  The mathematical structure of homogeneous loop quantum cosmology is analyzed,
starting with and taking into account the general classification of homogeneous
connections not restricted to be Abelian. As a first consequence, it is seen
that the usual approach of quantizing Abelian models using spaces of functions
on the Bohr compactification of the real line does not capture all properties
of homogeneous connections. A new, more general quantization is introduced
which applies to non-Abelian models and, in the Abelian case, can be mapped by
an isometric, but not unitary, algebra morphism onto common representations
making use of the Bohr compactification. Physically, the Bohr compactification
of spaces of Abelian connections leads to a degeneracy of edge lengths and
representations of holonomies. Lifting this degeneracy, the new quantization
gives rise to several dynamical properties, including lattice refinement seen
as a direct consequence of state-dependent regularizations of the Hamiltonian
constraint of loop quantum gravity. The representation of basic operators -
holonomies and fluxes - can be derived from the full theory specialized to
lattices. With the new methods of this article, loop quantum cosmology comes
closer to the full theory and is in a better position to produce reliable
predictions when all quantum effects of the theory are taken into account.
