  A general class of loop quantizations for anisotropic models is introduced
and discussed, which enhances loop quantum cosmology by relevant features seen
in inhomogeneous situations. The main new effect is an underlying lattice which
is being refined during dynamical changes of the volume. In general, this leads
to a new feature of dynamical difference equations which may not have constant
step-size, posing new mathematical problems. It is discussed how such models
can be evaluated and what lattice refinements imply for semiclassical behavior.
Two detailed examples illustrate that stability conditions can put strong
constraints on suitable refinement models, even in the absence of a fundamental
Hamiltonian which defines changes of the underlying lattice. Thus, a large
class of consistency tests of loop quantum gravity becomes available. In this
context, it will also be seen that quantum corrections due to inverse powers of
metric components in a constraint are much larger than they appeared recently
in more special treatments of isotropic, free scalar models where they were
artificially suppressed.
