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Constructive theory of the lower power locale

Published online by Cambridge University Press:  04 March 2009

M. Bunge  and
J. funk
M. Bunge
Affiliation:
McGill University, Deptartment of Mathematics and Statistics, 805 Sherbrooke Street West, Montréal, QC, Canada H3A 2K6 Email bunge@triples.math.mcgill.ca; funk@triples.math.mcgill.ca
J. funk
Affiliation:
McGill University, Deptartment of Mathematics and Statistics, 805 Sherbrooke Street West, Montréal, QC, Canada H3A 2K6 Email bunge@triples.math.mcgill.ca; funk@triples.math.mcgill.ca
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Abstract

This paper considers two main aspects of the lower power locale PL(X): first, its relation to the symmetric topos construction of Bunge and Carboni; and second, its points, which, it is shown, are equivalent to the weakly closed sublocales of X with open domain. This is done as part of a more general discussion of arbitrary weakly closed sublocales, including a new characterization using suplattice homomorphisms from o(X) to Sub(1), and a new proof of a theorem of Jibladze relating them to Ω-nuclei.

Information

Type
Research Article
Information
Mathematical Structures in Computer Science , Volume 6 , Issue 1 , February 1996 , pp. 69 - 83
Copyright
Copyright © Cambridge University Press 1996

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References

Börger, R. (1987) On the left adjoint from complete semi-lattices to frames, Haguen Preprint.Google Scholar
Bunge, M. and Carboni, A. (1994) The Symmetric Topos. To appear in Journal of Pure and Applied Algebra.CrossRefGoogle Scholar
Bunge, M. and Funk, J. (1995) Spreads and the Symmetric Topos. To appear in Journal of Pure and Applied Algebra.Google Scholar
Fox, R. H. (1957) Covering spaces with singularities. In: Fox, R. H.et al. (eds.) Algebraic Geometry and Topology (A symposium in honor of S. Lefshetz), Princeton University Press 243–257.CrossRefGoogle Scholar
Funk, J. (1995) The Display Locale of a cosheaf. Cahiers de Top. Géom Diff. Catégoriques 36 (1) 53–93.Google Scholar
Isbell, J. (1972) Atomless parts of spaces. Math. Scand. 31 5–32.CrossRefGoogle Scholar
Jibladze, M. and Johnstone, P. T. (1991) The frame of fiberwise closed nuclei. Cahiers de Top. Géom. Diff. Catégoriques 32 (2) 99–112.Google Scholar
Johnstone, P. T (1981) Factorization theorems for localic geometric morphisms, I. Cahiers de Top. Géom Diff. Catégoriques 22 3–17.Google Scholar
Johnstone, P. T. (1982) Stone Spaces, Cambridge University Press.Google Scholar
Johnstone, P. T. (1985) Vietoris Locales and Localic Semilattices. In: Hoffman, R. E. and Hoffman, K. H. (eds.) Continuous Lattices and Their Applications, Marcel Dekker 155–180.Google Scholar
Johnstone, P. T. (1989) A constructive ‘closed subgroup theorem’ for localic groups and groupoids. Cahiers de Top. Géom. Diff. Catégoriques 30 3–23.Google Scholar
Johnstone, P. T. (1992) Partial Products, Bagdomains and Hyperlocal Toposes. In: Fourman, M. P., Johnstone, P. T and Pitts, A. M. (eds.) Applications of Categories in Computer Science, London Mathematical Society Lecture Notes Series 177, Cambridge University Press 315–339.Google Scholar
Joyal, A. and Tierney, M. (1984) An extension of the Galois theory of Grothendieck. Memoirs Amer. Math. Soc. 309.Google Scholar
Kock, A. (1993) Monads for which structures are adjoint to units. To appear in Journal of Pure and Applied Algebra.Google Scholar
Kock, A. (1994) Some Extensive Quantities in Locale Theory - a comment on a theorem of Bunge and Funk. Talk given at the Peripatetic Seminar on Sheaves and Logic, Aarhus, December 1994 (unpublished).Google Scholar
Lawvere, F. W. (1983) Measures on Toposes. Lectures given at the Workshop on Category Theoretic Methods in Geometry, Aarhus, June 1983.Google Scholar
Macnab, D. S. (1981) Modal operators on Heyting algebras. Algebra Universalis 12 5–29.CrossRefGoogle Scholar
Pitts, A. M. (1985) On product and change of base for toposes. Cahiers de Top. Géom. Diff. Catégoriques 26 (1) 43–61.Google Scholar
Pitts, A. M. (1986) The lex reflection of a category with finite products, unpublished notes, December 1986.Google Scholar
Robinson, E. (1986) Power-Domains, Modalities and the Vietoris Monad, Technical Report No. 98, University of Cambridge Computer Laboratory.Google Scholar
Rosebrugh, R. and Wood, R. (1991) Constructive Complete Distributivity II. Math. Proc. Comb. Phil. Soc. 110 245–249.Google Scholar
Squire, R. (1989) Presheaf Toposes and Propositional Logic, Ph. D. Thesis, McMaster University. Vermeulen, J. J. C. (1989) Weak Compactness in Constructive Spaces, Technical Report RR079, Department of Mathematics, University of Cape Town.Google Scholar
Vickers, S. (1989) Topology via Logic, Cambridge Tracts in Theoretical Computer Science 5 Cambridge University Press.Google Scholar
Vickers, S. (1992) Geometric Theories and Databases. In: Fourman, M. P., Johnstone, P. T. and Pitts, A. M. (eds.) Applications of Categories in Computer Science, London Mathematical Society Lecture Notes Series 177, Cambridge University Press 288–314.Google Scholar
Wigner, D. (1979) Two notes on frames. J. Austral. Math. Soc. Ser. A 28 257–268.CrossRefGoogle Scholar