Cocomplete toposes whose exact completions are toposes
type: Item
instance of: scholarly article
title: Cocomplete toposes whose exact completions are toposes
zbMATH Open document ID: 1120.18002
published in: Journal of Pure and Applied Algebra
publication date: 2007-06-26T00:00:00Z
full work available at URL: http://sedici.unlp.edu.ar/handle/10915/83213
review text: The problem of characterization of the toposes whose exact completions are toposes is stated as an open problem in the last paragraph of p.131 of \textit{A. Carboni} [J. Pure Appl. Algebra 103, 117--148 (1995; Zbl 0839.18002)]. In this paper the author shows that if \(C\) is a cocomplete topos, then if the exact completion of \(C\) is a topos, then every indecomposable object in \(C\) is an atom. As a corollary the author characterizes the locally connected Grothendieck toposes whose exact completions are toposes. This result strengthens both the Lawvere-Schanuel characterization of Boolean presheaf toposes and Hofstra's characterization of the locally connected Grothendieck toposes whose exact completion is a Grothendieck topos. Also, the author shows that for any topological space \(X,\) the exact completion of \(SH(X)\) is a topos if and only if \(X\) is discrete. The corollary in this case characterizes the Grothendieck toposes with enough points whose exact completions are toposes.
Mathematics Subject Classification ID: 18A35
Mathematics Subject Classification ID: 18B25
zbMATH DE Number: 5167560
zbMATH Keywords: indecomposable objects
zbMATH Keywords: Grothendieck topos
zbMATH Keywords: complete topos
zbMATH Keywords: Sierpinski topos
zbMATH Keywords: functors
author: Matias Menni
reviewed by: Dimitru Buşneag
Wikidata QID: Q112120962
MaRDI profile type: Publication
OpenAlex ID: W1988667793
cites work: Molecular toposes
cites work: Some free constructions in realizability and proof theory
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cites work: Q4783274
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cites work: A characterization of the left exact categories whose exact completions are toposes
DOI: 10.1016/J.JPAA.2006.10.009
description: scientific article; zbMATH DE number 5167560