A duality between Lie algebroids and infinitesimal foliations

There are two natural analogues of algebraic foliations in derived algebraic geometry, called partition Lie algebroids and infinitesimal derived foliations, and both make sense in general characteristics. We construct an equivalence between these two notions under some finiteness conditions. Our method is refining the PD Koszul duality in \cite{BM}\cite{BCN} using the (completed) Hodge filtration.