We introduce a notion of a subtractive category. It generalizes the notion of a pointed subtractive variety of universal algebras in the sense of A. Ursini. Subtractive categories are closely related to Mal'tsev and additive categories: (i) a category C with finite limits is a Mal'tsev category if and only if for every object X in C the category Pt(X) = ((X, 1 X ) ↓ (C ↓ X)) of "points over X" is subtractive; (ii) a pointed category C with finite limits is additive if and only if C is subtractive and half-additive. (2000): 18C99, 18E05, 08B05.
Mathematics Subject Classifications
We introduce a notion of an extended operation which should serve as a new tool for the study of categories like Mal'tsev, unital, strongly unital and subtractive categories. However, in the present paper we are only concerned with subtractive categories, and accordingly, most of the time we will deal with extended subtractions, which are particular instances of extended operations. We show that these extended subtractions provide new conceptual characterizations of subtractive categories and moreover, they give an enlarged "algebraic tool" for working in a subtractive category-we demonstrate this by using them to describe the construction of associated abelian objects in regular subtractive categories with finite colimits. Also, the definition and some basic properties of abelian objects in a general subtractive category is given for the first time in the present paper.Keywords Mal'tsev category · Unital category · Strongly unital category · Subtractive category · Abelian objects · Natural operations Mathematics Subject Classifications (2000) 18C99 · 18D35 · 18E05 · 08B05
We extend the notion of an ‘‘ideal’’ from regular pointed categories\ud
to regular multi-pointed categories, and having ‘‘a good theory of ideals’’ will mean that\ud
there is a bijection between ideals and kernel pairs, which in the pointed case is the main\ud
property of ideal determined categories. The study of general categories with a good theory\ud
of ideals allows in fact a simultaneous treatment of ideal determined and Barr exact Goursat\ud
categories:weprove that in the case when all morphisms are chosen as null morphisms, the\ud
presence of a good theory of ideals becomes precisely the property for a regular category to\ud
be a Barr exact Goursat category
A pointed variety of universal algebras is protomodular in the sense of D. Bourn, if and only if it is classically ideal determined in the sense of A. Ursini (this result is due to D. Bourn and G. Janelidze). We prove a characterization theorem for pointed protomodular categories, which is a (pointed) categorical version of Ursini's characterization theorem for classically ideal determined varieties, involving classically 0-regular algebras. A suitable simplification of the property of a pair of relations, which is used to define a classically 0-regular algebra, yields a new closedness property of a single binary relation -we show that a finitely complete pointed category is protomodular if and only if every binary internal relation R → A 2 in it has this closedness property.
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