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) → ParOrd(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).

A new galois structure in the category of internal preorders

Finocchiaro C.;
2020-01-01

Abstract

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) → ParOrd(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).
2020
Alexandroff-discrete spaces
Galois theory
Internal preorders
Monotone-light factorization system
Partial orders
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11769/486897
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