Commutativity

TL;DR

This paper develops a general enriched category theory framework to unify various notions of commutativity across algebraic theories, monads, and operads, extending existing tensor product concepts.

Contribution

It introduces a tensor product for categories enriched over a normal duoidal category, generalizing previous notions of commutativity in algebraic and operadic contexts.

Findings

Unified framework for commutativity notions

Extended tensor product to enriched categories over duoidal categories

Reinterpreted classical tensor products within the new framework

Abstract

We describe a general framework for notions of commutativity based on enriched category theory. We extend Eilenberg and Kelly's tensor product for categories enriched over a symmetric monoidal base to a tensor product for categories enriched over a normal duoidal category; using this, we re-find notions such as the commutativity of a finitary algebraic theory or a strong monad, the commuting tensor product of two theories, and the Boardman-Vogt tensor product of symmetric operads.