Protomodular aspect of the dual of a topos
type: Item
instance of: scholarly article
title: Protomodular aspect of the dual of a topos
zbMATH Open document ID: 1047.18006
author: Dominique Bourn
published in: Advances in Mathematics
publication date: 2004-08-19T00:00:00Z
review text: The notion of protomodular category has been extensively investigated (principally by the author of this paper) in recent years, as a category having enough of the `intrinsic' algebraic structure of the category of groups to permit the development of much of the machinery of nonabelian homological algebra. It was first observed by the reviewer that the category of Heyting algebras is protomodular; indeed, it provides the best known example of a protomodular variety which does not admit a weakly associative Mal'cev operation. Since the contravariant power-object functor on any topos \(\mathcal E\) can be regarded as taking values in the category of internal Heyting algebras in \(\mathcal E\), it follows easily that the dual of any topos is protomodular (as well as being Barr-exact and congruence distributive). In the present paper, the author investigates the significance of the known results about Barr-exact protomodular categories as applied to the duals of toposes, and shows that they include several familiar exactness properties of toposes.
Mathematics Subject Classification ID: 18B25
Mathematics Subject Classification ID: 06D20
Mathematics Subject Classification ID: 08B05
Mathematics Subject Classification ID: 18C10
Mathematics Subject Classification ID: 18D30
zbMATH DE Number: 2093010
zbMATH Keywords: topos
zbMATH Keywords: fibrations
zbMATH Keywords: Mal'cev and protomodular categories
reviewed by: Peter Johnstone
MaRDI profile type: Publication
full work available at URL: j.aim.2003.09.004
OpenAlex ID: W2032520459
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DOI: 10.1016/J.AIM.2003.09.004
description: scientific article; zbMATH DE number 2093010