Minimality Notions via Factorization Systems

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 using factorization systems. We will find criteria on the category that ensure uniqueness, existence, and functoriality of the minimization aspects, where the proofs instantiate to those for reachability and bisimilarity 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.