On the Van Est homomorphism for Lie groupoids

The Van Est homomorphism for a Lie groupoid $G \rightrightarrows M$, as introduced by Weinstein-Xu, is a cochain map from the complex $C^\infty(BG)$ of groupoid cochains to the Chevalley-Eilenberg complex $C(A)$ of the Lie algebroid $A$ of $G$. It was generalized by Weinstein, Mehta, and Abad-Crainic to a morphism from the Bott-Shulman-Stasheff complex $\Omega(BG)$ to a (suitably defined) Weil algebra $W(A)$. In this paper, we will give an approach to the Van Est map in terms of the Perturbation Lemma of homological algebra. This approach is used to establish the basic properties of the Van Est map. In particular, we show that on the normalized subcomplex, the Van Est map restricts to an algebra morphism.