Deformations of VB-groupoids

VB-groupoids can be understood as vector bundles in the category of Lie groupoids. They encompass several classical objects, such as Lie group representations, 2-vector spaces, Lie group actions on vector bundles; moreover, they provide geometric pictures for 2-term representations up to homotopy of Lie groupoids, in particular the adjoint representation. In this talk, I will attach to every VB-groupoid a cochain complex controlling its deformations and discuss some features, such as Morita invariance, as well as some examples and applications. I will also compare it to its infinitesimal counterpart, the linear deformation complex of a VB-algebroid, via a Van Est map. This is joint work with Luca Vitagliano.