We introduce the category of I-Poisson manifolds: Its objects are just Poisson manifolds (P,Pi) together with appropriate ideals I---generalizing coisotropic submanifolds to the singular setting---but its morphisms are an important relaxation of Poisson maps. This permits one to consider an algebraic generalization of coisotropic reduction.

I-Poisson maps are now precisely those maps which induce morphisms of Poisson algebras between the corresponding reductions.

Every singular foliation on M gives rise to an I-Poisson manifold on P=T*M. We prove that Hausdorff-Morita equivalent singular foliations have isomorphic I-Poisson reductions. Further applications of the framework arise when adding Riemannian metrics into the game. We also provide examples of the Poisson algebras that arise in such a reduction, like the one for rotations.

In a second part, we present a construction which provides a graded-commutative algebra resolution for every module resolution of an ideal I in a commutative ring R. We show that this can be used for I-Poisson manifolds to construct 1. a graded Poisson manifold that can be identified with the BFV phase space for the singular constrained system and 2. a BFV charge whose degree zero cohomology provides a cohomological resolution of the I-Poisson reduction.

This is work in progress, the first part with H. Nahari, the second part with A. Hancharuk and with C. Laurent-Gengoux.