The Web Monoid and Opetopic Sets

TL;DR

This paper introduces a new definition of opetopic sets using fibrations and the web monoid, clarifying their construction and relating them to existing operadic frameworks.

Contribution

It presents a novel approach to defining opetopic sets with explicit fibrations and introduces the web monoid, connecting it to the Baez-Dolan slice construction.

Findings

Clarified the construction of opetopic sets using fibrations.

Introduced the web monoid concept.

Linked the web monoid to the Baez-Dolan slice construction.

Abstract

We develop a new definition of opetopic sets. There are two main technical ingredients. The first is the systematic use of fibrations, which are implicit in most of the approaches in the literature. Their explicit use leads to certain clarifications in the construction of opetopic sets and other constructions. The second is the "web monoid", which plays a role analogous to the "operad for operads" of Baez and Dolan, the "multicategory of function replacement" of Hermida, Makkai and Power. We demonstrate that the web monoid is closely related to the "Baez-Dolan slice construction" as defined by Kock, Joyal, Batanin and Mascari.