Decreasing diagrams and coherent presentations

TL;DR

This paper demonstrates how decreasing diagrams from rewriting systems can establish coherence theorems in category theory, exemplified by a coherent presentation of the 0-Hecke monoid of the symmetric group.

Contribution

It introduces a novel application of decreasing diagrams to prove coherence theorems and provides a coherent presentation of the 0-Hecke monoid.

Findings

Decreasing diagrams can be effectively used for coherence proofs.

A coherent presentation of the 0-Hecke monoid is constructed.

The method bridges rewriting systems and category theory coherence.

Abstract

We show how decreasing diagrams introduced in the theory of rewriting systems can be used to prove coherence type theorems in category theory. We apply this method to describe a coherent presentation of the $0$-Hecke monoid $\mathcal{H}(\Sigma_n)$ of the symmetric group $\Sigma_n$, i.e. a presentation by generators, relations, and relations between relations.