Abstract
We provide the appropriate common ‘(pre)framework’ for various central results of domain theory and topology, like the Lawson duality of continuous domains, the Hofmann–Lawson duality between continuous frames and locally compact sober spaces, the Hofmann–Mislove theorems about continuous semilattices of compact saturated sets, or the theory of stably continuous frames and their topological manifestations. Suitable objects for the pointfree approach are quasiframes, i.e., up-complete meet-semilattices with top, and preframes, i.e., meet-continuous quasiframes. We introduce the pointfree notion of locally compact well-filtered preframes, show that they are just the continuous preframes (using a slightly modified definition of continuity) and establish several natural dualities for the involved categories. Moreover, we obtain various characterizations of preframes having duality. Our results hold in ZF set theory without any choice principles.
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Adámek, J., Herrlich, H., Strecker, G.: Abstract and Concrete Categories. Wiley, New York (1990)
Banaschewski, B.: Coherent frames. In: Banaschewski, B., Hoffmann, R.-E. (eds.) Continuous Lattices, Lecture Notes in Math, vol 871. Springer, Berlin Heidelberg New York (1981)
Banaschewski, B.: Another look at the localic Tychonoff theorem. Comment. Math. Univ. Carolin. 29, 647–656 (1988)
Banaschewski, B., Brümmer, G.C.L.: Stably continuous frames. Math. Proc. Cambridge Philos. Soc. 104, 7–19 (1988)
Banaschewski, B., Erné, M.: On Krull’s separation lemma. Order 10, 253–260 (1993)
Erné, M.: Scott convergence and Scott topologies on partially ordered sets II. In: Banaschewski, B., Hoffmann, R.-E. (eds.) Continuous Lattices. Lecture Notes in Mathematics, vol. 871. Springer, Berlin Heidelberg New York (1981)
Erné, M.: Order, Topology and Closure. Lecture Notes, University of Hannover, Germany (1982)
Erné, M.: Distributivgesetze und die Dedekindsche Schnittvervollständigung. Abh. Braunschweig Wiss. Ges. 33, 117–145 (1982)
Erné, M.: Algebraic ordered sets and their generalizations. In: Rosenberg, I., Sabidussi, G. (eds.) Algebras and Orders, Proc. Montreal 1992. Kluwer, Amsterdam, Netherlands (1994)
Erné, M.: General Stone duality. In: Clementino, M.M. et al. (eds.) Proc. IV Iberoamerican Conf. on Topology, Coimbra 2001. Topology Appl. 137, 125–158 (2004)
Erné, M.: Minimal bases, ideal extensions, and basic dualities. Topology Proceedings 29, 445–489 (2005)
Erné, M.: Sober spaces, locally hypercompact spaces and quasicontinuous posets. University of Hannover, Germany (2005)
Erné, M.: Choicefree dualities for domains. University of Hannover, Germany (2005)
Gierz, G., Hofmann, K.H., Keimel, K., Lawson, J.D., Mislove, M., Scott, D.S.: A Compendium of Continuous Lattices. Springer, Berlin Heidelberg New York (1980)
Gierz, G., Hofmann, K.H., Keimel, K., Lawson, J.D., Mislove, M., Scott, D.S.: Continuous Lattices and Domains. Oxford University Press, London, UK (2003)
Grätzer, G.: General Lattice Theory. Birkhäuser, Basel, Switzerland (1978)
Hofmann, K.H.: Stably continuous frames and their topological manifestations. In: Bentley, H.L. et al. (eds.) Categorical Topology. Sigma Series in Pure Mathematics, vol. 3, pp. 282–307. Heldermann, Berlin, Germany (1984)
Hofmann, K.H., Lawson, J.: The spectral theory of distributive continuous lattices. Trans. Amer. Math. Soc. 246, 285–310 (1978)
Hofmann, K.H., Lawson, J.: On the order-theoretical foundation of a theory of quasicompactly generated spaces without separation axiom. J. Austral. Math. Soc. Ser. A 36, 194–212 (1984)
Hofmann, K.H., Mislove, M.: Local compactness and continuous lattices. In: Banaschewski, B., Hoffmann, R.-E. (eds.) Continuous Lattices. Lecture Notes in Mathematics, vol. 871, pp. 209–248. Springer, Berlin Heidelberg New York (1981)
Isbell, J.: Atomless parts of spaces. Math. Scand. 31, 5–32 (1972)
Johnstone, P.T.: Stone Spaces. Cambridge University Press, UK (1982)
Johnstone, P.T.: Vietoris locales and localic semilattices. In: Hoffmann, R.E., Hofmann, K.H. (eds.) Continuous Lattices and Their Applications, Bremen, 1982. Lecture Notes in Pure and Applied Mathematics, vol. 101. Marcel Dekker, New York (1985)
Johnstone, P.T.: The art of pointless thinking: the category of locales. In: Herrlich, H., Porst, H.-E. (eds.) Category Theory at Work, pp. 85–107. Heldermann, Berlin, Germany (1991)
Johnstone, P.T., Vickers, S.J.: Preframe presentations present. In: Carboni, A., Pedicchio, M.C., Rosalini, G. (eds.) Proceedings of the 1990 Como Category Theory Conference. Lecture Notes in Mathematics, vol. 1488, pp. 193–212. Springer, Berlin Heidelberg New York (1991)
Lawson, J.D.: The duality of continuous posets. Houston J. Math. 5, 357–386 (1979)
Pincus, D.: Adding dependent choice to the prime ideal theorem. Logic Colloquium 76, North-Holland, Amsterdam, The Netherlands (1977)
Porst, H.-E., Tholen, W.: Concrete dualities. In: Herrlich, H., Porst, H.-E. (eds.) Category Theory at Work, pp. 111–136. Heldermann, Berlin, Germany (1991)
Scott, D.S.: Prime ideal theorems for rings, lattices and Boolean algebras. Bull. Amer. Math. Soc. 60, 390(Abstract) (1954)
Tarski, A.: Prime ideal theorems for Boolean algebras and the axiom of choice. Bull. Amer. Math. Soc. 60, 390–391(Abstract) (1954)
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Erné, M. Choiceless, Pointless, but not Useless: Dualities for Preframes. Appl Categor Struct 15, 541–572 (2007). https://doi.org/10.1007/s10485-006-9029-4
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DOI: https://doi.org/10.1007/s10485-006-9029-4