Title: Quantifiers on languages and codensity monads,
Abstract: This paper contributes to the techniques of topo-algebraic recognition for
languages beyond the regular setting as they relate to logic on words. In
particular, we provide a general construction on recognisers corresponding to
adding one layer of various kinds of quantifiers and prove a related
Reutenauer-type theorem. Our main tools are codensity monads and duality
theory. Our construction hinges, in particular, on a measure-theoretic
characterisation of the profinite monad of the free S-semimodule monad for
finite and commutative semirings S, which generalises our earlier insight that
the Vietoris monad on Boolean spaces is the codensity monad of the finite
powerset functor.