Representations up to homotopy and Bottʼs spectral sequence for Lie groupoids

In this paper we study the notion of representation up to homotopy of a Lie groupoid and the resulting derived category, and show that the adjoint representation is well defined as a representation up to homotopy. As an application, we extend Bottʼs spectral sequence converging to the cohomology of classifying spaces of Lie groups to the case of Lie groupoids. We explain the relation of this construction with the models of Cartan and Getzler for equivariant cohomology. Our work is closely related to and inspired by Behrendʼs [3], Bottʼs [4], and Getzlerʼs [9].