Opetopic algebras I: Algebraic structures on opetopic sets

TL;DR

This paper introduces opetopic algebras, a new class of algebraic structures on opetopic sets, providing foundational tools for modeling higher algebraic structures using opetopic spaces.

Contribution

It defines opetopic algebras via two approaches and constructs an opetopic nerve functor that embeds these algebras into opetopic sets, advancing higher algebraic modeling.

Findings

Defined opetopic algebras including categories and operads

Constructed an opetopic nerve functor fully embedding opetopic algebras

Established fully faithful nerve functors for categories and operads

Abstract

We define a family of structures called "opetopic algebras", which are algebraic structures with an underlying opetopic set. Examples of such are categories, planar operads, and Loday's combinads over planar trees. Opetopic algebras can be defined in two ways, either as the algebras of a "free pasting diagram" parametric right adjoint monad, or as models of a small projective sketch over the category of opetopes. We define an opetopic nerve functor that fully embeds each category of opetopic algebras into the category of opetopic sets. In particular, we obtain fully faithful opetopic nerve functors for categories and for planar coloured Set-operads. This paper is the first in a series aimed at using opetopic spaces as models for higher algebraic structures.