Quantified functional analysis and seminormed spaces: A dual adjunction

In this work we investigate the natural algebraic structure that arises on dual spaces in the context of quantified functional analysis. We show that the category of absolutely convex modules is obtained as the category of Eilenberg–Moore algebras induced by the dualization functor [−,R][−,R] on locally convex approach spaces. We also establish a dual adjunction between the latter category and the category of seminormed spaces.