[Paper Review] Iterated icons

Eugenia Cheng, Nick Gurski|arXiv (Cornell University)|Aug 29, 2013

Homotopy and Cohomology in Algebraic Topology7 references4 citations

TL;DR

This paper introduces an iterated construction of weakly enriched categories using enriched icons as 2-cells in a symmetric monoidal bicategory. When applied twice to Cat, it yields a symmetric monoidal bicategory of partially strict tricategories, and restricting to doubly degenerate objects recovers the bicategory of 2-tuply monoidal categories, completing the Periodic Table of higher categories.

ABSTRACT

We study the totality of 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.

Motivation & Objective

• To develop a systematic method for iterated enrichment in symmetric monoidal bicategories using enriched icons as 2-cells.
• To resolve the missing bicategory of 2-tuply monoidal categories in earlier studies of the Periodic Table of higher categories.
• To construct a symmetric monoidal bicategory of partially strict tricategories via two-step enrichment starting from Cat.
• To generalize the construction to k-tuply monoidal n-categories for arbitrary k and n.

Proposed method

• Using enriched icons as 2-cells in weak enrichment over a monoidal bicategory to define higher-dimensional categorical structures.
• Applying the enrichment process iteratively, leveraging the symmetry of the underlying monoidal bicategory to ensure coherence.
• Focusing on the symmetric monoidal bicategory Cat as the initial category for enrichment.
• Restricting to doubly degenerate objects in the resulting structure to extract the bicategory of 2-tuply monoidal categories.
• Generalizing the construction to k-tuply monoidal n-categories by extending the iterated enrichment process.
• Using the coherence theorems of symmetric monoidal bicategories to ensure well-definedness and associativity of the resulting structures.

Experimental results

Research questions

• RQ1 How can enriched icons be used to iteratively construct higher-dimensional categorical structures in symmetric monoidal bicategories?
• RQ2 What structure emerges when the enrichment process is applied twice to the symmetric monoidal bicategory Cat?
• RQ3 Why is the bicategory of 2-tuply monoidal categories missing from prior formulations of the Periodic Table, and how can it be recovered?
• RQ4 What conditions ensure that iterated enrichment yields a well-defined symmetric monoidal bicategory?
• RQ5 How can the construction be generalized to k-tuply monoidal n-categories?

Key findings

• Iterated enrichment using enriched icons in a symmetric monoidal bicategory yields a well-defined symmetric monoidal bicategory of partially strict tricategories when applied twice to Cat.
• The restriction of the doubly enriched structure to doubly degenerate objects produces the correct bicategory of 2-tuply monoidal categories, resolving a gap in earlier work.
• The construction demonstrates that symmetry in the monoidal bicategory is essential for the iterated enrichment to be coherent and well-behaved.
• The resulting symmetric monoidal bicategory of partially strict tricategories provides a natural framework for studying higher categorical coherence and coherence theorems.
• The method generalizes to k-tuply monoidal n-categories, offering a systematic approach to constructing such structures via iterated enrichment.