# Van Est differentiation and integration

## TL;DR

This paper revisits the Van Est theory relating Lie group cohomology to Lie algebra cohomology, extending it to Lie groupoids and algebroids using homological algebra techniques to obtain precise cochain-level results.

## Contribution

It introduces a homological algebra approach to the Van Est theory, providing explicit homotopy inverses at the cochain level for differentiation and integration maps.

## Key findings

- Constructed homotopy inverses to Van Est differentiation maps
- Extended Van Est theory to Lie groupoids and algebroids
- Achieved precise cochain-level results

## Abstract

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. In this paper, continuing an idea from [18], we revisit the van Est theory using the Perturbation Lemma from homological algebra. Using this technique, we obtain precise results for the van Est differentiation and integrations maps at the level of cochains. Specifically, we construct homotopy inverses to the van Est differentiation maps that are right inverses at the cochain level.