Let PreOrd(C) be the category of internal preorders in an exact category C. We show that the pair (Eq(C),ParOrd(C)) is a pretorsion theory in PreOrd(C), where Eq(C) and ParOrd(C) are the full subcategories of internal equivalence relations and of internal partial orders in C, respectively. We observe that ParOrd(C) is a reflective subcategory of PreOrd(C) such that each component of the unit of the adjunction is a pullback-stable regular epimorphism. The reflector F: PreOrd(C) -> PardOrd(C) turns out to have stable units in the sense of Cassidy, Hébert and Kelly, thus inducing an admissible categorical Galois structure. In particular, when C is the category Set of sets, we show that this reflection induces a monotone-light factorization system (in the sense of Carboni, Janelidze, Kelly and Paré) in PreOrd(Set). A topological interpretation of our results in the category of Alexandroff-discrete spaces is also given, via the well-known isomorphism between this latter category and PreOrd(Set).
Keywords: Internal preorders, partial orders, Galois theory, monotone-light factorization system, Alexandroff-discrete spaces
2020 MSC: 18E50, 18A32, 18B35, 18E40, 06A15
Theory and Applications of Categories, Vol. 35, 2020, No. 11, pp 326-349.