On the Axioms of Arboreal Categories

2026-03-23

AI Summary

This work addresses a deficiency in the existing axiomatic framework of arboreal categories, where the “path-connectedness” condition proves insufficient for supporting logical model comparison within the game comonadic setting. To remedy this, the paper proposes “tree-connectedness” as a refined axiom, reconstructing the foundational structure of arboreal categories accordingly. It demonstrates that essential properties—such as the path functor forming a Street fibration—remain valid under this revised framework. By integrating techniques from category theory, comonad theory, and fibration theory, the study rectifies the original framework’s shortcomings while preserving its core semantic capabilities for logical model comparison games, thereby establishing a more robust and coherent axiomatic foundation.

Technology Category

• Knowledge Representation and Reasoning: Nonmonotonic Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

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: Foundation models and LLMs for Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Arboreal categories were introduced as an axiomatic framework for game comonads, which provide a comonadic view on many model-comparison games in logic. We demonstrate the inadequacy of the axiom stating that paths are connected. We then propose the notion of ``tree-connectedness'' to address this deficiency, and show that all the essential properties of arboreal categories that we are aware of remain valid under this new definition. Furthermore, we show that the path functor is a Street fibration.

Problem

• arboreal categories
• axioms
• path connectedness
• tree-connectedness
• game comonads

Innovation

• arboreal categories
• tree-connectedness
• game comonads
• Street fibration
• path functor