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Li-yao Xia (Nov 20 2020 at 17:12):I'm looking for generalizations/miscellaneous insights related to this idea, mostly out of curiosity: in the category of (labelled) state transition systems and (forward) simulations, consider "determinization", that thing from automata theory which transforms a possibly nondeterministic state machine S into a deterministic one D(S) whose states are elements of the powerset P(S). Well, D is a monad (or looks very much like one at least). The construction is interesting because its Kleisli arrows "S simulates D(S')" coincide with trace inclusion "Traces(S) is a subset of Traces(S')", so we can kinda say that D is a monad because set inclusion is transitive.
Can this idea be adapted to the setting where simulations are coalgebra morphisms?
Also there is another way to "determinize" a state machine, rather than P(S), is to construct a machine whose states are sets of traces. It's a bit scary from the point of view of automata since it's a priori not obvious that this preserves finiteness, but if you don't care about finiteness, that's (more or less) bisimilar to the "sets of states" construction. If there is a categorical generalization for the previous construction, is there a generalization of this other one, and are they the same?
Liang-Ting Chen (Nov 25 2020 at 07:49):sounds similar to the topic of https://arxiv.org/abs/1302.1046