Symmetry of Regular Diamonds, the Goursat Property, and Subtractivity

We investigate 3-permutability, in the sense of universal algebra, in an abstract categorical setting which unifies the pointed and the non-pointed contexts in categorical algebra. This leads to a unified treatment of regular subtractive categories and of regular Goursat categories, as well as of E-subtractive varieties (where E is the set of constants in a variety) recently introduced by the fourth author. As an application, we show that “ideals” coincide with “clots” in any regular subtractive category, which can be considered as a pointed analogue of a known result for regular Goursat categories. Introduction The concept of a category equipped with an idealN of morphisms in the sense of C. Ehresmann [6], which was used by M. Grandis in [9] in his “categorical foundation of homological and homotopical algebra”, turns out to have yet another interesting use in modern categorical algebra, where it gives a suitable general context for comparing and unifying results from pointed and non-pointed contexts. The pointed context is captured by choosing N to be the class of zero morphisms of a pointed category, while the non-pointed context, which we call the total context, is given when N is the class of all morphisms of a category. In [7] it was shown that the notion of an ideal determined category [12] can be conveniently extended from the pointed context to the context of a general N , so that in the total context it becomes the notion of a Barr exact [2] Goursat category [5, 4]. Such an extension is based on replacing the notion of a kernel from the pointed context, not with the standard notion of a kernel with respect to a class N (used in e.g. [9]), which trivializes in the total context, but with the notion of a “star-kernel” introduced in [7] (which was called a “kernel star” there), which in the total context becomes the notion of a kernel pair. In the present paper we study the Goursat property beyond Barr exactness and show that its pointed counterpart is precisely subtractivity [15]. In this process we establish a Research supported by F.N.R.S. grant Crédit aux chercheurs 1.5.016.10F, by South African National Research Foundation and by Georgian National Science Foundation (GNSF/ST09 730 3-105), by CMUC/FCT (Portugal) and the FCT Grant PTDC/MAT/120222/2010 through the European program COMPETE/FEDER, and by the Université catholique de Louvain. Received by the editors 2012-01-30 and, in revised form, 2012-07-17. Published on 2012-07-31 in the volume of articles from CT2011. 2000 Mathematics Subject Classification: 18D99, 18C99, 18C05, 08B05.