A unified treatment of commuting tensor products of categories, operads, symmetric multicategories and their bimodules

TL;DR

This paper unifies various commuting tensor products across categories, operads, and multicategories, extending their structure to bimodules and double categories, and providing a comprehensive framework for their interactions.

Contribution

It introduces a unified approach to commuting tensor products, extending them to bimodules and double categories, and generalizes existing results in the literature.

Findings

Constructs a double category of symmetric multicategories and bimodules.

Shows the symmetric oplax monoidal structure of the double category.

Provides a general construction of commuting tensor products on double categories.

Abstract

We provide a unified treatment of several commuting tensor products considered in the literature, including the tensor product of enriched categories and the Boardman-Vogt tensor product of operads and symmetric multicategories, subsuming work of Elmendorf and Mandell. We then show how a commuting tensor product extends to bimodules, generalising results of Dwyer and Hess. In particular, we construct a double category of symmetric multicategories, symmetric multifunctors and bimodules and show that it admits a symmetric oplax monoidal structure. These applications are obtained as instances of a general construction of commuting tensor products on double categories of monads, monad morphisms and bimodules.