A groupoid approach to spaces of generalized connections

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.