Representations up to homotopy and cohomology of classifying spaces

Abstract

The classifying space BG of a Lie group G can be thought of as the homotopy quotient of the least free action imaginable, the action of $G$ on a point. The cohomology of BG is related in a very precise sense to the representations on the polynomials in the Lie algebra. This relation makes it possible to construct infinitesimal models for the cohomology of BG, which in turn generalize to the models for equivariant cohomology. This thesis is the study of how these relations can be extended to more general homotopy quotients, classifying spaces of Lie groupoids. The conclusion is that most of the generalization goes through, provided one works in the context of representations up to homotopy.