### 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.

Original language | English |
---|---|

Pages (from-to) | 2690-2737 |

Number of pages | 48 |

Journal | Advances in Mathematics |

Volume | 224 |

Issue number | 6 |

DOIs | |

Publication status | Published - Aug 2010 |

## Fingerprint Dive into the research topics of 'Polynomial functors and opetopes'. Together they form a unique fingerprint.

## Cite this

*Advances in Mathematics*,

*224*(6), 2690-2737. https://doi.org/10.1016/j.aim.2010.02.012