Coherence for Categorified Operadic Theories

Abstract

Given an algebraic theory which can be described by a (possibly symmetric) operad $P$, we propose a definition of the \emph{weakening} (or \emph{categorification}) of the theory, in which equations that hold strictly for $P$-algebras hold only up to coherent isomorphism. This generalizes the theories of monoidal categories and symmetric monoidal categories, and several related notions defined in the literature. Using this definition, we generalize the result that every monoidal category is monoidally equivalent to a strict monoidal category, and show that the "strictification" functor has an interesting universal property, being left adjoint to the forgetful functor from the category of strict $P$-categories to the category of weak $P$-categories. We further show that the categorification obtained is independent of our choice of presentation for $P$, and extend some of our results to many-sorted theories, using multicategories.