Abstract
A complete latticeL isconstructively completely distributive, (CCD), when the sup arrow from down-closed subobjects ofL toL has a left adjoint. The Karoubian envelope of the bicategory of relations is biequivalent to the bicategory of (CCD) lattices and sup-preserving arrows. There is a restriction to order ideals and “totally algebraic” lattices. Both biequivalences have left exact versions. As applications we characterize projective sup lattices and recover a known characterization of projective frames. Also, the known characterization of nuclear sup lattices in set as completely distributive lattices is extended to yet another characterization of (CCD) lattices in a topos.
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References
B. Banaschewski and S. B. Niefield. Projective and supercoherent frames.Journal of Pure and Applied Algebra, 70:45–51, 1991.
A. Carboni, G. M. Kelly, and R. J. Wood. A 2-categorical approach to change of base and geometric morphisms 1.Cahiers de topologie et géometrie différentielle catégoriques, XXXII:47–95, 1991.
A. Carboni and R. Street. Order ideals in categories.Pacific Journal of Mathematics, 124(2):275–278, 1986.
C.J. Mulvey. Intuitionistic algebra and representation of rings.Memoirs of the AMS, 148:3–57, 1974.
R. Guitart et J. Riguet. Envelopes karoubiennes de catégories de kleisli.Cahiers de topologie et géometrie différentielle catégoriques, XXX:261–266, 1992.
B. Fawcett and R. J. Wood. Constructive complete distributivity I.Math. Proc. Cam. Phil. Soc., 107:81–89, 1990.
D. A. Higgs and K. A. Rowe. Nuclearity in the category of complete semi-lattices.Journal of Pure and Applied Algebra, 57:67–78, 1989.
R-E. Hoffmann. Continuous posets and adjoint sequences.Semigroup Forum, 18:173–188, 1979.
P. T. Johnstone.Stone Spaces. Cambridge University Press, 1982.
A. Joyal and M. Tierney.An Extension of the Galois Theory of Grothendieck. Memoirs of the American Mathematical Society No. 309, American Mathematical Society, 1984.
F. W. Lawvere. Metric spaces, generalized logic and closed categories.Rendiconti del Seminario Matematico e Fisico di Milano, 43:135–166, 1973.
G. N. Raney. A subdirect-union representation for completely distributive complete lattices.Proc. Amer. Math. Soc., 4:518–512, 1953.
R. Rosebrugh and R. J. Wood. Constructive complete distributivity II.Math. Proc. Camb. Phil. Soc., 110:245–249, 1991.
R. Rosebrugh and R. J. Wood. Constructive complete distributivity III.Canadian Mathematical Bulletin, 35:537–547, 1992.
R. Rosebrugh and R. J. Wood. Constructive complete distributivity V. in preparation, 1994.
Keith Rowe. Nuclearity.Canadian Mathematical Bulletin, 31:227–235, 1989.
R. Street and R. F. C. Walters. Yoneda structures on 2-categories.Journal of Algebra, 50:350–379, 1978.
S. Vickers. Information systems for continuous posets.Theoretical Computer Science, 114:201–229, 1993.
R. J. Wood. Proarrows 1.Cahiers de topologie et géometrie différentielle catégoriques, XXIII:279–290, 1982.
R. J. Wood. Proarrows 2.Cahiers de topologie et géometrie différentielle catégoriques, XXVI:135–168, 1985.
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Research partially supported by grants from NSERC Canada. Diagrams typeset using catmac.
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Rosebrugh, R., Wood, R.J. Constructive complete distributivity IV. Appl Categor Struct 2, 119–144 (1994). https://doi.org/10.1007/BF00873296
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DOI: https://doi.org/10.1007/BF00873296