Compact Hausdorff Spaces with Relations and Gleason Spaces
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We consider an alternate form of the equivalence between the category of compact Hausdorff spaces and continuous functions and a category formed from Gleason spaces and certain relations. This equivalence arises from the study of the projective cover of a compact Hausdorff space. This line leads us to consider the category of compact Hausdorff spaces with closed relations, and the corresponding subcategories with continuous and interior relations. Various equivalences of these categories are given extending known equivalences of the category of compact Hausdorff spaces and continuous functions with compact regular frames, de Vries algebras, and also with a category of Gleason spaces that we introduce. Study of categories of compact Hausdorff spaces with relations is of interest as a general setting to consider Gleason spaces, for connections to modal logic, as well as for the intrinsic interest in these categories.
KeywordsCompact Hausdorff space Gleason cover Closed relation Continuous relation Interior relation Compact regular frame De Vries algebra
Mathematics Subject Classification54D30 54G05 54E05 06D22
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We thank the referee for pointing out  to us, as well as for a number of useful suggestions, particularly involving adjunctions in order enriched categories.
- 1.Abramsky, S., Coecke, B.: A categorical semantics of quantum protocols In: Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science (LiCS-04), pp. 415–425. IEEE Computer Society, New York (2004) (Extended version at arXiv:quant-ph/0402130)
- 9.de Vries, H.: Compact spaces and compactifications. An algebraic approach, Ph.D. thesis, University of Amsterdam, (1962)Google Scholar
- 18.Johnstone, P.T.: Vietoris locales and localic semilattices, Continuous lattices and their applications (Bremen, 1982), pp. 155–180. Lecture Notes in Pure and Application Mathematics, vol. 101, Dekker, New York (1985)Google Scholar
- 21.Townsend, C.F.: Preframe techniques in constructive locale theory. Ph.D. thesis, University of London (1996)Google Scholar
- 22.Wyler, O.: Algebraic theories of continuous lattices. In: Continuous Lattices, pp. 390–413. Springer, Berlin (1981)Google Scholar