  Given a representation up to homotopy of a Lie algebroid on a 2-term complex
of vector bundles, we define the corresponding holonomy as a strict 2-functor
from a Weinstein path 2-groupoid to the gauge 2-groupoid of the underlying
2-term complex. We construct a corresponding transformation 2-groupoid and we
prove that the 1-truncation of this 2-groupoid is isomorphic to the Weinstein
groupoid of the VB-algebroid associated to a representation up to homotopy. As
applications, we describe alternative integration schemes for semi-direct
products of Lie 2-algebras and string algebras.
