§ Covering Spaces for Automata

• 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.