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I was wondering if there is a smooth (sophisticated) way to associate the algebra 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?

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  • $\begingroup$ I think the construction you describe is just $D^*O(N_{\bullet})$, where $D^*$ is the left adjoint to the denormalisation functor from cdgas to cosimplicial algebras. [This construction for turning higher Artin stacks into higher algebriods features in my papers on Poisson structures.] $\endgroup$ Feb 14, 2018 at 8:23

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