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A new Galois structure in the category of internal preorders. (English) Zbl 1435.18009

Summary: Let \(\mathsf{PreOrd}(\mathbb{C})\) be the category of internal preorders in an exact category \(\mathbb{C}\). We show that the pair (\(\mathsf{Eq}(\mathbb{C}),\,\mathsf{ParOrd}(\mathbb{C})\)) is a pretorsion theory in \(\mathsf{PreOrd}(\mathbb{C})\), where \(\mathsf{Eq}(\mathbb{C})\) and \(\mathsf{ParOrd}(\mathbb{C})\) are the full subcategories of internal equivalence relations and of internal partial orders in \(\mathbb{C}\), respectively. We observe that \(\mathsf{ParOrd}(\mathbb{C})\) is a reflective subcategory of \(\mathsf{PreOrd}(\mathbb{C})\) such that each component of the unit of the adjunction is a pullback-stable regular epimorphism. The reflector \(F\): \(\mathsf{PreOrd}(\mathbb{C}) \to \mathsf{ParOrd}(\mathbb{C})\) turns out to have stable units in the sense of Cassidy, Hébert and Kelly, thus inducing an admissible categorical Galois structure. In particular, when \(\mathbb{C}\) is the category \(\mathsf{Set}\) of sets, we show that this reflection induces a monotone-light factorization system (in the sense of [A. Carboni et al., Appl. Categ. Struct. 5, No. 1, 1–58 (1997; Zbl 0866.18003)]) in \(\mathsf{PreOrd(Set)}\). 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 \(\mathsf{PreOrd(Set)}\).

MSC:

18E50 Categorical Galois theory
18A32 Factorization systems, substructures, quotient structures, congruences, amalgams
18B35 Preorders, orders, domains and lattices (viewed as categories)
18E40 Torsion theories, radicals
06A15 Galois correspondences, closure operators (in relation to ordered sets)

Citations:

Zbl 0866.18003
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Full Text: arXiv Link

References:

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