This talk was based on Deligne's paper Action du groupe des tresses sur une catégorie, Invent. Math. 128:159-175, 1997.
For any monoid M one can consider pseudoactions of the monoid on a category; these satisfy the usual axioms to be an action only up to coherent isomorphism. If the monoid M is not given explicitly but only by some presentation, one would like to be able to define actions in terms of the presentation. In general it is not known how to do this, but if all relations have the form g.h=k then one can define a ``pseudoaction of the presentation', and the category of all such will be equivalent to the category of pseudoactions of the monoid if the presentation satisfies a contractibility condition.
A presentation of this kind is given for the monoid of positive braids on n strings; more generally such a presentation is given for the monoid of generalized positive braids corresponding to any finite Coxeter group.