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Categorical Extension of Dualities: From Stone to de Vries and Beyond, I

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Abstract

Propounding a general categorical framework for the extension of dualities, we present a new proof of the de Vries Duality Theorem for the category KHaus of compact Hausdorff spaces and their continuous maps, as an extension of a restricted Stone duality. Then, applying a dualization of the categorical framework to the de Vries duality, we give an alternative proof of the extension of the de Vries duality to the category Tych of Tychonoff spaces that was provided by Bezhanishvili, Morandi and Olberding. In the process of doing so, we obtain new duality theorems for both categories, KHaus and Tych.

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Notes

  1. Using Corollary 15, one can obtain as well the recent extension of the Stone Duality Theorem to the category of zero-dimensional Hausdorff spaces and continuous maps, as established in [17].

  2. We note that in [16], as the restriction of a duality involving the category of all locally compact Hausdorff spaces, another category dually equivalent to \({\mathbf{KHaus}}\) is presented. While its composition law may be considered to be more natural than that of the category \({\mathbf{deV}}\), its morphisms, which are special multi-valued maps, may not.

  3. Of course, formally we should have written \(\varepsilon ^{\mathrm{op}}:ST\rightarrow \mathrm{Id}_{{{\mathcal {A}}}^{\mathrm{op}}}\) for the counit of the dual adjunction and listed the triangular equalities as \(T\varepsilon ^{\mathrm{op}}\circ \eta T=1_T,\;\varepsilon ^{\mathrm{op}}S\circ S\eta =1_S\). But in this paper we will generally suppress the explicit use of the op-formalism for functors and natural transformations, in order to keep the notation simple and maintain the symmetric presentation of the units in \({{\mathcal {X}}}\) and the counits in \({{\mathcal {A}}}\).

References

  1. Adámek, J., Herrlich, H., Strecker, G.E.: Abstract and concrete categories: the joy of cats. In: Pure and Applied Mathematics (New York). Wiley, New York. xiv + 482. http://tac.mta.ca/tac/reprints/ articles/17/tr17abs.html. Republished in: Reprints in Theory and Applications of Categories, No. 17, pp. 1–507 (2006)

  2. Adámek, J., Herrlich, H., Rosický, J., Tholen, W.: Injective hulls are not natural. Algebra Univ. 48, 379–388 (2002)

    Article  MathSciNet  Google Scholar 

  3. Alexandroff, P.S.: Outline of Set Theory and General Topology. Nauka, Moskva (1977) (in Russian)

  4. Arhangel’skii, A.V., Ponomarev, V.I.: Fundamentals of General Topology: Problems and Exercises. Reidel, Dordrecht (1984) (Originally published by Izdatelstvo Nauka, Moscow (1974))

  5. Bezhanishvili, G.: Stone duality and Gleason covers through de Vries duality. Topol. Appl. 157, 1064–1080 (2010)

    Article  MathSciNet  Google Scholar 

  6. Bezhanishvili, G., Bezhanishvili, N., Sourabh, S., Venema, Y.: Irreducible equivalence relations, Gleason spaces, and de Vries duality. Appl. Categ. Struct. 25(3), 381–401 (2017)

    Article  MathSciNet  Google Scholar 

  7. Bezhanishvili, G., Morandi, P.J., Olberding, B.: An extension of De Vries duality to completely regular spaces and compactifications. Topol. Appl. 257, 85–105 (2019)

    Article  MathSciNet  Google Scholar 

  8. Bezhanishvili, G., Morandi, P.J., Olberding, B.: An extension of de Vries duality to normal spaces and locally compact Hausdorff spaces. J. Pure Appl. Algebra 224, 703–724 (2020)

    Article  MathSciNet  Google Scholar 

  9. Comfort, W., Negrepontis, S.: Chain Conditions in Topology. Cambridge University Press, Cambridge (1982)

    Book  Google Scholar 

  10. de Vries, H.: Compact spaces and compactifications, an algebraic approach. PhD thesis. Van Gorcum, The Netherlands (1962). https://www.illc.uva.nl/Research/Publications/Dissertations/HDS/

  11. Dimov, G.: Some generalizations of the Fedorchuk duality theorem—I. Topol. Appl. 156, 728–746 (2009)

    Article  MathSciNet  Google Scholar 

  12. Dimov, G.: A de Vries-type duality theorem for locally compact spaces—II, 1–37. arXiv:0903.2593v4

  13. Dimov, G.: A de Vries-type duality theorem for the category of locally compact spaces and continuous maps—I. Acta Math. Hung. 129, 314–349 (2010)

    Article  MathSciNet  Google Scholar 

  14. Dimov, G.: Some generalizations of the Stone duality theorem. Public. Math. Debr. 80, 255–293 (2012)

    Article  MathSciNet  Google Scholar 

  15. Dimov, G.: Proximity-type relations on boolean algebras and their connections with topological spaces. Doctor of Sciences (= Dr. Habil.) Thesis, Faculty of Mathematics and Informatics, Sofia University “St. Kl. Ohridski", Sofia, 1–292 (2013). https://www.fmi.uni-sofia.bg/bg/prof-dmn-georgi-dimov

  16. Dimov, G., Ivanova, E.: Yet another duality theorem for locally compact spaces. Houst. J. Math. 42(2), 675–700 (2016)

    MathSciNet  MATH  Google Scholar 

  17. Dimov, G., Ivanova-Dimova, E.: Two extensions of the Stone Duality to the category of zero-dimensional Hausdorff spaces. Filomat 35(6), (2021) (in press) (preprint in arXiv:1901.04537v3, 1–33)

  18. Dimov, G., Ivanova-Dimova, E., Tholen, W.: Extensions of dualities and a new approach to the Fedorchuk duality. Topol. Appl. 281, 107207 (2020)

    Article  MathSciNet  Google Scholar 

  19. Dimov, G., Ivanova-Dimova, E., Tholen, W.: Categorical extension of dualities: from Stone to de Vries and beyond, II (Work in progress)

  20. Dimov, G., Vakarelov, D.: Contact algebras and region-based theory of space: a proximity approach—I. Fundamenta Informaticae 74(2–3), 209–249 (2006)

    MathSciNet  MATH  Google Scholar 

  21. Düntsch, I., Vakarelov, D.: Region-based theory of discrete spaces: a proximity approach. Ann. Math. Artif. Intell. 49, 5–14 (2007)

    Article  MathSciNet  Google Scholar 

  22. Engelking, R.: General Topology. Sigma Series in Pure Mathematics, vol. 6, 2nd edn. Heldermann Verlag, Berlin (1989)

    MATH  Google Scholar 

  23. Fedorchuk, V.V.: Boolean \(\delta \)-algebras and quasi-open mappings Sibirsk. Mat. Ž. 14(5), 1088–1099 (1973) (English translation: Siberian Math. J. 14(1973), 759-767 (1974))

  24. Gleason, A.M.: Projective topological spaces. Ill. J. Math. 2, 482–489 (1958)

    MathSciNet  MATH  Google Scholar 

  25. Halmos, P.: Lectures on Boolean Algebras. Springer, New York (1974)

    Book  Google Scholar 

  26. Johnstone, P.T.: Stone Spaces. Cambridge University Press, Cambridge (1982)

    MATH  Google Scholar 

  27. Koppelberg, S.: Handbook on Boolean Algebras, vol. 1: General Theory of Boolean Algebras. North Holland (1989)

  28. Mac Lane, S.: Categories for the Working Mathematician, 2nd edn. Springer, New York (1998)

    MATH  Google Scholar 

  29. Manes, E.G.: A triple-theoretic construction of compact algebras. In: B. Eckmann (ed.), Seminar on Triples and Categorical Homology Theory. Lecture Notes in Mathematics, vol. 80, pp. 91–119. Springer, Berlin (1969)

  30. Mardešic, S., Papic, P.: Continuous images of ordered compacta, the Suslin property and dyadic compacta. Glasnik mat.-fis. i astronom. 17, 3–25 (1962)

    MATH  Google Scholar 

  31. Medvedev, M.. Ya..: Semiadjoint functors and Kan extensions. Sib. Math. J. 15, 674–676 (1975) (English translation of: Sib. Mat. Z. 15, 952–956 (1974))

  32. Naimpally, S., Warrack, B.: Proximity Spaces. Cambridge, London (1970)

    MATH  Google Scholar 

  33. Ponomarev, V.I.: Paracompacta: their projection spectra and continuous mappings. Mat. Sb. (N.S.) 60, 89–119 (1963). in Russian

    MathSciNet  Google Scholar 

  34. Ponomarev, V.I., Šapiro, L.B.: Absolutes of topological spaces and their continuous mappings. Uspekhi Mat. Nauk 31, 121–136 (1976). in Russian

    MathSciNet  MATH  Google Scholar 

  35. Porter, J.R., Woods, R.G.: Extensions and Absolutes of Hausdorff Spaces. Springer, New York (1988)

    Book  Google Scholar 

  36. Rump, W.: The absolute of a topological space and its application to abelian l-groups. Appl. Categ. Struct. 17(2), 153–174 (2009)

    Article  MathSciNet  Google Scholar 

  37. Stone, M.H.: The theory of representations for Boolean algebras. Trans. Am. Math. Soc. 40, 37–111 (1936)

    MathSciNet  MATH  Google Scholar 

  38. van Oosten, J.: Basic Category Theory. Department of Mathematics, Utrecht University, The Netherlands (2002)

  39. Vakarelov, D., Dimov, G., Düntsch, I., Bennett, B.: A proximity approach to some region-based theories of space. J. Appl. Non-Class. Log. 12, 527–559 (2002)

    Article  MathSciNet  Google Scholar 

  40. Walker, R.C.: The Stone–Čech Compactification. Springer, Berlin (1974)

    Book  Google Scholar 

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Acknowledgements

The authors are grateful for the referee’s careful reading of the paper and some useful suggestions.

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Authors and Affiliations

  1. Faculty of Mathematics and Inf., Sofia University, 5 J. Bourchier Blvd., 1164, Sofia, Bulgaria

    Georgi Dimov & Elza Ivanova-Dimova

  2. Department of Mathematics and Statistics, York University, Toronto, ON, M3J 1P3, Canada

    Walter Tholen

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  1. Georgi Dimov
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  2. Elza Ivanova-Dimova
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Correspondence to Walter Tholen.

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Communicated by Maria Manuel Clementino.

Dedicated to the memory of Professor Hendrik de Vries (November 5, 1932–May 13, 2021)

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The first author acknowledges the support by Bulgarian National Science Fund, contract no. DN02/15/ 19.12.2016. The second author acknowledges the support by the Bulgarian Ministry of Education and Science under the National Research Programme “Young scientists and postdoctoral students” approved by DCM # 577/17.08.2018. The third author acknowledges the support under Discovery Grant No. 501260 of the Natural Sciences and Engineering Council of Canada.

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Dimov, G., Ivanova-Dimova, E. & Tholen, W. Categorical Extension of Dualities: From Stone to de Vries and Beyond, I. Appl Categor Struct 30, 287–329 (2022). https://doi.org/10.1007/s10485-021-09658-6

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