poisson manifolds where its objects are just Poisson manifolds together with appropriate ideals of smooth functions, but its morphisms are an important relaxation of Poisson maps. We show that every singular foliation gives rise to an I-Poisson manifold and SRFs can be translated into this setting. This perspective permits us to construct an algebraic invariant of Hausdorff Morita equivalence for singular foliations. This is a joint work in progress with T. Strobl.