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Finite preorders and topological descent. I. (English) Zbl 1018.18004

[For part II, see ibid. 174, No. 3, 303-309 (2002; Zbl 1018.18005).]
In the following categories: \({\mathcal R}el\), \({\mathcal R}efl{\mathcal R}el\), \({\mathcal P}reord\), \({\mathcal F}in{\mathcal P}reord\), \({\mathcal F}in{\mathcal T}op\) of sets equipped with a relation, a reflexive relation, a preorder, of finite preordered sets, of finite topological spaces, respectively, the regular epimorphisms, pullback stable regular epimorphisms, effective descent morphisms, effective \(E\)-descent morphisms for various classes of morphisms \(E\), are characterized. Several counter-examples are given, e.g., a descent morphism that is not an effective étale-descent morphism, and converse. The link between topological spaces and preordered sets comes from the equivalence of the two last categories.

MSC:

18B30 Categories of topological spaces and continuous mappings (MSC2010)
18A20 Epimorphisms, monomorphisms, special classes of morphisms, null morphisms
18D30 Fibered categories
18A25 Functor categories, comma categories
18B35 Preorders, orders, domains and lattices (viewed as categories)

Citations:

Zbl 1018.18005
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References:

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