The higher algebra and geometry of monoidal bicategories

TL;DR

This paper characterizes braided, sylleptic, and symmetric monoidal bicategories as specific types of $ ext{E}_k$-monoids within a cartesian monoidal $( ext{infinity,1})$-category, using little cubes operads for computational management.

Contribution

It establishes a precise correspondence between certain monoidal bicategories and $ ext{E}_k$-monoids, employing operadic geometry to bridge 2D algebra and $ ext{infinity}$-categorical algebra.

Findings

Braided, sylleptic, and symmetric monoidal bicategories are $ ext{E}_k$-monoids.

Operadic geometry facilitates computations between 2D and $ ext{infinity}$-categorical algebra.

Provides a unified framework for understanding monoidal bicategories via higher algebra.

Abstract

We show that braided, sylleptic and symmetric monoidal bicategories are precisely the $\mathsf{E}_k$-monoids in the cartesian monoidal $(\infty,1)$-category of bicategories for respective integers $k$. To manage the underlying computations, we use the geometry of the little cubes operads to mediate between the 2-dimensional algebra underlying the former and the $\infty$-categorical algebra underlying the latter.