Coherent presentations of Artin monoids

Stéphane Gaussent 1 Yves Guiraud 2, 3 Philippe Malbos 1, 3
1 AGL - Algèbre, géométrie, logique
ICJ - Institut Camille Jordan [Villeurbanne]
3 PI.R2 - Design, study and implementation of languages for proofs and programs
PPS - Preuves, Programmes et Systèmes, Inria Paris-Rocquencourt, UPD7 - Université Paris Diderot - Paris 7, CNRS - Centre National de la Recherche Scientifique : UMR7126
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.
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Stéphane Gaussent, Yves Guiraud, Philippe Malbos. Coherent presentations of Artin monoids. Compositio Mathematica, Foundation Compositio Mathematica, 2015, 151 (5), pp.957-998. ⟨10.1112/S0010437X14007842⟩. ⟨hal-00682233v4⟩

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