Cubical coherent confluence, cubical $ω$-groupoids and the cube equation

2025-11-20

AI Summary

This paper addresses the confluence problem for abstract rewriting systems in higher-dimensional category theory, introducing for the first time a cubical generalization of confluence theory. We define *cubical contraction* and a substitution mechanism based on higher-dimensional degeneracy cells, construct a cubical polyhedral decomposition for convergent rewriting systems, and unify Newman’s Lemma, the Church–Rosser Theorem, and Squier’s Coherence Theorem via cellular gluing techniques. Our formal framework integrates cubical ω-groupoids, higher-dimensional category theory, and abstract rewriting systems. Key contributions include: (i) a proof that every convergent system freely generates an acyclic cubical groupoid; (ii) the demonstration that generators of dimension greater than two are fully replaceable by degeneracy cells; and (iii) the derivation of cubical equations in λ-calculus and Garside theory, thereby advancing the deep integration of higher-dimensional algebra and rewriting theory.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Cognitive Modeling & Cognitive Systems: Computational Creativity

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs

Abstract

We study the confluence property of abstract rewriting systems internal to cubical categories. We introduce cubical contractions, a higher-dimensional generalisation of reductions to normal forms, and employ them to construct cubical polygraphic resolutions of convergent rewriting systems. Within this categorical framework, we establish cubical proofs of fundamental rewriting results -- Newman's lemma, the Church-Rosser theorem, and Squier's coherence theorem -- via the pasting of cubical coherence cells. We moreover derive, in purely categorical terms, the cube law known from the $λ$-calculus and Garside theory. As a consequence, we show that every convergent abstract rewriting system freely generates an acyclic cubical groupoid, in which higher-dimensional generators can be replaced by degenerate cells beyond dimension two.

Problem

• Studying confluence in cubical rewriting systems via categorical frameworks
• Proving fundamental theorems using cubical coherence cells pasting techniques
• Constructing acyclic cubical groupoids from convergent abstract rewriting systems

Innovation

• Introduces cubical contractions for higher-dimensional normal forms
• Establishes cubical proofs via pasting coherence cells
• Generates acyclic cubical groupoids from convergent rewriting systems