A categorical analogue of the monoid semiring construction

TL;DR

This paper generalizes the monoid semiring construction to categories, unifying algebraic and analytical summation notions, and explores its categorical properties and computational interpretations.

Contribution

It introduces a categorical analogue of the monoid semiring construction, unifying algebraic and analytical summation within a categorical framework.

Findings

The construction generalizes to categories for monoids and semirings.

It exhibits natural categorical properties.

It has meaningful computational interpretations.

Abstract

This paper introduces and studies a categorical analogue of the familiar monoid semiring construction. By introducing an axiomatisation of summation that unifies notions of summation from algebraic program semantics with various notions of summation from the theory of analysis, we demonstrate that the monoid semiring construction generalises to cases where both the monoid and the semiring are categories. This construction has many interesting and natural categorical properties, and natural computational interpretations.