Every étendue comes from a local equivalence relation
type: Item
instance of: scholarly article
title: Every étendue comes from a local equivalence relation
zbMATH Open document ID: 0779.18005
DOI: 10.1016/0022-4049(92)90118-Y
author: Anders Kock
author: Ieke Moerdijk
published in: Journal of Pure and Applied Algebra
publication date: 1993-05-16T00:00:00Z
review text: A topos \({\mathcal E}\) is an étendue iff there is an object \(U\) in \({\mathcal E}\) covering the terminal object such that the topos \({\mathcal E}/U\) is a localic topos, in other words equivalent to a topos of sheaves on a locale. By an equivalence relation on a locale \(L\) one means a sublocale \(R\subseteq L\times L\) satisfying the expected properties of reflexivity, symmetry and transitivity, together with the requirement that the projections \(R\to L\) are both open maps of locales. The equivalence relations on \(L\) form a sheaf and a global section of this sheaf is referred to as a local equivalence relation. Given such a local equivalence relation \(r\), there is the notion of an \(r\)-sheaf and one obtains a category \(\text{sh}(L,r)\) of \(r\)-sheaves on \(L\). If \(r\) is a locally simply connected local equivalence relation, then \(\text{sh}(L,r)\) is an étendue and moreover there is the main result of the article, namely that every étendue \({\mathcal E}\) is equivalent to a topos of the form \(\text{sh}(L,r)\), where \(r\) is locally simply connected.
reviewed by: K. I. Rosenthal
Mathematics Subject Classification ID: 18B25
zbMATH DE Number: 166141
zbMATH Keywords: étendue
zbMATH Keywords: localic topos
zbMATH Keywords: topos of sheaves on a locale
zbMATH Keywords: global section
zbMATH Keywords: local equivalence relation
MaRDI profile type: Publication
cites work: Seminar of algebraic geometry du Bois-Marie 1963--1964. Topos theory and étale cohomology of schemes (SGA 4). Vol. 1: Topos theory. Exp. I--IV
cites work: Atomic toposes
cites work: Q3926769
cites work: Molecular toposes
cites work: Q5580926
cites work: Toposes as homotopy groupoids
cites work: An extension of the Galois theory of Grothendieck
cites work: A Godement theorem for locales
cites work: Q3986545
cites work: Spaces with local equivalence relations, and their monodromy
cites work: Colimits of accessible categories
cites work: An elementary proof of the descent theorem for Grothendieck toposes
cites work: Morita equivalence for continuous groups
cites work: The Classifying Topos of a Continuous Groupoid. I
cites work: Q5185230
description: scientific article; zbMATH DE number 166141