论文信息 - Locales are Not Pointless

Locales are Not Pointless

The Kripke-Joyal semantics is used to interpret the fragment of intuitionistic logic containing ^;! and 8 in the category of locales. An axiomatic theory is developed that can be interpreted soundly in two ways, using either lower or upper powerlocales, so that pairs of separate results can be proved as single formal theorems. Openness and properness of maps between locales are characterized by descriptions using the logic, and it is proved that a locale is open ii its lower powerlocale has a greatest point. The entire account is constructive and holds for locales over any topos.

Steven J. Vickers | S. Vickers

[1] I. Moerdijk. Descent theory for toposes , 1989 .

[2] Jimmie D. Lawson,et al. On the order-theoretical foundation of a theory of quasicompactly generated spaces without separation axiom , 1984, Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics.

[3] Bernhard Banaschewski,et al. Another look at the localic Tychonoff theorem , 1988 .

[4] Michael W. Mislove,et al. Local compactness and continuous lattices , 1981 .

[5] P. T. Johnstone,et al. Partial products, bagdomains and hyperlocal toposes , 1992 .

[6] Steven J. Vickers,et al. Quantales, observational logic and process semantics , 1993, Mathematical Structures in Computer Science.

[7] A. Kock. Monads for which Structures are Adjoint to Units , 1995 .

[8] The symmetric topos , 1995 .

[9] A. Joyal,et al. An extension of the Galois theory of Grothendieck , 1984 .

[10] Edmund Robinson,et al. A characterization of pie limits , 1991 .

[11] J.J.C. Vermeulen,et al. Proper maps of locales , 1994 .

[12] A. Carboni,et al. PREFRAME PRESENTATIONS PRESENT , 1991 .

[13] M. Bunge,et al. Constructive theory of the lower power locale , 1996, Mathematical Structures in Computer Science.