On the cohomology of representations up to homotopy of Lie groupoids - Carvalho, Fernando Studzinski (2019)
We study the concept of representations up to homotopy of Lie groupoids. Our main result is the proof that the cohomology of a Lie groupoid with coefficients in a representation up to homotopy is a Morita invariant of the groupoid. This can be interpreted as a way to provide cohomological invariants for orbifolds and more generally for differentiable stacks, which are spaces with singularities whose isomorphism classes are in one-to-one correspondence with Morita equivalence classes of Lie groupoids. To prove this result, we rely on the theory of simplicial objects in smooth categories e.g. simplicial manifolds, sim- plicial vector bundles, and equivalences between them which are defined in terms of maps called hypercovers. We also prove results on the invariance of the simplicial cohomology of these spaces under hypercovers.