Weak algebraic structures are increasingly common, both in computer science (e.g. complete bicategories as models for the lambda calculus, oo-topoi as models for homotopy type theory) and in mathematical practice (e.g. Aoo or Eoo-algebras). The most well-known is probably the notion of monoidal category, which is a categorification of the notion of monoid. Nevertheless, finding a suitable weakening of an algebraic structure (encoded e.g. as an operad) is often complicated: while in good cases standard techniques exist, they often yield objects which are too large to be usable in practice.

In this talk, we start by motivating why it is natural to consider weakened versions of algebraic structures. We then explain the general setting in which categorification takes place, and what makes a weak structure the "right" categorification of a classical notion. Finally we show how to effectively compute small categorifications of structures given by generators and relations, using rewriting methods.