scratch
§ Covering Spaces for Automata
created 2025-04-02
- Idea: think of the alphabet as a bouquet, with one petal for each letter in the alphabet.
- Now, an automaton is a covering space of this bouquet! See that each state "locally" looks like the bouquet (potentially unfolded).
- See that if we take a string, then that is a path in the bouquet, and the "path lift", starting at the initial state (the basepoint of the automata) gives us the final state.
- See that this immediately also gives us monodromylike behaviour via path lifting.
- See that from monodromy, we can view automata morphisms as covering spaces morphisms.
- TODO: Think about Nielsen Transformations in this context.
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