  For the minimization of state-based systems (i.e. the reduction of the number
of states while retaining the system's semantics), there are two obvious
aspects: removing unnecessary states of the system and merging redundant states
in the system. In the present article, we relate the two aspects on coalgebras
by defining an abstract notion of minimality. The abstract notion minimality
and minimization live in a general category with a factorization system. We
will find criteria on the category that ensure uniqueness, existence, and
functoriality of the minimization aspects. The proofs of these results
instantiate to those for reachability and observability minimization in the
standard coalgebra literature. Finally, we will see how the two aspects of
minimization interact and under which criteria they can be sequenced in any
order, like in automata minimization.
