  A framework for higher gauge theory based on a 2-group is presented, by
constructing a groupoid of connections on a manifold acted on by a 2-group of
gauge transformations, following previous work by the authors where the general
notion of the action of a 2-group on a category was defined. The connections
are discretized, given by assignments of 2-group data to 1- and 2-cells coming
from a given cell structure on the manifold, and likewise the gauge
transformations are given by 2-group assignments to 0-cells. The 2-cells of the
manifold are endowed with a bigon structure, matching the 2-dimensional algebra
of squares which is used for calculating with 2-group data. Showing that the
action of the 2-group of gauge transformations on the groupoid of connections
is well-defined is the central result. The effect, on the groupoid of
connections, of changing the discretization is studied, and partial results and
conjectures are presented around this issue. The transformation double category
that arises from the action of a 2-group on a category, as defined in previous
work by the authors, is described for the case at hand, where it becomes a
transtormation double groupoid. Finally, examples of the construction are given
for simple choices of manifold: the circle, the 2-sphere and the torus.
