Polynomial functors and opetopes

Joachim Kock, André Joyal, Michael Batanin, Jean François Mascari

Research output: Contribution to journalArticleResearchpeer-review

27 Citations (Scopus)


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. © 2010 Elsevier Inc.
Original languageEnglish
Pages (from-to)2690-2737
JournalAdvances in Mathematics
Issue number6
Publication statusPublished - 1 Aug 2010


  • Monad
  • Opetope
  • Polynomial functor
  • Tree


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