Abstract: 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.