Finite preorders and topological descent. II: étale descent
type: Item
instance of: scholarly article
title: Finite preorders and topological descent. II: étale descent
zbMATH Open document ID: 1018.18005
DOI: 10.1016/S0022-4049(02)00046-4
author: George Janelidze
author: Manuela Sobral
published in: Journal of Pure and Applied Algebra
publication date: 2003-02-05T00:00:00Z
review text: [For part I, see ibid. 175, 187-205 (2002; Zbl 1018.18004).] This paper provides a characterization of effective étale-descent morphisms of finite topological spaces, equivalently of finite preordered sets. More precisely, let \(p:E\to B\) be a continuous map of finite topological spaces, \({\mathcal E}(E)\) and \({\mathcal E}(B)\) the categories of finite étale bundles over \(E\) and \(B\) respectively, and \(p^*\): \({\mathcal E}(B)\to{\mathcal E} (E)\) the pullback functor along \(p\). It is proved that \(p\) is an effective étale-descent morphism, that is \(p\) is a monadic functor, if and only if the three following properties are fullfilled: (1) The map \(p:E\to p(E)\) is a regular quotient of topological spaces. (2) The universal category \({\mathcal Z} (Eq(p))\) associated to the double category associated to the equivalent relation \(Eq(p)\) on \(E\) determined by \(p\), is a preorder. (3) The map \(p\) is essentially surjective, i.e., up to specialization and generization.
Mathematics Subject Classification ID: 18B30
Mathematics Subject Classification ID: 18A20
Mathematics Subject Classification ID: 18B35
Mathematics Subject Classification ID: 18C20
Mathematics Subject Classification ID: 18D05
zbMATH DE Number: 1864811
zbMATH Keywords: finite topological descent
zbMATH Keywords: effective étale-descent
reviewed by: Yves Diers
MaRDI profile type: Publication
cites work: Facets of descent. I
cites work: Facets of descent. II
cites work: Effective descent maps of topological spaces
full work available at URL: s0022-4049(02)00046-4
OpenAlex ID: W1995014195
description: scientific article; zbMATH DE number 1864811