Commutative monads as a theory of distributions

TL;DR

This paper develops a categorical framework using commutative monads to unify the theory of distributions, Schwartz distributions, and probability distributions, providing formal connections and new insights.

Contribution

It explicitly links the theory of commutative monads with Schwartz distributions and probability distributions, expanding the categorical understanding of these concepts.

Findings

Established a formal connection between monads and Schwartz distributions

Extended the framework to include probability distributions

Provided proofs of key properties within the categorical setting

Abstract

The theory of commutative monads on cartesian closed categories provides a framework where aspects of the theory of distributions and other extensive quantities can be formulated and some results proved. We make explicit a link between our theory and the theory of Schwartz distributions of compact support. We also discuss probability distributions.