Formal geometry and Tamarkin--Tsygan calculi of dg manifolds

The main goal of this paper is to study the formal geometry of dg manifolds à la Fedosov. For any dg manifold $(\mathcal{M},Q)$, we construct a Fedosov dg foliation (or dg Lie algebroid) ${\mathcal{F}}_Q\to {\mathcal{N}}_Q$. We establish homotopy contractions between their respective spaces of polyvector fields, differential forms, polydifferential operators, and polyjets. As a consequence, we prove that their respective Cartan calculi and noncommutative calculi, in the sense of Tamarkin–Tsygan, are isomorphic.