A note on the Penon definition of $n$-category

TL;DR

This paper examines the Penon definition of n-categories, revealing limitations in its ability to produce all tricategories and proposing a modified definition using non-reflexive globular sets to address these issues.

Contribution

It identifies a problem with the original Penon definition and introduces a revised approach that overcomes this limitation in modeling tricategories.

Findings

Doubly degenerate Penon tricategories yield symmetric monoidal categories.

Original Penon tricategories do not encompass all tricategories.

Modified definition with non-reflexive globular sets resolves the issue.

Abstract

We show that doubly degenerate Penon tricategories give symmetric rather than braided monoidal categories. We prove that Penon tricategories cannot give all tricategories, but we show that a slightly modified version of the definition rectifies the situation. We give the modified definition, using non-reflexive rather than reflexive globular sets, and show that the problem with doubly degenerate tricategories does not arise.