A taxonomy of categories for relations

2025-02-14

arXiv.org

AI Summary

This paper addresses the lack of a unified classification framework for the structural properties of relational abstractions. Methodologically, it introduces the first hierarchical and systematic taxonomy of relational categories, grounded in the Kleisli category of the symmetric monoidal monad as a unifying generative mechanism. This framework subsumes diverse relational structures—including relational database schemas, program semantics models, and relational representations in AI—along with their enriched variants, within a single categorical setting. The key contribution is the identification of a common origin: all major relational categories in the literature arise as instances of this monadic Kleisli construction. By exposing this deep structural unity, the taxonomy enhances theoretical coherence and conceptual clarity. It provides a rigorous, general mathematical foundation applicable across program semantics, database theory, and AI-based relational modeling.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• Knowledge Representation and Reasoning: Nonmonotonic Reasoning
• Machine Learning: Statistical Relational/Logic Learning

Application Category

• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data

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.

Problem

• Develops a taxonomy for categories abstracting relational structures
• Presents enriched categories for relations and their monadic origins
• Organizes diverse concepts in relational category literature

Innovation

• Categories abstracting structural properties of relations
• Enriched categories and Kleisli categories of monads
• Taxonomy organizing concepts and frameworks in literature