Functional distribution monads in functional-analytic contexts

We give a general categorical construction that yields several monads of measures and distributions as special cases, alongside several monads of filters. The construction takes place within a categorical setting for generalized functional analysis, called a functional-analytic context, formulated in terms of a given monad or algebraic theory T enriched in a closed category V. By employing the notion of commutant for enriched algebraic theories and monads, we define the functional distribution monad associated to a given functional-analytic context. We establish certain general classes of examples of functional-analytic contexts in cartesian closed categories V, wherein T is the theory of R-modules or R-affine spaces for a given ring or rig R in V, or the theory of R-convex spaces for a given preordered ring R in V. We prove theorems characterizing the functional distribution monads in these contexts, and on this basis we establish several specific examples of functional distribution monads.