Polynomial functors and opetopes

TL;DR

This paper introduces a combinatorial, tree-based definition of opetopes, connects it to polynomial monads and classical definitions, and explores their properties and computational aspects.

Contribution

It provides a new elementary combinatorial approach to opetopes, relates it to polynomial monads, and introduces the concept of stable opetopes and their fixpoints.

Findings

Opetopes are defined via trees for easy manipulation.

The new definition aligns with Leinster's classical approach.

The calculus of opetopes is suitable for machine implementation.

Abstract

We give an elementary and direct combinatorial definition of opetopes in terms of trees, well-suited for graphical manipulation and explicit computation. To relate our definition to the classical definition, we recast the Baez-Dolan slice construction for operads in terms of polynomial monads: our opetopes appear naturally as types for polynomial monads obtained by iterating the Baez-Dolan construction, starting with the trivial monad. We show that our notion of opetope agrees with Leinster's. Next we observe a suspension operation for opetopes, and define a notion of stable opetopes. Stable opetopes form a least fixpoint for the Baez-Dolan construction. A final section is devoted to example computations, and indicates also how the calculus of opetopes is well-suited for machine implementation.