A good theory of ideals in regular multi-pointed categories

By a multi-pointed category we mean a category C equipped with an ideal of null morphisms, i.e. a class N of morphisms satisfying f∈N∨g∈N⇒fg∈N for any composable pair f,g of morphisms. Such categories are precisely the categories enriched in the category of pairs X=(X,N) where X is a set and N is a subset of X, whereas a pointed category has the same enrichment, but restricted to those pairs X=(X,N) where N is a singleton. We extend the notion of an “ideal” from regular pointed categories to regular multi-pointed categories, and having “a good theory of ideals” will mean that there is a bijection between ideals and kernel pairs, which in the pointed case is the main property of ideal determined categories. The study of general categories with a good theory of ideals allows in fact a simultaneous treatment of ideal determined and Barr exact Goursat categories: we prove that in the case when all morphisms are chosen as null morphisms, the presence of a good theory of ideals becomes precisely the property for a regular category to be a Barr exact Goursat category. Among other things, this allows to obtain a unified proof of the fact that lattices of effective equivalence relations are modular both in the case of Barr exact Goursat categories and in the case of ideal determined categories.