  The quantum completion of the space of connections in a manifold can be seen
as the set of all morphisms from the groupoid of the edges of the manifold to
the (compact) gauge group. This algebraic construction generalizes an analogous
description of the gauge-invariant quantum configuration space of Ashtekar and
Isham, clarifying the relation between the two spaces. We present a description
of the groupoid approach which brings the gauge-invariant degrees of freedom to
the foreground, thus making the action of the gauge group more transparent.
