The authors clarify the role of Hofmann's Axiom in the old-style definition of a semi-abelian category. By removing this axiom they obtain the categorical counterpart of the notion of an ideal determined variety of universal algebras - which they therefore call an ideal determined category. Using known counter-examples from universal algebra they conclude that there are ideal determined categories which fail to be Mal'tsev. They also show that there are ideal determined Mal'tsev categories which fail to be semi-abelian.
Janelidze, G., Marki, L., Tholen, W., Ursini, A. (2010). Ideal-determined categories. CAHIERS DE TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE CATÉGORIQUES, 51(2), 115-125.
Ideal-determined categories
URSINI, ALDO
2010-01-01
Abstract
The authors clarify the role of Hofmann's Axiom in the old-style definition of a semi-abelian category. By removing this axiom they obtain the categorical counterpart of the notion of an ideal determined variety of universal algebras - which they therefore call an ideal determined category. Using known counter-examples from universal algebra they conclude that there are ideal determined categories which fail to be Mal'tsev. They also show that there are ideal determined Mal'tsev categories which fail to be semi-abelian.
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https://hdl.handle.net/11365/24607
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