  Lie theory for the integration of Lie algebroids to Lie groupoids, on the one
hand, and of Poisson manifolds to symplectic groupoids, on the other, has
undergone tremendous developements in the last decade, thanks to the work of
Mackenzie-Xu, Moerdijk-Mrcun, Cattaneo-Felder and Crainic-Fernandes, among
others. In this thesis we study - part of - the categorified version of this
story, namely the integrability of LA-groupoids (groupoid objects in the
category of Lie algebroids), to double Lie groupoids (groupoid objects in the
category of Lie groupoids) providing a first set of sufficient conditions for
the integration to be possible. Mackenzie's double Lie structures arise
naturally from lifting processes, such as the cotangent lift or the path
prolongation, on ordinary Lie theoretic and Poisson geometric objects and we
use them to study the integrability of quotient Poisson bivector fields, the
relation between "local" and "global" duality of Poisson groupoids and Lie
theory for Lie bialgebroids and Poisson groupoids.
