Coherent presentations of Artin monoids. (22nd December 2014)

• Record Type: Journal Article
• Authors: Gaussent, Stéphane; Guiraud, Yves; Malbos, Philippe
• Is Part Of: Compositio mathematica. Volume 151:Number 5(2015:Sep.)
• Page Start: 957
• Page End: 998
• Publication Date: 2014-12-22
• Subjects: 18D05, -- 20F36, -- 68Q42 (primary); Artin monoid, -- presentation, -- coherence, -- higher-dimensional rewriting, -- completion, -- action of monoids on categories; Mathematics -- Periodicals; 510
• DOI: 10.1112/S0010437X14007842
• Languages: English
• ISSNs: 0010-437X

Abstract

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.