This talk is based on joint work with Matias del Hoyo. The idea is to explain an explicit, direct, and rigorous construction of the higher Lie algebroid underlying any higher Lie groupoid. We identify a key ideal of cochains which has the information of what differentiation means in simplicial and geometric terms. This part of the results entails two main theorems: a normal form one and one characterizing the quotient by the ideal, also yielding a novel generalization of the van Est differentiation map. Finally, we describe our third theorem which is a generalization to the higher context of the classical "van Est isomorphism theorem" in cohomology.