Riguet and Generalized Congruences on a Category: Relationships and Applications

Abstract

We investigate Riguet congruences and generalized congruences on a category, focusing on their interrelations from both lattice-theoretic and category-theoretic perspectives. We also characterize functors that are full and surjective on objects in terms of regular epimorphisms, extremal epimorphisms and in terms of strong and regular generalized congruences. On the lattice-theoretic side, we prove that for a category $\mathsf{C}$, the set $\mathrm{RCgr}(\mathsf{C})$ of all Riguet congruences, ordered by inclusion, is a bounded directed-complete ordered set, while the set $\mathrm{GCgr}(\mathsf{C})$ of all generalized congruences is an algebraic lattice. We establish a bridge between these structures via a Scott continuous morphism. From a category-theoretic standpoint, we lift these results to relative adjunctions between the categories $\mathsf{RCgr}(\mathsf{C})$ and $\mathsf{GCgr}(\mathsf{C})$ associated to the above ordered sets, as well as between the categories $\mathsf{RCCat}$, of Riguet classified categories, and $\mathsf{GCCat}$, of generalized classified categories. Furthermore, within Manes' framework of categories of $\mathsf{K}$-objects with structure, we investigate the relationship between the wide subcategory $\mathsf{RCCat}_{\mathrm{full}}$ of $\mathsf{RCCat}$, whose morphisms are the full morphisms of $\mathsf{RCCat}$, and $\mathsf{GCCat}$, relating these constructions to the Grothendieck theory of fibrations. Finally, we present applications of Riguet congruences across various mathematical fields.