The theory of opetopes via Kelly-Mac Lane graphs

TL;DR

This paper presents a novel construction of opetopes, fundamental shapes in higher category theory, by utilizing Kelly-Mac Lane graphs to provide a clearer and more explicit framework.

Contribution

It introduces a new method for constructing opetopes using Kelly-Mac Lane graphs, linking two previous approaches for better understanding.

Findings

Opetopes are constructed explicitly using Kelly-Mac Lane graphs.

The approach simplifies the understanding of opetopic shapes.

The method connects tree-based and graph-based representations of opetopes.

Abstract

This paper follows from two earlier works. In the first we gave an explicit construction of opetopes, the underlying cell shapes in the theory of opetopic n-categories; at the heart of this construction is the use of certain trees. In the second we gave a description of trees using Kelly-Mac Lane graphs. In the present paper we apply the latter to the former, to give a construction of opetopes using Kelly-Mac Lane graphs.