Rewriting modulo in diagrammatic algebras and application to categorification

TL;DR

This paper develops a rewriting framework for diagrammatic algebras, enabling systematic study of their structures and proving a basis theorem for graded rak{gl}_2-foams, with applications in categorification and quantum topology.

Contribution

It introduces a novel rewriting theory for diagrammatic algebras, including rewriting modulo categorical properties, and provides the first basis proof for graded rak{gl}_2-foams.

Findings

Established a basis theorem for graded rak{gl}_2-foams.

Developed an algorithmic rewriting approach combining linear, higher, and modulo rewriting.

Provided foundational tools for confluence and termination in diagrammatic algebra rewriting.

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.