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Subtractive Categories and Extended Subtractions

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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.

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Authors and Affiliations

  1. Laboratoire de Mathématiques Pures et Appliquées, Université du Littoral, Bat. H. Poincaré, 50 Rue F. Buisson, BP 699, 62228, Calais, France

    Dominique Bourn

  2. Department of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch 7701, Cape Town, South Africa

    Zurab Janelidze

Authors
  1. Dominique Bourn
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  2. Zurab Janelidze
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Corresponding author

Correspondence to Dominique Bourn.

Additional information

The second author acknowledges the support of Claude Leon Foundation, INTAS (06-1000017-8609) and Georgian National Science Foundation (GNSF/ST06/3-004).

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Bourn, D., Janelidze, Z. Subtractive Categories and Extended Subtractions. Appl Categor Struct 17, 317–343 (2009). https://doi.org/10.1007/s10485-008-9182-z

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  • DOI: https://doi.org/10.1007/s10485-008-9182-z

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