Representations up to homotopy of Lie groupoids were introduced by Arias Abad and Crainic offering a conceptually clear setting to define the adjoint representation of a Lie groupoid. A representation up to homotopy (ruth) is defined as a certain differential graded module, so one can talk about the cohomology of a ruth. A result of Gracia Saz and Mehta says that any ruth on a 2-term graded vector bundle corresponds a geometric object called a VB-groupoid. In this seminar, we will show that the cohomology of a 2-term ruth of a Lie groupoid is a Morita invariant, hence yielding a cohomology of the associated differentiable stack. Then I will discuss how this extends to ruth’s on arbitrary graded bundles. The talk is based on a joint work with M. del Hoyo (Rio de Janeiro) and ongoing project with M. Del Hoyo (Rio de Janeiro) and Fernando Studzinski (Sao Paulo).

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