In a series of papers, P. Dehornoy has investigated certain algebraic invariants associated with balanced equational varieties. In general, these invariants form inverse monoids, known as "structural monoids", but particularly nice cases result in groups. Dehornoy has shown that Richard Thompson's groups F and V, each of which is finitely presentable, infinite and simple, arise as the invariants associated to the varieties of semigroups and of commutative semigroups, respectively. He subsequently obtained presentations of these groups utilising the pentagon and hexagon coherence axioms for coherently associative and commutative bifunctors. This talk will show how to systematise this observation. To every balanced equational variety E, we will associate a collection of categorifications. Each categorification yields a presentation of an inverse monoid, which is isomorphic to the structural monoid of E if and only if the categorification is coherent. The resulting presentations reflect some nice combinatorial features of the structural monoids. We may also briefly see how to extend this construction to any rewriting 2-theory, which is a Cat-enriched Lawvere theory definable by variables, with the level of detail being proportional to the time until drinks...