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 Li-Bland and Meinrenken (Enseign Math 61(1–2):93–137, 2015), 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.

中文翻译：

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Van Est 差异化和整合

经典的 Van Est 理论将李群的平滑上同调与相关的李代数或其相关版本的上同调联系起来。该理论的某些方面推广到李群群及其李代数。在本文中，我们延续 Li-Bland 和 Meinrenken（Enseign Math 61(1-2):93-137, 2015）的想法，使用同调代数中的微扰引理重新审视 van Est 理论。使用这种技术，我们获得了 cochains 级别的 van Est 分化和集成图的精确结果。具体来说，我们构建了 van Est 微分图的同伦逆，这是在 Cochain 级别的右逆。