### Abstract

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 |
---|---|

Pages (from-to) | 32-114 |

Journal | Cahiers de Topologie et Géométrie Différentielle Catégoriques |

Volume | LX |

Issue number | 1 |

Publication status | Published - 2019 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Algebra and Number Theory
- Geometry and Topology

### Cite this

**A study of Penon weak n-categories Part 2: A multisimplicial nerve construction.** / Cottrell, Thomas.

Research output: Contribution to journal › Article

*Cahiers de Topologie et Géométrie Différentielle Catégoriques*, vol. LX, no. 1, pp. 32-114.

}

TY - JOUR

T1 - A study of Penon weak n-categories Part 2: A multisimplicial nerve construction

AU - Cottrell, Thomas

PY - 2019

Y1 - 2019

N2 - In this paper we take the first step towards a comparison betweenan 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 nerveconstruction, we recall a nerve construction for bicategories proposed by Leinster, and prove that the nerve of a bicategory given by this construction isa Tamsamani–Simpson weak 2-category. We then define our nerve functorfor 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 thisresult holds for higher n.

AB - In this paper we take the first step towards a comparison betweenan 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 nerveconstruction, we recall a nerve construction for bicategories proposed by Leinster, and prove that the nerve of a bicategory given by this construction isa Tamsamani–Simpson weak 2-category. We then define our nerve functorfor 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 thisresult holds for higher n.

KW - n-category

KW - higher-dimensional category

KW - nerve construction

M3 - Article

VL - LX

SP - 32

EP - 114

JO - Cahiers de Topologie et Géométrie Différentielle Catégoriques

JF - Cahiers de Topologie et Géométrie Différentielle Catégoriques

SN - 1245-530X

IS - 1

ER -