Subtractive Categories

Abstract

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

Categorical (binary) difference terms and protomodularity

“…Categorical terms were used in [12] and in [33] for studying certain aspects of subtractive categories [26], which generalize (pointed) subtractive varieties of universal algebras [47]. In [13] pointed protomodular categories were axiomatized via categorical terms (under suitable completeness and cocompleteness assumptions).…”

Fibration of points and congruence modularity

“…Finally Section 7 studies the regular and exact contexts where new examples emerge. We show in particular that the regular unital categories, [3], are punctually congruence hyperextensible, the regular subtractive categories, [12], are punctually congruence hypoextensible, and that the exact finitely cocomplete congruence modular categories are Gumm.…”

“…Let us recall the following definition, see [12]: Definition 7.6. A pointed category C is subtractive when any split left punctual relation (see Definition 5.4) is punctual.…”

“…g are equal: Proof. According to [12], a category C is strongly unital if and only if it is both unital and subtractive. Consequently the result follows from Remark 5.2 and Propositions 7.4 and 7.8.…”

3 × 3 lemma for star-exact sequences

“…There is also a middle 3 × 3 lemma, which states that if the composite of the two morphisms in the middle row is null and if the top and the bottom rows are short exact, then the middle row is also short exact. It was recently proved by the second author [18] that, in any normal category (i.e., a pointed regular category where every regular epimorphism is a normal epimorphism), the upper and the lower 3 × 3 lemmas are equivalent, and they hold precisely when the normal category is subtractive [17]. The middle 3 × 3 lemma turns out to be stronger than the other two, and it is equivalent to protomodularity [3].…”