Between Markov and restriction: Two more monads on categories for relations

2025-08-27

AI Summary

This paper addresses the insufficient abstraction of relational categorical taxonomies by introducing two novel instances of gs-monoidal categories, whose structure is axiomatized via the concepts of “arrow quality” and “domain”, positioning them strictly between Markov and restriction categories. Methodologically, it pioneers the use of quality- and domain-preserving monads; leveraging symmetric monoidal monads, Kleisli constructions, and the gs-monoidal framework, it models and verifies semiring-weighted relations. The main contributions are threefold: (i) a unifying generalization of partial-function domain semantics and weighted relational models; (ii) a rigorous proof that the induced Kleisli categories satisfy both quality conservation and domain conservation laws; and (iii) a substantial extension of the abstraction level and expressive power of existing relational categorical frameworks.

Technology Category

• Knowledge Representation and Reasoning: Qualitative Reasoning
• Reasoning under Uncertainty: Relational Probabilistic Models
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Search and Retrieval-Augmented AI: Vertical and domain-specific search

Abstract

The study of categories abstracting the structural properties of relations has been extensively developed over the years, resulting in a rich and diverse body of work. A previous paper offered a survey providing a modern and comprehensive presentation of these ``categories for relations'' as instances of gs-monoidal categories, showing how they arise as Kleisli categories of suitable symmetric monoidal monads. The end result was a taxonomy that organised numerous related concepts in the literature, including in particular Markov and restriction categories. This paper further enriches the taxonomy: it proposes two categories that are once more instances of gs-monoidal categories, yet more abstract than Markov and restriction categories. They are characterised by an axiomatic notion of mass and domain of an arrow, the latter one of the key ingredient of restriction categories, which generalises the domain of partial functions. The paper then introduces mass and domain preserving monads, proving that the associated Kleisli categories in fact preserve the corresponding equations and that these monads arise naturally for the categories of semiring-weighted relations.

Problem

• Proposing two new abstract gs-monoidal categories
• Characterizing arrows by axiomatic mass and domain notions
• Introducing monads preserving mass and domain equations

Innovation

• Introduces mass and domain preserving monads
• Proposes two abstract gs-monoidal categories
• Generalizes Markov and restriction categories structure