Applied Categorical Structures

, Volume 17, Issue 4, pp 317–343 | Cite as

Subtractive Categories and Extended Subtractions

Article

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 

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Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques Pures et AppliquéesUniversité du LittoralCalaisFrance
  2. 2.Department of Mathematics and Applied MathematicsUniversity of Cape TownCape TownSouth Africa

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