(joint with Daniel Berwick-Evans) In the Lie groupoid literature there are two standard constructions that associate a Lie algebra to a Lie groupoid: global sections of its Lie algebroid and Mackenzie and Xu's multiplicative vector fields. Neither of these Lie algebras is preserved under Morita equivalence of Lie groupoids.
We show this pair of Lie algebras assemble into a Lie 2-algebra that is Morita invariant: given two Morita equivalent Lie groupoids their Lie 2-algebras are related by a (version of) Noohi's butterfly. Furthermore, the underlying category of this Lie 2-algebra is naturally equivalent to Hepworth's category of vector fields on the stack associated to the Lie groupoid.