Mal'cev conditions revisited
type: Item
instance of: scholarly article
title: Mal'cev conditions revisited
zbMATH Open document ID: 1171.18002
published in: Applied Categorical Structures
publication date: 2009-07-13T00:00:00Z
review text: A full subcategory \({\mathcal L}\) of a locally presentable category \({\mathcal K}\) is a cosieve if for every \({\mathcal K}\)-morphism \(f:L\to K\) from \(L\in {\mathcal L}\) it follows \(K\in {\mathcal L}\). A cosieve \({\mathcal L}\) is set generated if there exists a set \({\mathcal X}\) of objects from \({\mathcal L}\) such that \(K\in {\mathcal L}\) exactly when there exist \(X\in {\mathcal X}\) and a \({\mathcal K}\)-morphism from \(X\) into \(K\). If \({\mathcal X}\) consists of exactly one object then \({\mathcal L}\) is principal. The Vopěnka principle says that there exists no large discrete full subcategory of a locally presentable category and a semiweak Vopěnka principle says that for no locally presentable category \({\mathcal K}\) there exists a family \(\{A_i\mid i\) is an ordinal\} of \({\mathcal K}\)-objects such that \(\Hom_{{\mathcal K}}(A_i,A_j)\neq\emptyset\) if and only if \(i\geq j\). It is proved that for a cosieve \({\mathcal L}\) of a locally presentable category are equivalent: \({\mathcal L}\) is set generated; \({\mathcal L}\) is closed under \(\lambda\)-pure subobjects for some regular cardinal \(\lambda\); \({\mathcal L}\) is an accessible category. The Vopěnka principle is equivalent with the fact that every cosieve of a locally presentable category is set generated and the semiweak Vopěnka principle is equivalent to the fact that every cosieve of a locally presentable category closed under products is principal. Set generated cosieves of locally presentable categories are investigated. Let \({\mathcal T}h\) be the category of algebraic theories then \({\mathcal T}h\) is locally presentable and the consequences for cosieves of \({\mathcal T}h\) are presented and their relation to Mal'cev conditions is discussed.
reviewed by: Václav Koubek
Mathematics Subject Classification ID: 18C10
Mathematics Subject Classification ID: 08B05
Mathematics Subject Classification ID: 18C35
zbMATH DE Number: 5577701
zbMATH Keywords: sieve
zbMATH Keywords: locally presentable category
zbMATH Keywords: algebraic theory
zbMATH Keywords: Mal'cev condition
zbMATH Keywords: Mal'cev term
author: Dominique Bourn
author: Jiří Rosický
MaRDI profile type: Publication
full work available at URL: s10485-007-9114-3
OpenAlex ID: W1975930203
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DOI: 10.1007/S10485-007-9114-3
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description: scientific article; zbMATH DE number 5577701