Functional distribution monads in functionalanalytic 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 $\textit{functionalanalytic context}$, 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 functionalanalytic context. We establish certain general classes of examples of functionalanalytic 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.
 Publication:

arXiv eprints
 Pub Date:
 January 2017
 arXiv:
 arXiv:1701.08152
 Bibcode:
 2017arXiv170108152L
 Keywords:

 Mathematics  Category Theory;
 Mathematics  Functional Analysis