Operations on Languages and Codensity Monads
summary
Presents a concrete duality route from languages to recognizers: use Stone duality to turn the Boolean algebra of quotients into a Stone space with a dense internal monoid and recover the syntactic monoid; shows existential quantification corresponds to the topological construction X × Vietoris X and a unary Schützenberger product on monoids; identifies that any commutative set monad admits a co-density monad lift to Stone and to monoids, so power set and semimodule monads produce recognizers that actually work for existential and modular counting.
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