Categorical extension of dualities: from Stone to de Vries and beyond. I. (English) Zbl 1492.54013
Given a Stone-type dual equivalence \(\mathcal{A}^{\mathrm{op}} \longleftrightarrow \mathcal{X}\) and a full embedding \(\mathcal{X} \hookrightarrow \mathcal{Y}\), the authors propose a natural construction for a category \(\mathcal{B}\), into which \(\mathcal{A}\) may be fully embedded, allowing for an extension to a dual equivalence \(\mathcal{B}^{\mathrm{op}} \longleftrightarrow \mathcal{Y}\). A key ingredient to the definition of \(\mathcal{B}\) as a quotient of a comma category \(C(\mathcal{A},\mathcal{P},\mathcal{X})/\sim\) is the notion of an \(\mathcal{X}\)-covering class \(\mathcal{P}\) of morphisms in \(\mathcal{Y}\), which is an abstraction of projective covers (absolutes). The framework is applied to establish new proofs for known dualities, eventually presenting “mediating” categories of independent interest: (1) The approach to the de Vries duality between de Vries algebras and compact Hausdorff spaces starts from the duality \(\textbf{CBoo}^{\mathrm{op}} \longleftrightarrow \textbf{EKH}\) of complete Boolean algebras and extremally disconnected compact Hausdorff spaces, and yields an intermediate equivalent category \(\textbf{deV}^{\mathrm{op}} \longleftrightarrow (\textbf{deVBoo}/\sim)^{\mathrm{op}} \longleftrightarrow \textbf{KHaus}\) of de Vries algebras with handier morphisms. (2) The Fedorchuk duality \(\textbf{Fed}^{\mathrm{op}} \longleftrightarrow \textbf{KHaus}_{\mathrm{q-open}}\) is obtained from the duality \((\textbf{CBoo}_{\mathrm{sup}})^{\mathrm{op}} \longleftrightarrow \textbf{EKH}_{\mathrm{open}}\) having restrictions to suprema-preserving homomorphisms and open continuous mappings, as outlined in [G. Dimov et al., Topology Appl. 281, Article ID 107207, 26 p. (2020; Zbl 1457.54016)]. (3) Employing a hom representation of the de Vries duality and a construction essentially dual to that of (1), a duality theorem for the category of Tychonoff spaces from [G. Bezhanishvili et al., ibid. 257, 85–105 (2019; Zbl 1412.54034)] is obtained as the composite of the equivalences \(\textbf{UBdeV}^{\mathrm{op}} \longleftrightarrow \textbf{UdeV}^{\mathrm{op}} \longleftrightarrow D(\mathcal{A},\mathcal{J},\mathcal{X})^{\mathrm{op}} \longleftrightarrow \textbf{Tych}\), where \(\textbf{UBdeV}\) denotes the category of so-called universal Boolean de Vries extensions and \(\textbf{UdeV}\) that of universal de Vries pairs.
Reviewer: Alexander Vauth (Lübbecke)
MSC:
| 54D30 | Compactness |
| 54D15 | Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.) |
| 18A40 | Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.) |
| 54E05 | Proximity structures and generalizations |
| 54C10 | Special maps on topological spaces (open, closed, perfect, etc.) |
| 54G05 | Extremally disconnected spaces, \(F\)-spaces, etc. |
| 06E15 | Stone spaces (Boolean spaces) and related structures |
| 54H10 | Topological representations of algebraic systems |
| 03G05 | Logical aspects of Boolean algebras |
Keywords:
compact Hausdorff space; Tychonoff space; Stone space; regular closed/open set; irreducible map; quasi-open map; projective cover; (complete) Boolean algebra; (normal) contact algebra; de Vries algebra; ultrafilter; cluster; right/left lifting of a dual adjunction; semi-right adjoint functor; covering class; Stone duality; Tarski duality; de Vries duality; (universal) de Vries pair; Booleanization of a de Vries pair; (universal) Boolean de Vries extensionReferences:
| [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) · Zbl 1113.18001 |
| [2] | Adámek, J.; Herrlich, H.; Rosický, J.; Tholen, W., Injective hulls are not natural, Algebra Univ., 48, 379-388 (2002) · Zbl 1061.18010 · doi:10.1007/s000120200006 |
| [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) · Zbl 1190.54015 · doi:10.1016/j.topol.2010.01.007 |
| [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) · Zbl 1425.54006 · doi:10.1007/s10485-016-9434-2 |
| [7] | Bezhanishvili, G.; Morandi, PJ; Olberding, B., An extension of De Vries duality to completely regular spaces and compactifications, Topol. Appl., 257, 85-105 (2019) · Zbl 1412.54034 · doi:10.1016/j.topol.2019.02.007 |
| [8] | Bezhanishvili, G.; Morandi, PJ; Olberding, B., An extension of de Vries duality to normal spaces and locally compact Hausdorff spaces, J. Pure Appl. Algebra, 224, 703-724 (2020) · Zbl 1422.54012 · doi:10.1016/j.jpaa.2019.06.005 |
| [9] | Comfort, W.; Negrepontis, S., Chain Conditions in Topology (1982), Cambridge: Cambridge University Press, Cambridge · Zbl 0488.54002 · doi:10.1017/CBO9780511897337 |
| [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) · Zbl 1175.54035 · doi:10.1016/j.topol.2008.09.010 |
| [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) · Zbl 1240.54088 · doi:10.1007/s10474-010-0035-5 |
| [14] | Dimov, G., Some generalizations of the Stone duality theorem, Public. Math. Debr., 80, 255-293 (2012) · Zbl 1265.18008 · doi:10.5486/PMD.2012.4814 |
| [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) · Zbl 1354.18002 |
| [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) · Zbl 1457.54016 · doi:10.1016/j.topol.2020.107207 |
| [19] | Dimov, G., Ivanova-Dimova, E., Tholen, W.: Categorical extension of dualities: from Stone to de Vries and beyond, II (Work in progress) · Zbl 1457.54016 |
| [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) · Zbl 1111.68122 |
| [21] | Düntsch, I.; Vakarelov, D., Region-based theory of discrete spaces: a proximity approach, Ann. Math. Artif. Intell., 49, 5-14 (2007) · Zbl 1124.68105 · doi:10.1007/s10472-007-9064-3 |
| [22] | Engelking, R., General Topology (1989), Berlin: Heldermann Verlag, Berlin · Zbl 0684.54001 |
| [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)) · Zbl 0285.54031 |
| [24] | Gleason, AM, Projective topological spaces, Ill. J. Math., 2, 482-489 (1958) · Zbl 0083.17401 |
| [25] | Halmos, P., Lectures on Boolean Algebras (1974), New York: Springer, New York · Zbl 0285.06010 · doi:10.1007/978-1-4612-9855-7 |
| [26] | Johnstone, PT, Stone Spaces (1982), Cambridge: Cambridge University Press, Cambridge · Zbl 0499.54001 |
| [27] | Koppelberg, S.: Handbook on Boolean Algebras, vol. 1: General Theory of Boolean Algebras. North Holland (1989) · Zbl 0676.06019 |
| [28] | Mac Lane, S., Categories for the Working Mathematician (1998), New York: Springer, New York · Zbl 0906.18001 |
| [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) · Zbl 0186.02901 |
| [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) · Zbl 0119.17906 |
| [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)) · Zbl 0317.18002 |
| [32] | Naimpally, S.; Warrack, B., Proximity Spaces (1970), London: Cambridge, London · Zbl 0206.24601 |
| [33] | Ponomarev, VI, Paracompacta: their projection spectra and continuous mappings, Mat. Sb. (N.S.), 60, 89-119 (1963) · Zbl 0138.17504 |
| [34] | Ponomarev, VI; Šapiro, LB, Absolutes of topological spaces and their continuous mappings, Uspekhi Mat. Nauk, 31, 121-136 (1976) · Zbl 0341.54048 |
| [35] | Porter, JR; Woods, RG, Extensions and Absolutes of Hausdorff Spaces (1988), New York: Springer, New York · Zbl 0652.54016 · doi:10.1007/978-1-4612-3712-9 |
| [36] | Rump, W., The absolute of a topological space and its application to abelian l-groups, Appl. Categ. Struct., 17, 2, 153-174 (2009) · Zbl 1170.54002 · doi:10.1007/s10485-008-9133-8 |
| [37] | Stone, MH, The theory of representations for Boolean algebras, Trans. Am. Math. Soc., 40, 37-111 (1936) · JFM 62.0033.04 |
| [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) · Zbl 1185.68682 · doi:10.3166/jancl.12.527-559 |
| [40] | Walker, RC, The Stone-Čech Compactification (1974), Berlin: Springer, Berlin · Zbl 0292.54001 · doi:10.1007/978-3-642-61935-9 |
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