We propose elementary definitions of opetopic sets and opetopes. We define opetopic sets by a simple structure on a direct category 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.
Keywords: opetopic sets, opetopes, substitution, grafting, polynomial, polynomial monad
2020 MSC: 18N20, 18N30, 18C15
Theory and Applications of Categories, Vol. 45, 2026, No. 6, pp 206-245.
Published 2026-01-27.
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