Coherent presentations of Artin monoids

TL;DR

This paper develops a method to produce coherent presentations of Artin monoids using higher-dimensional rewriting, extending classical presentations with relations between relations, and provides new proofs of existing theorems.

Contribution

It introduces a homotopical completion-reduction method for Artin monoids and demonstrates that Tits-Zamolodchikov 3-cells extend Artin's presentation into a coherent presentation.

Findings

Coherent presentations of Artin monoids are constructed using higher-dimensional rewriting.

Tits-Zamolodchikov 3-cells extend Artin's presentation coherently.

A new constructive proof of Deligne's theorem on Artin monoid actions is provided.

Abstract

We compute coherent presentations of Artin monoids, that is presentations by generators, relations, and relations between the relations. For that, we use methods of higher-dimensional rewriting that extend Squier's and Knuth-Bendix's completions into a homotopical completion-reduction, applied to Artin's and Garside's presentations. The main result of the paper states that the so-called Tits-Zamolodchikov 3-cells extend Artin's presentation into a coherent presentation. As a byproduct, we give a new constructive proof of a theorem of Deligne on the actions of an Artin monoid on a category.