Cohomology of Lie Groupoid Modules and the Generalized van Est Map

Abstract The van Est map is a from Lie groupoid cohomology (with respect to sheaf taking values in representation) algebroid cohomology. We generalize the allow for more general sheaves, namely sheaves of sections (smooth or holomorphic) $G$-module, where $G$-modules are structures, which differentiate representations. Many geometric structures involving groupoids and stacks classified by not representations, including $S^1$-groupoid extensions equivariant gerbes. Examples such $\mathcal{O}^*$ $\mathcal{O}^*(*D)\,,$ latter invertible meromorphic functions with poles along divisor $D\,.$ show that there an infinitesimal description corresponding then define generalized relating these cohomologies study its kernel image. Applications include integration several extensions, actions on gerbes, certain $\infty $-algebroids.