Abstract
Using the parallel between the preframe and the suplattice approach to locale theory it is shown that the patch construction, as an action on topologies, is the same thing as the process of recovering a discrete poset from its algebraic dcpo (ideal completion).
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Banaschewski, B., Brümmer, G.C.L.: Stably continuous frames. Math. Proc. Cambridge Philos. Soc. 104(7), 7–19 (1988)
Birkhoff, G., Fink, O.: Representations of lattices by sets. Trans. Amer. Math. Soc. 87, 299–316 (1948)
Escardó, M.: Properly injective spaces and function spaces. Topology Appl. 89, 75–120 (1984)
Escardó, M.: The regular-locally-compact coreflection of a stably locally compact locale. J. Pure Appl. Algebra 157, 41–55 (2001)
Gierz, G.K., Hofmann, K.H., Keimel, J., Lawson, J.D., Mislove, M., Scott, D.: A Compendium of Continuous Lattices. Springer, Berlin (1980)
Hoffmann, R.-E.: The fell compactification revisited. In: Hoffmann, R.-E., Hofmann, K.H. (eds.) Continuous Lattices and their Applications. Proceedings of the Third Conference on Categorical and Topological Aspects of Continuous Lattices (Bremen 1982). Lecture Notes in Pure and Applied Mathematics, vol. 101, pp. 57–116. Marcel-Dekker, New York (1985)
Hofmann, K.H.: Stably continuous frames and their topological manifestations. In: Bentley, H.L., Herrlich, H., Rajagopalan, M., Wolff, H. (eds.) Categorical Topology. Proceedings of the 1983 Conference in Toledo. Sigma Series in Pure and Applied Mathematics, vol. 5, pp. 282–307. Heldermann, Berlin (1984)
Johnstone, P.T.: Stone spaces. In: Cambridge Studies in Advanced Mathematics, vol. 3. Cambridge University Press, Cambridge (1982)
Johnstone, P.T.: Sketches of an Elephant: A Topos Theory Compendium, vols. 1, 2. Oxford Logic Guides 43, 44. Oxford Science Publications, Oxford (2002)
Johnstone, P.T., Vickers, S.J.: Preframe presentations present. In: Carboni, A., Pedicchio, M.C., Rosolini, G. (eds.) Category Theory—Proceedings, Como, 1990. Lecture Notes in Mathematics, vol. 1488, pp. 193–212. Springer, New York (1991)
Joyal, A., Tierney, M.: An extension of the Galois theory of Grothendieck. Mem. Amer. Math. Soc. 309, 1–155 (1984)
Jung, A., Sümderhauf, Ph.: On the duality of compact vs. open. In: Andima, S., Flagg, R.C., Itzkowitz, G., Misra, P., Kong, Y., Kopperman, R. (eds.) Papers on General Topology and Applications: Eleventh Summer Conference at the University of Southern Maine, vol. 806, pp. 214–230. Annals of the New York Academy of Sciences. New York Academy of Sciences, New York (1996)
Jung, A., Kegelmann, M., Moshier, A.: Stably compact spaces and closed relations. Electron. Notes Theor. Comput. Sci. 45, 1–23 (2001)
Karazeris, P.: Compact topologies on locally presentable categories. Cahiers Topologie Géom. Différentielle Catég. 38(3), 227–255 (1997)
Künzi, H.P.A., Brümmer, G.C.L.: Sobrification and bicompletion of totally bounded quasi-uniform spaces. Math. Proc. Cambridge Philos. 101, 237–246 (1987)
Lawson, J.D.: The duality of continuous posets. Proc. Amer. Math. Soc. 78, 477–81 (1979)
Manes, E.G.: A triple miscellany: some aspects of the theory of algebras over a triple. Ph.D. Thesis, Wesleyan University (1967)
Manes, E.G.: A triple theoretic construction of compact algebras. In: Seminar on Triples and Categorical Homology Theory. Lecture Note Mathematics, vol. 80, pp. 91–118. Springer, New York (1969)
Priestley, H.: Representation of distributive lattices by means of ordered stone spaces. Bull. London Math. Soc. 2, 186–90 (1970)
Rudin, W.: Real and Complex Analysis 3rd Edition. McGraw-Hill, New York (1966)
Scott, D.: Domains for denotational semantics. Lecture Notes in Comput. Sci., Springer 144, 577–613 (1982)
Smyth, M.B.: Stable compactification I. J. London Math. Soc. 45, 321–340 (1992)
Stone, M.: Topological representation of distributive lattices and Brouwerian logics. Časopis Pešt. Mat. Fys. 67, 1–25 (1937)
Taylor, P.: Geometric and higher order logic in terms of abstract stone duality. Theory Appl. Categ. 7(15), 284–338 (2000)
Townsend, C.F.: Preframe techniques in constructive locale theory. Ph.D. thesis, Imperial College, London (1996)
Townsend, C.F.: An axiomatic account of weak localic triquotient assignments (2003, submitted)
Townsend, C.F.: A categorical account of the Hofmann-Mislove theorem. Math. Proc. Cambridge Philos. Soc. 139, 441–456 (2005)
Townsend, C.F.: On the parallel between the suplattice and preframe approaches to locale theory. Ann. Pure Appl. Logic 137, 391–412 (2006) (Proceedings of the Second Workshop in Formal Topology, Venice 2002, Special issue)
Vermeulen, J.J.C.: Some constructive results related to compactness and the (strong) Hausdorff property for locales. Lecture Notes in Math., Springer 1488, 401–409 (1991)
Vickers, S.J.: Topology Via Logic. Cambridge University Press, Cambridge (1989)
Vickers, S.J.: Information systems for continuous posets. Theoret. Comput. Sci. 114, 201–229 (1993)
Vickers, S.J.: Constructive points of powerlocales. Math. Proc. Cambridge Philos. Soc. 122, 207–222 (1997)
Wyler, O.: Compact ordered spaces and prime Wallman compactifcations. In: Category Theory. Proc. Conference Toledo, Ohio 1983. Sigma Series in Pure Math, vol. 3, pp. 618–635. Sigma, Wilmslow (1984)
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Townsend, C.F. The Patch Construction is Dual to Algebraic DCPO Representation. Appl Categor Struct 19, 61–92 (2011). https://doi.org/10.1007/s10485-008-9177-9
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DOI: https://doi.org/10.1007/s10485-008-9177-9