Abstract
This paper contributes to the techniques of topoalgebraic 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 yields, in particular, a new 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.