Multisets and Distributions

TL;DR

This paper introduces a simplified, combinatorics-free construction of distributive laws for multisets and distributions, leveraging a general 2-category theorem and the Parikh map to transfer properties from lists to multisets.

Contribution

It provides a new, lightweight method to derive distributive laws for multisets and distributions using 2-category theory and the Parikh map, avoiding complex combinatorics.

Findings

Constructed a distributive law for lists and distributions.

Transferred properties from lists to multisets via a 2-category theorem.

Simplified the understanding of distributive laws for multisets and distributions.

Abstract

We give a lightweight alternative construction of Jacobs's distributive law for multisets and distributions that does not involve any combinatorics. We first give a distributive law for lists and distributions, then apply a general theorem on 2-categories that allows properties of lists to be transferred automatically to multisets. The theorem states that equations between 2-cells are preserved by epic 2-natural transformations. In our application, the appropriate epic 2-natural transformation is defined in terms of the Parikh map, familiar from formal language theory, that takes a list to its multiset of elements.