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Decreasing diagrams and coherent presentations

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 $\mathcal{H}(Σ_n)$ of the symmetric group $Σ_n$, i.e. a presentation by generators, relations, and relations between relations.