Subtractive Categories and Extended Subtractions

Abstract

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

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).…”

“…For example, K. Fichtner's condition [21] characterizing 0-regular varieties, and the condition defining A. Ursini's subtractive varieties [47] (which first appears in [23]). The (pointed) categorical versions of these two conditions were considered in [13] and [12], respectively.…”

“…We call such a pair (C, E) an extended category, as we think of the elements of E(X, Y ) as "extra morphisms" added to C (an extended category is also the same as a "one-dimensional" equipment [15]). The term "extended category" was first used in [12] with a more restricted meaning; namely, in the definition of an extended category given in [12] we required hom C to be a subfunctor of E. Dropping this requirement turned out to be a crucial step in generalizing the characterization of pointed protomodularity obtained in [13] to the non-pointed context, which is one of the main goals of the present paper.…”

“…Then δ is the subtraction of an abelian group structure on hom C (see e.g. [7], [30], [12]), which makes C into an additive category. Remark 7.16.…”

Split Short Five Lemma for Clots and Subtractive Categories

“…Again, in the pointed case this notion has a categorical counterpart, called a subtractive category [19] (see also [23] and [12]). In the present paper we propose another pointed categorical version of the notion of a subtractive variety, which we call a homologically subtractive category, and which is defined by a special weakening of the split short five lemma.…”