## Abstract

Classical definitions of weak higher-dimensional categories are given inductively, for example, a bicategory has a set of objects and hom categories, and a tricategory has a set of objects and hom bicategories. However, more recent definitions of weak n-categories for all natural numbers n, or of weak -categories, take more sophisticated approaches, and the nature of the 'hom is often not immediate from the definitions'. In this paper, we focus on Leinster's definition of weak -category based on an earlier definition by Batanin and construct, for each weak -category, an underlying (weak -category)-enriched graph consisting of the same objects and for each pair of objects x and y, a hom weak -category. We also show that our construction is functorial with respect to weak -functors introduced by Garner.

Original language | English |
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Pages (from-to) | 420-441 |

Number of pages | 22 |

Journal | Mathematical Structures in Computer Science |

Volume | 32 |

Issue number | 4 |

DOIs | |

Publication status | Published - 12 Apr 2022 |

### Bibliographical note

Funding Information:We gratefully acknowledge the support of Royal Society grant IE160402. The second author is supported by ERATO HASUO Metamathematics for Systems Design Project (No. JPMJER1603), JST. No data were generated in association with this paper.

## Keywords

- Weak ω-category
- identity type
- intensional Martin-Löf type theory
- operad
- weak ω-functor
- weak ω-groupoid

## ASJC Scopus subject areas

- Mathematics (miscellaneous)
- Computer Science Applications