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Covering morphisms in categories of relational algebras. (English) Zbl 1306.18002

Professor George Janelidze posed the problem whether the study of classical topological coverings via categorical Galois Theory could be extended to the setting of relational algebras (see, e.g., [G. Dzhanelidze, J. Algebra 132, No. 2, 270–286 (1990; Zbl 0702.18006)]. It is the purpose of this paper to give a first contribution to the investigation of coverings in the realm of relational algebras. In particular the authors obtain new characterizations for effective descent maps in the categories of \(M\)-ordered sets, for a given monoid \(M,\) and of multi-ordered sets.

MSC:

18B30 Categories of topological spaces and continuous mappings (MSC2010)
54D05 Connected and locally connected spaces (general aspects)
54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)
14E20 Coverings in algebraic geometry
54C10 Special maps on topological spaces (open, closed, perfect, etc.)

Citations:

Zbl 0702.18006
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References:

[1] Barr, M., Relational algebras, 39-55 (1970), Berlin · Zbl 0204.33202 · doi:10.1007/BFb0060439
[2] Borceux, F., Janelidze, G.: Galois theories. In: Cambridge Studies in Advanced Mathematics, 72. Cambridge University Press (2001) · Zbl 0978.12004
[3] Cassidy, C., Hébert, M., Kelly, G.M.: Reflective subcategories, localizations and factorization systems. J. Aust. Math. Soc. Ser. A 38, 287-329 (1985) · Zbl 0573.18002 · doi:10.1017/S1446788700023624
[4] Clementino, M.M., Hofmann, D.: Triquotient maps via ultrafilter convergence. Proc. Amer. Math. Soc. 130, 3423-3431 (2002) · Zbl 1008.54011 · doi:10.1090/S0002-9939-02-06472-9
[5] Clementino, M.M., Hofmann, D.: Topological features of lax algebras. Appl. Categ. Struct. 11, 267-286 (2003) · Zbl 1024.18003 · doi:10.1023/A:1024274315778
[6] Clementino, M.M., Hofmann, D.: Effective descent morphisms in categories of lax algebras. Appl. Categ. Struct. 12, 413-425 (2004) · Zbl 1078.18008 · doi:10.1023/B:APCS.0000049310.37773.fa
[7] Clementino, M.M., Hofmann, D.: Descent morphisms and a van Kampen Theorem in categories of lax algebras. Topol. Appl. 159, 2310-2319 (2012) · Zbl 1253.18004 · doi:10.1016/j.topol.2011.07.031
[8] Clementino, M.M., Hofmann, D., Janelidze, G.: Local homeomorphisms via ultrafilter convergence. Proc. Amer. Math. Soc. 133, 917-922 (2005) · Zbl 1061.54012 · doi:10.1090/S0002-9939-04-07569-0
[9] Clementino, M.M., Hofmann, D., Janelidze, G.: On exponentiability of étale algebraic homomorphisms. J. Pure Appl. Algebra 217, 1195-1207 (2013) · Zbl 1409.18004 · doi:10.1016/j.jpaa.2012.10.013
[10] Clementino, M.M., Hofmann, D., Janelidze, G.: The Monads of Classical Algebra are Seldom Weakly Cartesian. J. Homotopy Relat. Struct. doi:10.1007/s40062-013-0063-2 (2012) · Zbl 1309.18004
[11] Clementino, M.M., Hofmann, D., Tholen, W.: Exponentiability in categories of lax algebras. Theory Appl. Categories 11, 337-352 (2003) · Zbl 1032.18002
[12] Clementino, M.M., Janelidze, G.: A note on effective descent morphisms of topological spaces and relational algebras. Topol. Appl. 158, 2431-2436 (2011) · Zbl 1242.18008
[13] Clementino, M.M., Tholen, W.: Metric, topology and multicategory—a common approach. J. Pure Appl. Algebra 179, 13-47 (2003) · Zbl 1015.18004 · doi:10.1016/S0022-4049(02)00246-3
[14] Dyckhoff, R., Tholen, W.: Exponentiable morphisms, partial products and pullback complements. J. Pure Appl. Algebra 49, 103-116 (1987) · Zbl 0659.18003 · doi:10.1016/0022-4049(87)90124-1
[15] Janelidze, G.: The fundamental theorem of Galois theory. Math USSR Sbornik 64, 359-374 (1989) · Zbl 0677.18003 · doi:10.1070/SM1989v064n02ABEH003313
[16] Janelidze, G.: Pure Galois theory in categories. J. Algebra 132, 270-286 (1990) · Zbl 0702.18006 · doi:10.1016/0021-8693(90)90130-G
[17] Janelidze, G., Categorical Galois theory: Revision and some recent developments, 139-171 (2004), Dordrecht · Zbl 1072.18003 · doi:10.1007/978-1-4020-1898-5_2
[18] Janelidze, G.: Descent and Galois Theory, Lecture Notes of the Summer School in Categorical Methods in Algebra and Topology, Haute Bodeux, Belgium. available at http://www.math.yorku.ca/ tholen/ (2007)
[19] Janelidze, G., Tholen, W.: How algebraic is the change of base functor? Springer Lect. Notes Math. 1488, 174-186 (1991) · Zbl 0802.18005 · doi:10.1007/BFb0084219
[20] Janelidze, G., Sobral, M.: Finite preorders and topological descent I. J. Pure Appl. Algebra 175, 187-205 (2002) · Zbl 1018.18004 · doi:10.1016/S0022-4049(02)00134-2
[21] Mahmoudi, M., Schubert, C., Tholen, W.: Universality of coproducts in categories of lax algebras. Appl. Categ. Struct. 14, 243-249 (2006) · Zbl 1103.18007 · doi:10.1007/s10485-006-9019-6
[22] Manes, E.: Taut monads and T0-spaces. Theoret. Comput. Sci. 275, 79-109 (2002) · Zbl 1033.18001 · doi:10.1016/S0304-3975(00)00415-1
[23] Reiterman, J., Tholen, W.: Effective descent maps of topological spaces. Top. Appl. 57, 53-69 (1994) · Zbl 0829.54011 · doi:10.1016/0166-8641(94)90033-7
[24] Xarez, J.J.: The monotone-light factorization for categories via preordered and ordered sets, PhD Thesis. University Aveiro (2003)
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