Foliations, even if regular, can have a topologically very bad quotient space of leaves. An alternative would be a homotopy quotient and appropriate versions of multi-vector fields and forms thereon. Historically, this first occurred in 1983 with the work of Batalin-Fradkin-Vilkovisky on the foliation of a constraint surface in a symplectic manifold W by a set of first-class constraints. A key ingredient of the BFV construction is a resolution $K_{A/I}$ of $A/I$ for the `constraint' ideal $I$ in the commutative algebra $A = C^{\infty} (W)$ together with a Chevalley-Eilenberg complex $\kii$ for $I/I^{\ 2}$ as a Lie algebra.

This led to the strong homotopy version of Lie-Rinehart algebras, introduced by Kjeseth (1996)He constructed such an $\infty$-Lie-Rinehart algebra structure on $(K_{A/I},\kii)$.. Much later, such algebras were used to study general regular foliations by Huebschmann (2003) and then Vitagliano (2012).

On the other hand, singular foliations have been studied beginning around 2006, primarily from a holonomy point of view. Most recently, singular foliations were studied in Lavau's thesis (2016) from a homotopy point of view closer to that of BFV.

Our goal is to clarify the relation between these various constructions `up to homotopy'