Coherent Presentations of Monoidal Categories

Presentations of categories are a well-known algebraic tool to providedescriptions of categories by means of generators, for objects and morphisms,and relations on morphisms. We generalize here this notion, in order toconsider situations where the objects are considered modulo an equivalencerelation, which is described by equational generators. When those form aconvergent (abstract) rewriting system on objects, there are three very naturalconstructions that can be used to define the category which is described by thepresentation: one consists in turning equational generators into identities(i.e. considering a quotient category), one consists in formally addinginverses to equational generators (i.e. localizing the category), and oneconsists in restricting to objects which are normal forms. We show that, undersuitable coherence conditions on the presentation, the three constructionscoincide, thus generalizing celebrated results on presentations of groups, andwe extend those conditions to presentations of monoidal categories