Descent for regular epimorphisms in Barr exact Goursat categories
The paper begins with a characterization of a Goursat category and goes on to give a lemma on effective (global) descent morphisms. Another lemma then discusses a relationship between the category ER(\(\mathcal C\)) of equivalence relations in \(\mathcal C\) (\(\mathcal C\) is assumed to be a Barr exact Goursat category), the category RR(\(\mathcal C\)) of reflexive relations in \(\mathcal C\) and RG(\(\mathcal C\)), the category of reflexive graphs in \(\mathcal C\). This then leads to the main result that every regular epimorphism in ER(\(\mathcal C\)) is an effective descent morphism (Lemma 2.2, and its translation into the language of regular epimorphism, Theorem 2.3). The paper ends with a few comments and open questions; one of them is how to describe effective descent morphisms in ER(\(\mathcal C\))\(\sim\)RE(\(\mathcal C\)), for an arbitrary Barr exact category \(\mathcal C\).
scientific article; zbMATH DE number 5871638