The Catalan simplicial set II

TL;DR

This paper extends the classification of skew-monoidal categories using the Catalan simplicial set to skew monoidales in monoidal bicategories and further characterizes monoidal bicategories via maps from the Catalan set.

Contribution

It generalizes the classification of skew-monoidal structures to higher categorical contexts using the Catalan simplicial set.

Findings

Skew-monoidales in monoidal bicategories are classified by maps from the Catalan set.

Monoidal bicategories are characterized by maps from the Catalan set to nerves of Bicat.

A new definition of skew-monoidal bicategory consistent with existing theory is provided.

Abstract

The Catalan simplicial set $\mathbb{C}$ is known to classify skew-monoidal categories in the sense that a map from $\mathbb{C}$ to a suitably defined nerve of $\mathrm{Cat}$ is precisely a skew-monoidal category \cite{Catalan1}. We extend this result to the case of skew monoidales internal to any monoidal bicategory $\mathcal{B}$. We then show that monoidal bicategories themselves are classified by maps from $\mathbb{C}$ to a suitably defined nerve of $\mathrm{Bicat}$ and extend this result to obtain a definition of skew-monoidal bicategory that aligns with existing theory.