Weak n-categories: opetopic and multitopic foundations

TL;DR

This paper extends the theory of weak n-categories by defining opetopes and multitopes using symmetric operads with categories of objects, providing new constructions and relationships between these structures.

Contribution

It generalizes opetopic and multitopic frameworks for weak n-categories using symmetric operads with categories, and introduces a slice construction for multitopes.

Findings

Defined opetopes from symmetric operads with categories of objects

Embedded 1-level generalized multicategories into symmetric operads

Constructed multitopes via a slice construction analogous to Baez-Dolan

Abstract

We generalise the concepts introduced by Baez and Dolan to define opetopes constructed from symmetric operads with a category, rather than a set, of objects. We describe the category of 1-level generalised multicategories, a special case of the concept introduced by Hermida, Makkai and Power, and exhibit a full embedding of this category in the category of symmetric operads with a category of objects. As an analogy to the Baez-Dolan slice construction, we exhibit a certain multicategory of function replacement as a slice construction in the multitopic setting, and use it to construct multitopes. We give an explicit description of the relationship between opetopes and multitopes.