Rewriting modulo in diagrammatic algebras and application to categorification

TL;DR

This work develops a flexible, higher-dimensional rewriting framework tailored to diagrammatic algebras, combining linear and modulo rewriting within linear Gray (3-sesqui) settings. The central innovation is the notion of succ-tamed congruence, which allows confluence reasoning in contexts where planar isotopies or pivotal structures are captured modulo relations, and the derivation of a Basis-from-Convergence principle that reduces basis construction to normal forms under modulo rewriting. The methodology is then applied to graded ${ m gl}_2$-foams, producing a convergent presentation and proving a basis theorem for their hom-spaces, demonstrating the practical viability of rewriting modulo in quantum topology and higher representation theory. The results provide intrinsic, algorithmic tools for understanding deformations and coherence in diagrammatic algebras, with potential for computer implementation and extensions to deformation theory and higher structures. Overall, the paper establishes a robust bridge between rewriting theory and diagrammatic categorification, enabling systematic analysis of bases, coherence data, and higher symmetries.

Abstract

We develop a rewriting theory suitable for diagrammatic algebras and lay down the foundations of a systematic study of their higher structures. In this paper, we focus on the question of finding bases. As an application, we give the first proof of a basis theorem for graded $\mathfrak{gl}_2$-foams, a certain diagrammatic algebra appearing in categorification and quantum topology. Our approach is algorithmic, combining linear rewriting, higher rewriting and rewriting modulo another set of rules -- for diagrammatic algebras, the modulo rules typically capture a categorical property, such as pivotality. In the process, we give novel approaches to the foundations of these theories, including to the notion of confluence. Other important tools include termination rules that depend on contexts, rewriting modulo invertible scalars, and a practical guide to classifying branchings modulo. This article is written to be accessible to experts on diagrammatic algebras with no prior knowledge on rewriting theory, and vice-versa.

Figures (7)

• Figure 1.1: A relation in the Temperley--Lieb category. Thanks to the diagrammatics, it is easily understood as "evaluate the closed loop to $q+q^{-1}$, and apply a planar isotopy".
• Figure 1.2: Critical branchings in $\mathsf{P}_{\mathfrak{G}_3}$.
• Figure 1.3: In a 3-category, a Gray category or a 3-sesquicategory, the interchange law for 2-morphisms respectively holds strictly, holds weakly via interchangers, or does not hold (a priori).
• Figure 4.1: 3-cells in ${\mathsf{R}_{}}_3$. Recall the convention that we omit objects: all the 3-cells can be labelled by any object, as long as the label is legal (\ref{['defn:legal_diagram']}). In the last two cases, the wiggly lines are only visual aids, and are not part of the data. Each 3-cell has its notation, depicted above the arrow, so that ${\mathsf{R}_{}}_3=\{\mathsf{dd},\mathsf{dm},\mathsf{bb}_\circlearrowleft,\mathsf{bb}_\circlearrowright,\mathsf{nc},\mathsf{sq}\}$.
• Figure 4.2: Critical branching between types ${\mathsf{B}_{}}$ and type $\mathsf{nc}$.

Theorems & Definitions (165)

• Lemma 1.1: linear Newmann's lemma
• Lemma 1.2: characterization of rewriting steps in ${}^\circ \mathsf{S}$
• Lemma 1.3: classification of branchings in ${}^\circ \mathsf{S}$
• Remark 1.4
• Remark 2.2: low-dimensional cases
• Remark 2.3
• Remark 2.4
• Remark 2.5: low-dimensional cases
• Remark 2.6: low-dimensional case
• Definition 2.7: FM_RewritingGrayCategories_2022