  This paper serves as a preparation of work that focuses on extracting
cosmological sectors from Loop Quantum Gravity. We start with studying the
extraction of subsystems from classical systems. A classical Hamiltonian system
can be reduced to a subsystem of ''relevant observables'' using the pull-back
under the Poisson-embedding of the ''relevant part of phase space'' into full
phase space. Since a quantum theory can be thought of as a noncommutative phase
space, one encounters the problem of embedding noncommutative spaces. We solve
this problem for a physically interesting set of quantum systems and embeddings
by constructing the noncommutative analogue of the construction of an embedding
as the projection to the base space of an embedding of fibre bundles over the
involved spaces. This paper focuses on the physical ideas that enter our
programme of reduction of quantum theories and tries to explain these on
examples rather than abstractly, which will be the focus of a forthcoming
paper.
