  Applications of Riemannian quantum geometry to cosmology have had notable
successes. In particular, the fundamental discreteness underlying quantum
geometry has led to a natural resolution of the big bang singularity. However,
the precise mathematical structure underlying loop quantum cosmology and the
sense in which it implements the full quantization program in a symmetry
reduced model has not been made explicit. The purpose of this paper is to
address these issues, thereby providing a firmer mathematical and conceptual
foundation to the subject.
