The Categorification of a Symmetric Operad is Independent of Signature

TL;DR

This paper introduces a notion of categorification for symmetric operads based on a signature and proves that this process is independent of the signature choice, exemplified by symmetric monoidal categories.

Contribution

It defines a general categorification of symmetric operads and proves its independence from the signature used, extending the understanding of operad weakening.

Findings

Categorification of symmetric operads is well-defined up to equivalence.

For the operad of commutative monoids, categorification yields symmetric monoidal categories.

The process is invariant under different choices of generating signatures.

Abstract

Given a symmetric operad $P$, and a signature (or generating sequence) $\Phi$ for $P$, we define a notion of the "categorification" (or "weakening") of $P$ with respect to $\Phi$. When $P$ is the symmetric operad whose algebras are commutative monoids, with the standard signature, we recover the notion of symmetric monoidal categories. We then show that this categorification is independent (up to equivalence) of the choice of signature.