Simplicial methods for higher categories. Segal-type models of weak \(n\)-categories
type: Item
instance of: scholarly article
title: Simplicial methods for higher categories. Segal-type models of weak \(n\)-categories
zbMATH Open document ID: 1428.18001
DOI: 10.1007/978-3-030-05674-2
author: Simona Paoli
published in: Algebra and Applications
publication date: 2019-02-25T00:00:00Z
review text: \textbf{The setting.} The appropriate notion of weak \(n\)-category has been a subject of forefront research in the recent years. Various authors in the works \begin{itemize} \item[-] \textit{R. Street}, J. Pure Appl. Algebra 49, 283--335 (1987; Zbl 0661.18005) \item[-] \textit{A. Joyal}, ``Disks, duality and \(\Theta\)-categories'', unpublished preprint (1997) \item[-] \textit{M. A. Batanin}, Adv. Math. 136, No. 1, 39--103 (1998; Zbl 0912.18006) \item[-] \textit{J. C. Baez} and \textit{J. Dolan}, Adv. Math. 135, No. 2, 145--206 (1998; Zbl 0909.18006) and \item[] \textit{C. Hermida}, \textit{M. Makkai} and \textit{J. Power}, J. Pure Appl. Algebra 154, No. 1--3, 221--246 (2000; Zbl 0971.18005); ibid. 157, No. 2--3, 247--277 (2001; Zbl 0985.18006) and ibid. 166, No. 1--2, 83--104 (2002; Zbl 0992.18005) \item[-] \textit{J. Penon}, Cah. Topologie Géom. Différ. Catégoriques 40, No. 1, 31--80 (1999; Zbl 0918.18006) \item[-] \textit{Z. Tamsamani}, \(K\)-Theory 16, No. 1, 51--99 (1999; Zbl 0934.18008) and \item[] \textit{C. Simpson}, Homotopy theory of higher categories. From Segal categories to \(n\)-categories and beyond. Cambridge: Cambridge University Press (2012; Zbl 1232.18001) \item[-] \textit{T. Trimble,} whose ideas were published first in [\textit{T. Leinster}, Theory Appl. Categ. 10, 1--70 (2002; Zbl 0987.18007)] \item[-] \textit{T. Leinster}, Theory Appl. Categ. 12, 73--194 (2004; Zbl 1065.18006) \item[-] \textit{J. Kock}, IMRP, Int. Math. Res. Pap. 2006, No. 1, 54 p. (2006; Zbl 1158.18008) \end{itemize} proposed different definitions that were motivated by, and applied in different situations. In the research monograph under review not only a basically new variant and its study can be found but also a principally new approach to the subject. This is based on the particularly simple structure of iterated internal categories (that is, \(n\)-fold categories). One of the main results of the book is that the new approach is homotopically equivalent to the Tamsamani-…
Mathematics Subject Classification ID: 18A05
Mathematics Subject Classification ID: 18-02
Mathematics Subject Classification ID: 18N10
Mathematics Subject Classification ID: 18N40
Mathematics Subject Classification ID: 55-02
Mathematics Subject Classification ID: 55U40
Mathematics Subject Classification ID: 18N50
Mathematics Subject Classification ID: 55P10
Mathematics Subject Classification ID: 55U35
zbMATH DE Number: 7029566
zbMATH Keywords: weak \(n\)-category
zbMATH Keywords: \(n\)-fold category
zbMATH Keywords: weak globularity
zbMATH Keywords: Tamsamani-Simpson model
zbMATH Keywords: multinerve
zbMATH Keywords: Segal condition
zbMATH Keywords: homotopically discrete category
zbMATH Keywords: homotpy hypothesis
reviewed by: Gabriella Böhm
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-3-030-05674-2
OpenAlex ID: W4239678534
description: scientific article; zbMATH DE number 7029566