Perspectives on the Bott spectral sequence for Lie groupoids

The cohomology of a differentiable stack can be modelled by the Bott-Shulman-Stasheff bicomplex. However, computing its cohomology is still difficult in many examples. Improving on this requires additional methods such as using spectral sequences and finding models for the cohomology of each column of the bicomplex. This paper provides a first attempt of relating and unifying two approaches of this form. Firstly, Arias Abad has generalised the Bott spectral sequence to Lie groupoids by introducing representations up to homotopy. The second approach is the recent work on a model for the cohomology of differentiable stacks associated to proper and regular Lie groupoids by this author.