Rewriting modulo in diagrammatic algebras and application to categorification
arXiv:2502.03028 · doi:10.1016/j.aim.2026.111198
Abstract
We develop a rewriting theory modulo suitable for higher linear structures. In particular, the theory is suited for diagrammatic algebras as they appear in categorification, representation theory and quantum topology. As an application, we use the theory to prove the basis conjecture of a certain super-2-category related to odd Khovanov homology. Our approach combines linear rewriting, higher rewriting and rewriting modulo. For diagrammatic algebras, the modulo rules typically capture a categorical property, such as pivotality. In the process, we revisit the foundations of these theories, including the notion of confluence. Other important tools include termination rules that depend on contexts, rewriting modulo invertible scalars, and a method to classify branchings modulo. This article includes an introduction to rewriting theory for non-experts.
85 pages, Part II of the author's PhD thesis (arXiv:2410.11405), comments welcome! v2: exposition has been improved. final version. to appear in Advances in Mathematics