Simple homotopy theory and nerve theorem for categories
type: Item
instance of: scholarly article
title: Simple homotopy theory and nerve theorem for categories
zbMATH Open document ID: 1461.55013
author: Kohei Tanaka
published in: Topology and its Applications
publication date: 2021-03-09T00:00:00Z
review text: The paper develops some concepts of simple homotopy theory in the context of combinatorial topology. The discussion concentrates in particular on \(\Delta\)-complexes and loop-free categories. The former is a general form of a semi-simplicial set, while the latter is a category with trivial endomorphism sets, and where the existence of a morphism \(x\to y\) excludes the existence of a morphism the other way. Both concepts are well used, specifically in topological data analysis. The paper extends work of \textit{J. A. Barmak} and \textit{E. G. Minian} [Adv. Math. 218, No. 1, 87--104 (2008; Zbl 1146.57034)]. In particular a version of Quillen's theorem A. The paper also gives two generalisations of the classical nerve theorem. The first is to what is referred to as quasi-good covers, i.e. open covers such that every non-empty finite intersection has contractible connected components. The second generalisation is to \(\pi_1\)-good covers, i.e. covers such that every non-empty finite intersection is a \(K(\pi,1)\) space. The paper also offers an application of the nerve theorem to topological data analysis.
reviewed by: Ran Levi
Mathematics Subject Classification ID: 55U05
Mathematics Subject Classification ID: 05E18
Mathematics Subject Classification ID: 05E45
Mathematics Subject Classification ID: 57Q10
Mathematics Subject Classification ID: 55N31
Mathematics Subject Classification ID: 57Q70
Mathematics Subject Classification ID: 06A06
zbMATH DE Number: 7318664
zbMATH Keywords: simplicial complex
zbMATH Keywords: \(\Delta\)-complex
zbMATH Keywords: poset
zbMATH Keywords: small category
zbMATH Keywords: simple homotopy
zbMATH Keywords: nerve theorem
describes a project that uses: Perseus
MaRDI profile type: Publication
full work available at URL: j.topol.2021.107609
OpenAlex ID: W3123086240
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DOI: 10.1016/J.TOPOL.2021.107609
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description: scientific article; zbMATH DE number 7318664