Deformations of vector bundles in the categories of Lie algebroids and groupoids

This thesis deals with deformations of VB-algebroids and VB-groupoids. They can be considered as vector bundles in the categories of Lie algebroids and groupoids and encompass several classical objects, including Lie algebra and Lie group representations, 2-vector spaces and the tangent and the cotangent algebroid (groupoid) to a Lie algebroid (groupoid). Moreover, they are geometric models for some kind of representations of Lie algebroids (groupoids), namely 2-term representations up to homotopy. Finally, it is well known that Lie groupoids are concrete incarnations of differentiable stacks, hence VB-groupoids can be considered as representatives of vector bundles over differentiable stacks, and VB-algebroids their infinitesim