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This thesis deals with deformations of VB-algebroids and VB-groupoids. Theycan be considered as vector bundles in the categories of Lie algebroids andgroupoids and encompass several classical objects, including Lie algebra andLie group representations, 2-vector spaces and the tangent and the cotangentalgebroid (groupoid) to a Lie algebroid (groupoid). Moreover, they aregeometric models for some kind of representations of Lie algebroids(groupoids), namely 2-term representations up to homotopy. Finally, it is wellknown that Lie groupoids are "concrete" incarnations of differentiable stacks,hence VB-groupoids can be considered as representatives of vector bundles overdifferentiable stacks, and VB-algebroids their infinitesimal versions. In thiswork, we attach to every VB-algebroid and VB-groupoid a cochain complexcontrolling its deformations, their linear deformation complex. Moreover, thedeformation complex of a VB-algebroid is equipped with a DGLA structure. Thebasic properties of these complexes are discussed: their relationship with thedeformation complexes of the total spaces and the base spaces, particular casesand generalizations. The main theoretical results are a linear van Est theorem,that gives conditions for the linear deformation cohomology of a VB-groupoid tobe isomorphic to that of the corresponding VB-algebroid, and a Moritainvariance theorem, that implies that the linear deformation cohomology of aVB-groupoid is really an algebraic invariant of the associated vector bundle ofdifferentiable stacks. Finally, several examples are discussed, showing how thelinear deformation cohomologies are related to other well-known cohomologies.

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