Subtractive categories and extended subtractions (Q835752)

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.