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Monday, October 27, 2014

**Abstract:** The notion of representation for a Lie groupoid has the annoying problem that it isn't generally possible to define a good adjoint representation. To fix this problem, Arias Abad and Crainic introduced the notion of "representation up to homotopy". In this talk, I will show how 2-term representations up to homotopy are related to linear groupoid structures, which play the role of semidirect products. There is a one-to-one correspondence at the level of isomorphism classes, but at the level of objects, the correspondence is noncanonical, so it is possible for certain constructions to be "natural" in one perspective but not the other. A key example that illustrates the value of linear groupoids is the adjoint representation. To define the adjoint representation up to homotopy of a Lie groupoid G, one needs to choose a distribution transverse to the source fibers. On the other hand, the linear groupoid that corresponds to the adjoint representation is canonical; it is simply the tangent bundle TG. This talk is based on joint work with Alfonso Gracia-Saz (arXiv:1007.3658).