Rewriting in higher dimensional linear categories and application to the affine oriented Brauer category

TL;DR

This paper develops a rewriting theory for linear monoidal categories, introduces linear (n,p)-categories and polygraphs, and applies these concepts to prove a basis theorem for the affine oriented Brauer category using a new rewriting method.

Contribution

It introduces a novel rewriting framework for linear (n,p)-categories and applies it to establish a basis theorem for the affine oriented Brauer category.

Findings

Rewriting theory for linear monoidal categories developed.

Linear (n,p)-categories and polygraphs defined and utilized.

A basis theorem for the affine oriented Brauer category proved.

Abstract

In this paper, we introduce a rewriting theory of linear monoidal categories. Those categories are a particular case of what we will define as linear (n, p)-categories. We will also define linear (n, p)-polygraphs, a linear adapation of n-polygraphs, to present linear (n -- 1, p)-categories. We focus then on linear (3, 2)-polygraphs to give presentations of linear monoidal categories. We finally give an application of this theory in linear (3, 2)-polygraphs to prove a basis theorem on the category AOB with a new method using a rewriting property defined by van Ostroom: decreasingness.