Rewriting modulo isotopies in pivotal linear $(2,2)$-categories

TL;DR

This paper develops a rewriting framework for pivotal linear (2,2)-categories, enabling the computation of bases for 2-cell vector spaces, with applications to affine oriented Brauer diagrams.

Contribution

It introduces linear (3,2)-polygraphs modulo for rewriting in linear (2,2)-categories, specifically addressing pivotal categories with isotopy relations.

Findings

Computed bases of 2-cell vector spaces in affine oriented Brauer categories

Developed a symbolic method for rewriting modulo algebraic axioms

Reconstructed normally ordered dotted oriented Brauer diagrams

Abstract

In this paper, we study rewriting modulo a set of algebraic axioms in categories enriched in linear categories, called linear~$(2,2)$-categories. We introduce the structure of linear~$(3,2)$-polygraph modulo as a presentation of a linear~$(2,2)$-category by a rewriting system modulo algebraic axioms. We introduce a symbolic computation method in order to compute linear bases for the vector spaces of $2$-cells of these categories. In particular, we study the case of pivotal $2$-categories using the isotopy relations given by biadjunctions on $1$-cells and cyclicity conditions on $2$-cells as axioms for which we rewrite modulo. By this constructive method, we recover the bases of normally ordered dotted oriented Brauer diagrams in te affine oriented Brauer linear~$(2,2)$-category.