Representations up to homotopy and Bott's spectral sequence for Lie groupoids

Our aim here is to introduce and study the notion of representation up to homotopy of Lie groupoids, the resulting derived category, and to show that the adjoint representation is well defined as a representation up to homotopy. As an application, we extend Bott’s spectral sequence converging to the cohomology of classifying spaces from Lie groups to Lie groupoids. Our work is closely related to and inspired by Behrend [3], Bott [5], and Getzler [10].