Subtractive categories and extended subtractions
type: Item
instance of: scholarly article
title: Subtractive categories and extended subtractions
zbMATH Open document ID: 1176.18005
author: Dominique Bourn
author: Zurab Janelidze
published in: Applied Categorical Structures
publication date: 2009-08-31T00:00:00Z
review text: A pointed, finitely complete category \({\mathcal C}\) is subtractive if for every binary relation \(R\to X\times Y\) the corresponding relation \(\Hom_{{\mathcal C}}(C,R)\to\Hom_{{\mathcal C}} (C,X)\times\Hom_{{\mathcal C}}(C,Y)\) satisfies \((x,0_{C,Y})\in R\) whenever \((x,y)\in R\) and \((0_{C,X}, y)\in R\). A morphism \(s:X\times X\to X\) is a subtraction on \(X\) if \(s\circ u= 1_X\) and \(s\circ\Delta =0_X\) where \(u,\Delta :X\to X\times X\) are \({\mathcal C}\)-morphisms with \(\pi_1\circ u=1_X\), \(\pi_2\circ u=0_X\), \(\pi_1\circ\Delta =\pi_2\circ \Delta =1_X\) (\(\pi_1,\pi_2:X\times X\to X\) are projections). Let \(Ab({\mathcal C})\) be a full subcategory of \({\mathcal C}\) formed by all objects with subtraction. A sufficient condition under that \(Ab({\mathcal C})\) is a reflective subcategory of \({\mathcal C}\) is given. An extended operation is introduced. It is a tool for a categorical formulation of term equations of Malt'sev type. As a consequence an approximate subtraction, an extended subtraction and an extended category are defined and characterized. For example strongly unital categories are described. An extended operation is an `algebraic tool' for an investigation of subtractive categories. A relation to universal algebra is shown.
reviewed by: Václav Koubek
Mathematics Subject Classification ID: 18D35
Mathematics Subject Classification ID: 18E05
Mathematics Subject Classification ID: 18C99
Mathematics Subject Classification ID: 08B05
zbMATH DE Number: 5600111
zbMATH Keywords: Malt'sev category
zbMATH Keywords: strongly unital category
zbMATH Keywords: subtractive category
zbMATH Keywords: subtraction
MaRDI profile type: Publication
full work available at URL: s10485-008-9182-z
OpenAlex ID: W2094562030
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DOI: 10.1007/S10485-008-9182-Z
description: scientific article; zbMATH DE number 5600111