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

摘要

This belongs to a series of papers devoted to the study of the cohomology ofclassifying spaces of Lie groupoids. Our aim here is to introduce and study thenotion of representation up to homotopy of Lie groupoids, the resulting derivedcategory, and to show that the adjoint representation is well defined as arepresentation up to homotopy. As an application, we extend Bott's spectralsequence converging to the cohomology of classifying spaces from Lie groups toLie groupoids.