I was wondering if there is a smooth (sophisticated) way to associate algebras of global functions of formal groupoid associated to Lie-Rinehart algebra (considered as 1-stack) to its Chevalley-Eilenberg complex, that could be generalized to Lie algebroids and even Lie infinity algebroids.

So far, (at least when a Lie-Rinehart A-algebra L is free as A-module) I believe I can prove that they are quasi-isomorphic chain complexes by hand. I take the nerve $ N_\bullet$ of the formal groupoid, and use the cosimplicial Dold-Kan equivalence on $ \mathcal{O}(N_\bullet)$ to compute its homotopy colimit. I get A in degree zero and the (n-th tensor product of the) dual $ U_A(L)^\vee$ of the universal enveloping algebra in degree one (or $ n$ respectively). Then I can project this to the CE(L) by moding out the higher order guys in $ U_A(L)^\vee$ , and this projection seems to be qis.

However, this approach seems very rough and and uneasy to extend. Is anybody aware of a better way to do this, or even a simple argument proving that the two chain complexes are indeed quasi-isomorphic?