# Functional distribution monads in functional-analytic contexts

## TL;DR

This paper introduces a categorical framework for constructing monads of measures and distributions within functional analysis, generalizing classical concepts through enriched category theory and algebraic theories.

## Contribution

It develops a unified categorical construction of distribution monads in functional-analytic contexts using enriched algebraic theories and the notion of commutants.

## Key findings

- Defines the functional distribution monad in enriched categorical settings.
- Provides examples involving R-modules, R-affine spaces, and R-convex spaces.
- Characterizes the properties of these monads in various contexts.

## Abstract

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 $\textit{functional-analytic context}$, formulated in terms of a given monad or algebraic theory $\mathcal{T}$ enriched in a closed category $\mathcal{V}$. By employing the notion of $\textit{commutant}$ for enriched algebraic theories and monads, we define the $\textit{functional distribution monad}$ associated to a given functional-analytic context. We establish certain general classes of examples of functional-analytic contexts in cartesian closed categories $\mathcal{V}$, wherein $\mathcal{T}$ is the theory of $R$-modules or $R$-affine spaces for a given ring or rig $R$ in $\mathcal{V}$, or the theory of $\textit{$R$-convex spaces}$ for a given preordered ring $R$ in $\mathcal{V}$. We prove theorems characterizing the functional distribution monads in these contexts, and on this basis we establish several specific examples of functional distribution monads.