Understand
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 $\mathcal{V}$.