Coherence for Categorified Operadic Theories

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.

💡 Research Summary

The paper develops a general framework for “categorifying’’ algebraic theories that are presented by operads. Starting from an operad (P) (possibly symmetric) described by a presentation (\langle\Sigma\mid R\rangle), the authors define a weak (P)-category: each generator in (\Sigma) is interpreted as a 1‑cell, while each relation in (R) is realized only up to a coherent isomorphism, together with higher coherence 2‑cells that satisfy natural compatibility conditions. This construction abstracts the familiar passage from monoids to monoidal categories and from commutative monoids to symmetric monoidal categories, but works for any operadic theory.

The central technical achievement is a strictification theorem that generalizes Mac Lane’s coherence result: every weak (P)-category is equivalent, via a suitably defined 2‑functor, to a strict (P)-category in which all operadic equations hold on the nose. The strictification functor (S\colon\mathbf{Wk},P\text{-Cat}\to\mathbf{St},P\text{-Cat}) is built using coends in the free 2‑category generated by the operad, flattening the coherence isomorphisms into identities. Moreover, (S) is left adjoint to the forgetful functor (U\colon\mathbf{St},P\text{-Cat}\to\mathbf{Wk},P\text{-Cat}). This adjunction gives a universal property: (S) is the most free way to turn a weak structure into a strict one, while (U) simply forgets the strictness.

A significant concern is whether the construction depends on the chosen presentation of (P). The authors prove that if two presentations generate the same operad, the resulting 2‑categories of weak (P)-categories are biequivalent. Hence the categorification is intrinsic to the operad itself, not to any particular set of generators or relations.

The framework is then extended to many‑sorted theories by moving from operads to multicategories. In this setting each sort corresponds to a colour, and the weak multicategorical structure again admits strictification with the same adjoint relationship. The paper supplies concrete examples: ordinary monoidal categories, symmetric monoidal categories, braided monoidal categories, and more exotic operadic algebras, showing how each fits into the general picture and how the strictification recovers known coherence theorems.

Finally, the authors discuss future directions, including strictification for infinite operads, interactions with higher monads, and applications to type theory and programming language semantics where operadic signatures model computational effects. Overall, the work provides a unified, presentation‑independent theory of coherence for categorified operadic algebras, extending classical results to a broad class of algebraic structures and laying groundwork for further exploration in higher‑dimensional category theory and its applications.