An alternative characterisation of universal cells in opetopic n-categories

TL;DR

This paper introduces a new way to characterize universal cells in opetopic n-categories by defining composition through spans of (n-k)-categories and identifying universal cells via equivalences of these spans.

Contribution

It provides a novel characterization of universal cells in opetopic n-categories using spans and equivalences, addressing non-uniqueness of composition.

Findings

Composition in opetopic weak n-categories is generally non-unique.

Universal cells are characterized by their composition spans being equivalences.

The approach clarifies the structure of compositions in opetopic n-categories.

Abstract

We address the fact that composition in an opetopic weak n-category is in general not unique and hence is not a well-defined operation. We define composition with a given k-cell in an n-category by a span of (n-k)-categories. We characterise such a cell as universal if its composition span gives an equivalence of (n-k)-categories.