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

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.

💡 Research Summary

This paper presents a novel recasting of fundamental concepts from abstract rewriting theory within the framework of cubical higher-dimensional algebra. The central aim is to study confluence—the property that divergent computations can eventually reconverge—as an internal property of cubical categories, moving beyond the traditional globular (ω-categorical) setting often used in higher-dimensional rewriting.

The authors introduce a key new concept: cubical contractions. This is a higher-dimensional generalization of reduction strategies to normal forms. A contraction is defined recursively as a family of lax transformations on a cubical (ω, p)-category, built upon a “section” that chooses representatives (like normal forms) in a quotient of the p-dimensional cells. A cubical category equipped with such a structure is called a contracting category. The main technical result of the foundational section (Theorem 3.2.5) proves that every contracting (ω, 0)-category—that is, every cubical ω-groupoid defined this way—is acyclic. This means that every boundary with a “cubical hole” can be filled with a cell, providing a constructive method for proving higher-dimensional coherence.

Within this categorical framework, the paper revisits and provides new, diagrammatic proofs of cornerstone results from rewriting theory. These include variants of Newman’s Lemma (local confluence + termination imply confluence), the Church-Rosser theorem, and Squier’s coherence theorem (relating convergent presentations to finite coherence data). The proofs are executed via the pasting of cubical coherence cells in two and three dimensions. Notably, the proof of Newman’s Lemma in three cubical directions is achieved without explicitly assuming the Cube Law; instead, the law emerges from the geometric axioms of cubical categories. Furthermore, using contractions, the authors derive the Cube Law—known from λ-calculus and Garside theory—in purely categorical terms, without invoking coherence 3-cells.

The final major contribution lies in the construction of polygraphic resolutions. The paper shows how to build an acyclic cubical ω-groupoid freely generated from any convergent (confluent and terminating) abstract rewriting system. This construction, detailed in Theorem 5.3, uses a normalization strategy based on contractions and involves an analysis of the n-branchings of the rewriting system. A refinement of this construction (Theorem 5.3.2) ensures that this freely generated ω-groupoid produces no non-trivial generating cells in dimensions greater than 2. This result structurally confirms that, in this setting, the Cube Law does not require explicit coherence 3-cells and implies that, homotopically speaking, “all cubes are empty” for abstract rewriting systems.

In summary, this work successfully bridges rewriting theory and higher-dimensional category theory by using cubical categories as a natural habitat for confluence diagrams. It generalizes normalization to higher dimensions via contractions, proves classic results in a new geometric language, and constructs algebraic invariants (acyclic ω-groupoids) from computational properties (convergence), thereby offering a categorical answer to long-standing questions about the place of confluence and the Cube Law in abstract algebra.