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

Research output: Contribution to journalArticle

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.
Original languageEnglish
Pages (from-to)32-114
JournalCahiers de Topologie et Géométrie Différentielle Catégoriques
VolumeLX
Issue number1
Publication statusPublished - 2019

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Nerve
Bicategory
Functor
Prototype

Keywords

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

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Geometry and Topology

Cite this

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title = "A study of Penon weak n-categories Part 2: A multisimplicial nerve construction",
abstract = "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.",
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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.

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