Linear higher rewriting
and applications to diagrammatic algebras

Rewriting theory is the study of normal forms in structures presented by generators and relations. It has roots in the word problem for groups, and in the linear case, to the resolution of polynomial equations (so-called Gröbner bases). In this talk, I will discuss how to extend the (linear) theory higher: to (linear) monoidal categories, presented by generators and relations using string diagrams. This builds on the theory of polygraphs, see e.g. [2]. Emphasizes will be put on how various categorical constructions naturally appear, out of necessity—this includes semistrict categories, double categories, and spans.

Our approach is driven by examples, and in particular by diagrammatic algebras as they appear in representation theory and low-dimensional topology (e.g. Kac–Moody 2-categories or Hecke categories): we will give an explicit application to a conjecture in these fields. This talk also aims at introducing these problems to category theorists, and explain why a categorical viewpoint is becoming increasingly necessary. If time permits, we will discuss future development, such as implementation or $\mathrm{\infty }$-refinement.

This talk is based on [1].