Fedosov dg manifolds associated with Lie pairs

Given any pair (L, A) of Lie algebroids, we construct a differential graded manifold (L[1] ⊕ L/ A, Q) , which we call Fedosov dg manifold. We prove that the homological vector field Q constructed on L[1] ⊕ L/ A by the Fedosov iteration method arises as a byproduct of the Poincaré–Birkhoff–Witt map established in [18]. Finally, using the homological perturbation lemma, we establish a quasi-isomorphism of Dolgushev–Fedosov type: the differential graded algebras of functions on the dg manifolds (A[1] , d_A) and (L[1] ⊕ L/ A, Q) are homotopy equivalent.