Symmetry shifting for monoidal bicategories

TL;DR

This paper extends a classic theorem by showing how braidings in monoidal bicategories induce corresponding structures in their bicategories of monoids, utilizing advanced $ abla$-operadic methods.

Contribution

It generalizes Joyal and Street's theorem from monoidal categories to monoidal bicategories using $ abla$-operadic techniques, avoiding complex calculations.

Findings

Monoidal bicategories with braidings induce monoidal structures on their bicategories of monoids.

Sylleptic or symmetric braidings lead to braided or symmetric structures in the bicategory of monoids.

The proof leverages the $ abla$-operadic Additivity Theorem, simplifying the derivation.

Abstract

We show that every braiding on a monoidal bicategory induces a monoidal structure on its bicategory of monoids, such that if the former is sylleptic or symmetric then the latter is braided or symmetric, respectively. This extends a classic theorem of Joyal and Street for monoidal categories. The proof presented in this paper is an application of the $\infty$-operadic Additivity Theorem and thereby averts any considerable calculations.