A new Galois structure in the category of internal preorders

Alberto Facchini, Carmelo Finocchiaro and Marino Gran

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.

Published 2020-03-05.


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