Mathematics – Category Theory
Scientific paper
2012-03-23
Mathematics
Category Theory
66 pages
Scientific paper
We study Deligne's notion of action of a monoid on a category and, in particular, the piece of data that corresponds to the coherence relations that such an action should satisfy. We prove that actions of a monoid are equivalent to 2-functors from a 2-categorical cofibrant replacement of the monoid into the 2-category of categories. One way to compute such a cofibrant replacement is to consider the 2-category presented by a coherent presentation of the monoid: this is a presentation extended with a homotopy basis, that is, a set of relations between the relations that identifies any two proofs of the same equality in the monoid. Using higher-dimensional rewriting, in the polygraphic setting, we combine and extend Squier's theorem and Knuth-Bendix completion procedure into a "reduced homotopical completion" procedure that, when successful, transforms a given presentation into a relatively small coherent presentation. In particular, when used on Deligne's presentation of Artin-Tits groups of spherical type, the procedure computes the coherence conditions that Deligne finds with geometric methods.
Gaussent Stéphane
Guiraud Yves
Malbos Philippe
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