Van Est maps and the perturbation lemma

This talk is intended to be an educational lecture to learn about the Perturbation Lemma and the Van Est maps. The classical Van Est theory relates the smooth cohomology of Lie groups with the cohomology of the associated Lie algebra, or its relative versions. Some aspects of this theory generalize to Lie groupoids and their Lie algebroids. We will revisit the van Est theory using the Perturbation Lemma from homological algebra. Using this technique, we will obtain precise results for the van Est differentiation and integrations maps at the level of cochains. Specifically, we will construct homotopy inverses to the van Est differentiation maps that are right inverses at the cochain level.