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In differential geometry, Lie algebroids generalize on one hand Lie algebras, on the other hand the tangent bundle of a manifold: they are vector bundles equipped with an anchor map, i.e. a vector bundle morphism to the tangent bundle, and a Lie algebra structure on the space of sections subject to certain Leibniz rules. The integrated version of a Lie algebroid is a Lie groupoid. A purely algebraic version is a Lie-Rinehart algebra.
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Continuous and smooth Lie groupoid cohomology
Sorry for answering so late, I just was pointed to this question. Before answering your questions in the case of a Lie group (i.e., one object) let me point out that the difference between $\mathbb{R …