### Abstract

In this paper we take the first step towards a comparison between

an algebraic and a non-algebraic definition of weak n-category. This comparison takes the form of a nerve functor, the established method of moving from the algebraic setting to the non-algebraic setting. The algebraic definition we use is that due to Penon, and the non-algebraic definition we use is Simpson’s variant of Tamsamani’s definition. As a prototype for our nerve

construction, we recall a nerve construction for bicategories proposed by Leinster, and prove that the nerve of a bicategory given by this construction is

a Tamsamani–Simpson weak 2-category. We then define our nerve functor

for Penon weak n-categories. We prove that the nerve of a Penon weak 2-

category is a Tamsamani–Simpson weak 2-category, and conjecture that this

result holds for higher n.

an algebraic and a non-algebraic definition of weak n-category. This comparison takes the form of a nerve functor, the established method of moving from the algebraic setting to the non-algebraic setting. The algebraic definition we use is that due to Penon, and the non-algebraic definition we use is Simpson’s variant of Tamsamani’s definition. As a prototype for our nerve

construction, we recall a nerve construction for bicategories proposed by Leinster, and prove that the nerve of a bicategory given by this construction is

a Tamsamani–Simpson weak 2-category. We then define our nerve functor

for Penon weak n-categories. We prove that the nerve of a Penon weak 2-

category is a Tamsamani–Simpson weak 2-category, and conjecture that this

result holds for higher n.

Original language | English |
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Pages (from-to) | 32-114 |

Journal | Cahiers de Topologie et Géométrie Différentielle Catégoriques |

Volume | LX |

Issue number | 1 |

Publication status | Published - 2019 |

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### Keywords

- n-category
- higher-dimensional category
- nerve construction

### ASJC Scopus subject areas

- Algebra and Number Theory
- Geometry and Topology