### Abstract

We give a definition of weakn-categories based on the theory of operads. We work with operads having an arbitrary setSof types, or "S-operads," and given such an operadO, we denote its set of operations by elt(O). Then for anyS-operadOthere is an elt(O)-operadO^{+}whose algebras areS-operads overO. LettingIbe the initial operad with a one-element set of types, and definingI^{0+}=I,I^{(i+1)+}=(I^{i+})^{+}, we call the operations ofI^{(n-1)+}the "n-dimensional opetopes." Opetopes form a category, and presheaves on this category are called "opetopic sets." A weakn-category is defined as an opetopic set with certain properties, in a manner reminiscent of Street's simplicial approach to weakω-categories. Similarly, starting from an arbitrary operadOinstead ofI, we define "n-coherentO-algebras," which arentimes categorified analogs of algebras ofO. Examples include "monoidaln-categories," "stablen-categories," "virtualn-functors" and "representablen-prestacks." We also describe hown-coherentO-algebra objects may be defined in any (n+1)-coherentO-algebra.

Original language | English |
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Pages (from-to) | 145-206 |

Number of pages | 62 |

Journal | Advances in Mathematics |

Volume | 135 |

Issue number | 2 |

DOIs | |

Publication status | Published - 10 May 1998 |

Externally published | Yes |

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## Cite this

*Advances in Mathematics*,

*135*(2), 145-206. https://doi.org/10.1006/aima.1997.1695