  Given a Lie algebroid we discuss the existence of a smooth abelian
integration of its abelianization. We show that the obstructions are related to
the extended monodromy groups introduced recently in \cite{CFMb}. We also show
that this groupoid can be obtained by a path-space construction, similar to the
Weinstein groupoid of \cite{CF1}, but where the underlying homotopies are now
supported in surfaces with arbitrary genus. As an application, we show that the
prequantization condition for a (possibly non-simply connected) manifold is
equivalent to the smoothness of an abelian integration. Our results can be
interpreted as a generalization of the classical Hurewicz theorem.
