  It is the goal of this article to extend the notion of quantization from the
standard interpretation focused on non-commuting observables defined starting
from classical analogues, to the topological equivalents defined in terms of
coefficient groups in (co)homology. It is shown that the commutation relations
between quantum observables become (non)compatibility relations between
coefficient groups. The main result is the construction of a new, higher-level
form of quantization, as seen from the perspective of the universal coefficient
theorem. This idea brings us closer to a consistent quantization of gravity,
allows for a systematic description of topology changing string interactions
but also gives new, quantum-topological degrees of freedom in discussions
involving quantum information. On the practical side, a possible connection to
the fractional quantum Hall effect is explored.
