We will describe a quasi-isomorphism between two cochain complexes: a recently introduced one computing the linear cohomology of the tensor products of vector bundles over groupoids, and the complex of an associated tensor product of representations up to homotopy. The construction is inspired by and generalizes the analogous quasi-isomorphism for the complexes of differential forms on a Lie groupoid.
This is based on joint work with Lennart Obster and Luca Vitagliano.