## 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)

Zbl 1018.18005
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### References:

 [1] Clementino, M.M., On finite triquotient maps, J. pure appl. algebra, 168, 387-389, (2002) · Zbl 0985.54015 [2] Day, B.J.; Kelly, G.M., On topological quotient maps preserved by pullbacks, Proc. Cambridge philos. soc., 67, 553-558, (1970) · Zbl 0191.20801 [3] Giraud, J., Méthode de la descente, Bull. soc. math. France mem., 2, (1964) · Zbl 0211.32902 [4] M. Gran, Central extensions for internal groupoids in Maltsev categories, Ph.D. Thesis, Louvain-la-Neuve, Belgium, 1999. · Zbl 0974.18007 [5] G. Janelidze, M. Sobral, Finite preorders and topological descent, Preprint 99-10, Department of Mathematics, University of Coimbra. · Zbl 1018.18004 [6] G. Janelidze, M. Sobral, Finite preorders and topological descent II: étale descent, J. Pure Appl. Algebra, to appear. · Zbl 1018.18005 [7] G. Janelidze, W. Tholen, How algebraic is the change-of-base functor?, Lecture Notes in Mathematics, Vol. 1448, Springer, Berlin, 1991, pp. 157-173. · Zbl 0802.18005 [8] Janelidze, G.; Tholen, W., Facets of descent I, Appl. cat. struct., 2, 245-281, (1994) · Zbl 0805.18005 [9] Janelidze, G.; Tholen, W., Facets of descent II, Appl. cat. struct., 5, 229-248, (1997) · Zbl 0880.18007 [10] A. Kock, I. Moerdijk, Local equivalence relations and their sheaves, Aarhus Preprint Series, Vol. 19, 1991. · Zbl 0973.54012 [11] I. Le Creurer, Descent for internal categories, Ph.D. Thesis, Louvain-la-Neuve, Belgium, 1999. [12] Moerdijk, I., Descent theory for toposes, Bull. soc. math. belg., 41, 373-391, (1989) · Zbl 0688.18003 [13] Plewe, T., Localic triquotient maps are effective descent morphisms, Math. proc. Cambridge philos. soc., 122, 17-43, (1997) · Zbl 0878.54005 [14] Reiterman, J.; Tholen, W., Effective descent maps of topological spaces, Topology appl., 57, 53-69, (1994) · Zbl 0829.54011 [15] Sobral, M., Some aspects of topological descent, Appl. cat. struct., 4, 97-106, (1996) · Zbl 0884.54008 [16] Sobral, M., Another approach to topological descent theory, Appl. cat. struct., 9, 505-516, (2001) · Zbl 1001.54017 [17] M. Sobral, W. Tholen, Effective descent morphisms and effective equivalence relations, CMS Conference Proceedings, Vol. 13, American Mathematical Society, Providence, RI, 1992, pp. 421-433. · Zbl 0792.18002 [18] Vermeulen, J.J.C., Proper maps of locales, J. pure appl. algebra, 92, 79-107, (1994) · Zbl 0789.54019
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