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We study various aspects of the structure and representation theory ofsingular Artin monoids. This includes a number of generalizations of thedesingularization map and explicit presentations for certain finite quotientmonoids of diagrammatic nature. The main result is a categorification of theclassical desingularization map for singular Artin monoids associated to finiteWeyl groups using BGG category $\mathcal{O}$.

Finite quotients of singular Artin monoids and categorification of the desingularization map