A taxonomy of categories for relations

TL;DR

This paper presents a modern taxonomy of categories for relations, including their enriched versions, and explains their emergence as Kleisli categories of symmetric monoidal monads to clarify related concepts.

Contribution

It offers a structured classification of categories for relations, connecting them to Kleisli categories of symmetric monoidal monads for better understanding.

Findings

Provides a comprehensive taxonomy of categories for relations.

Shows how these categories arise as Kleisli categories of symmetric monoidal monads.

Clarifies the relationships among various related concepts in the literature.

Abstract

The study of categories that abstract the structural properties of relations has been extensively developed over the years, resulting in a rich and diverse body of work. This paper strives to provide a modern presentation of these ``categories for relations'', including their enriched version, further showing how they arise as Kleisli categories of symmetric monoidal monads. The resulting taxonomy aims at bringing clarity and organisation to the many related concepts and frameworks occurring in the literature.