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Coherent presentations of Artin monoids

Part of: Theory of computing Special aspects of infinite or finite groups Categories with structure

Published online by Cambridge University Press:  22 December 2014

Stéphane Gaussent ,
Yves Guiraud  and
Philippe Malbos
Stéphane Gaussent
Affiliation:
Université de Lyon, Institut Camille Jordan, CNRS UMR 5208, France Université Jean Monnet, 42023 Saint-Étienne Cedex 2, France email stephane.gaussent@univ-st-etienne.fr
Yves Guiraud
Affiliation:
INRIA, Laboratoire PPS, CNRS UMR 7126, France Université Paris 7, Case 7014, 75205 Paris Cedex 13, France email yves.guiraud@pps.univ-paris-diderot.fr
Philippe Malbos
Affiliation:
Université de Lyon, Institut Camille Jordan, CNRS UMR 5208, France Université Claude Bernard Lyon 1, 43 boulevard du 11 novembre 1918, 69622 Villeurbanne Cedex, France email malbos@math.univ-lyon1.fr
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Abstract

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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.

Keywords

Artin monoidpresentationcoherencehigher-dimensional rewritingcompletionaction of monoids on categories

MSC classification

Primary: 18D05: Double categories, $2$-categories, bicategories and generalizations 20F36: Braid groups; Artin groups 68Q42: Grammars and rewriting systems
Type
Research Article
Information
Compositio Mathematica , Volume 151 , Issue 5 , May 2015 , pp. 957 - 998
Copyright
© The Authors 2014 

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