Geodesic
A new Galois structure in the category of internal preorders
Theory and applications of categories, Tome 35 (2020), pp. 326-349

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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).

Publié le :
Classification : 18E50, 18A32, 18B35, 18E40, 06A15
Keywords: Internal preorders, partial orders, Galois theory, monotone-light factorization system, Alexandroff-discrete spaces 
@article{TAC_2020_35_a10,
     author = {Alberto Facchini and Carmelo Finocchiaro and Marino Gran},
     title = {A new {Galois} structure in the category of internal preorders},
     journal = {Theory and applications of categories},
     pages = {326--349},
     publisher = {mathdoc},
     volume = {35},
     year = {2020},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TAC_2020_35_a10/}
}
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Alberto Facchini; Carmelo Finocchiaro; Marino Gran. A new Galois structure in the category of internal preorders. Theory and applications of categories, Tome 35 (2020), pp. 326-349. http://geodesic.mathdoc.fr/item/TAC_2020_35_a10/