An elementary definition of opetopic sets
Taichi Uemura
TL;DR
The paper delivers an elementary, axiomatic framework for opetopic sets and opetopes, providing definitions that avoid heavy prerequisites and proving equivalence with the established polynomial-monad and generator-relations formulations. It shows that the category of opetopic sets is a presheaf category over the opetope category $\mathbb{O}$, and that $\mathbb{O}$ is terminal in $\mathbf{OSet}$ with $\mathbf{OSet} \simeq \mathbf{DFib}(\mathbb{O})$, enabling straightforward colimit computations. By developing boundaries, pasting diagrams, substitution, and grafting, the work unifies multiple viewpoints and demonstrates that the elementary opetopes coincide with the KJBM definition and with Ho Thanh’s presentation. The framework is constructive within Univalent Foundations, providing a clear, accessible path to higher-dimensional categorical structures and their presheaf semantics. Overall, the paper offers a concise, rigorous synthesis that clarifies and connects diverse definitions of opetopes and opetopic sets, with practical tools for computation and comparison across theories.
Abstract
We propose elementary definitions of opetopes and opetopic sets. We directly define opetopic sets by a simple structure and several axioms. Opetopes are then opetopic sets satisfying one more axiom. We show that our definition is equivalent to the polynomial monad definition given by Kock, Joyal, Batanin, and Mascari. We also show that our category of opetopes is equivalent to the one given by Ho Thanh.