Iterated icons

TL;DR

This paper explores the iterative process of constructing higher-dimensional categories using enriched icons within symmetric monoidal bicategories, leading to models of partially strict tricategories and insights into the Periodic Table of n-categories.

Contribution

It introduces a method to iterate enriched icon constructions in symmetric monoidal bicategories, producing models of higher categories and connecting to the Periodic Table of n-categories.

Findings

Iterating enriched icon construction yields symmetric monoidal bicategories of partially strict tricategories.

Restricting to doubly degenerate cases recovers the bicategory of 2-tuply monoidal categories.

The approach generalizes to all k-tuply monoidal n-categories.

Abstract

We study the totality of categories weakly enriched in a monoidal bicategory using a notion of enriched icon as 2-cells. We show that when the monoidal bicategory in question is symmetric then this process can be iterated. We show that starting from the symmetric monoidal bicategory Cat and performing the construction twice yields a convenient symmetric monoidal bicategory of partially strict tricategories. We show that restricting to the doubly degenerate ones immediately gives the correct bicategory of "2-tuply monoidal categories" missing from our earlier studies of the Periodic Table. We propose a generalisation to all k-tuply monoidal n-categories.