Polynomial functors and opetopes

Joachim Kock*, André Joyal, Michael Batanin, J. F. Mascari

*Corresponding author for this work

    Research output: Contribution to journalArticle

    21 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.

    Original languageEnglish
    Pages (from-to)2690-2737
    Number of pages48
    JournalAdvances in Mathematics
    Issue number6
    Publication statusPublished - Aug 2010

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    Kock, J., Joyal, A., Batanin, M., & Mascari, J. F. (2010). Polynomial functors and opetopes. Advances in Mathematics, 224(6), 2690-2737. https://doi.org/10.1016/j.aim.2010.02.012