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Self-duality, Minimal Invariant Objects and Karoubi Invariance in Information Categories

  • Abbas Edalat
Conference paper
Part of the Workshops in Computing book series (WORKSHOPS COMP.)

Abstract

We introduce IP-categories by enriching the notion of I-category (information category) such that every inclusion morphism has a right adjoint or projection. Categories of information systems for various domains in semantics are all examples of IP-categories. We show. that a weak notion of substructure relation between two Scott information systems induces general adjunctions between them. An IP-category with a zero object has the self-dual property that its opposite is again an IP-category. In a complete IP-category with a zero object, limits and colimits of chains of projections and inclusions coincide. As a consequence of the self-duality, a simple characterisation of minimal invariant objects of contravariant and mixed functors on IP-categories is obtained. IP-categories are closed under taking the Karoubi envelope. We use the arrow category of an effectively given IP-category to solve domain equations in categories of continuous information systems effectively.

Keywords

Information Category Domain Equation Invariant Object Projection Morphism General Adjunction 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    S. Abramsky. Domain theory in logical form. Annals of Pure and Applied Logic, 1991Google Scholar
  2. [2]
    G. Berry and P.-L. Curien. Sequential algorithms on concrete data structures. Technical report, Report of Ecole Nationale Superieure des Mines de Paris, Centre de Mathematiques Appliquées, Sophia Antipolis, 1981.Google Scholar
  3. [3]
    A. Edalat. Self duality in information categories. Technical Report Doc 91/42, Department of Computing, Imperial College, 1991.Google Scholar
  4. [4]
    A. Edalat. Continuous I-categories. In A. Nerode and M. Taitslin, editors, Logical Foundations of Computer Science, volume 620 of Lecture Notes in Computer Science, pages 127–138. Springer-Verlag, 1992.Google Scholar
  5. [5]
    A. Edalat and M. B. Smyth. Categories of information systems. In D. H. Pitt, P. L. Curien, S. Abramsky, A. M. Pitts, A. Poigne, and D. E. Rydeheard, editors, Category theory in computer science, pages 37–52. Springer-Verlag, 1991.CrossRefGoogle Scholar
  6. [6]
    A. Edalat and M. B. Smyth. Compact metric information systems. In 1992 Rex workshop on semantics of programming languages. Springer-Verlag, 1993. to appear.Google Scholar
  7. [7]
    A. Edalat and M. B. Smyth. I-categories as a framework for solving domain equations. Theoretical Computer Science,1993. to appear.Google Scholar
  8. [8]
    A. Edalat and M. B. Smyth. Information Categories. Applied Categorical Structures,1993. to appear.Google Scholar
  9. [9]
    P. J. Freyd. Algebraically complete categories. In A. Carboni et al, editor, Proc. 1990 Como Category theory conference, volume 1488 of Lec. Notes in Maths, pages 95–104. Springer Verlag, 1991.Google Scholar
  10. [10]
    P. J. Freyd. Remarks on algebraically compact categories. In A. M. Pitts M. P. Fourman, P. T. Johnstone, editor, Applications of categories in computer science, number 177 in L.M.S. Lecture notes series, pages 95–106. Cambridge University Press, 1992.Google Scholar
  11. [11]
    C. Gunter. Profinite Solutions for Recursive Domain Equations. PhD thesis, University of Wisconsin, Madison, 1985.Google Scholar
  12. [12]
    A. Jung. Cartesian Closed Categories of Domains. PhD thesis, Technische Hochschule Darmstadt, 1988.MATHGoogle Scholar
  13. [13]
    K. G. Larsen and G. Winskel. Using information systems to solve recursive domain equations effectively. In D. B. MacQueen G. Kahn and G. Plotkin, editors, Semantics of Data Types, pages 109–130, Berlin, 1984. Springer-Verlag. Lecture Notes in Computer Science Vol. 173.CrossRefGoogle Scholar
  14. [14]
    D. J. Lehmann and M. B. Smyth. Algebraic specification of data types: A synthetic approach. Mathematical Systems Theory, 14: 97–139, 1981.MathSciNetMATHCrossRefGoogle Scholar
  15. [15]
    S. Mac Lane. Categories for the Working Mathematician. Springer-Verlag, Berlin, 1971.MATHCrossRefGoogle Scholar
  16. [16]
    D. S. Scott. Domains for denotational semantics. In M. Nielson and E. M. Schmidt, editors, Automata, Languages and Programming: Proceedings 1982. Springer-Verlag, Berlin, 1982. Lecture Notes in Computer Science 140.Google Scholar
  17. [17]
    M. B. Smyth. I-categories and duality. In M. P. Fourman, P. T. Johnstone, and A. M. Pitts, editors, Application of categories in computer science, volume 177 of London Mathematical Society Lecture Note Series, pages 270–287. Cambrdge University Press, 1992.Google Scholar
  18. [18]
    M. B. Smyth and G. D. Plotkin. The category-theoretic solution of recursive domain equations. SIAM J. Computing, 11: 761–783, 1982.MathSciNetMATHCrossRefGoogle Scholar
  19. [19]
    P. Taylor. Recursive Domains, Indexed category Theory and Polymorphism. PhD thesis, Cambridge University, 1986.Google Scholar
  20. [20]
    S. J. Vickers. Geometric theories and databases. In P. Johnstone M. P. Fourman and A. M. Pitts, editors, Applications of Categories in Computer Science, pages 288–314. Cambridge University Press, 1991.Google Scholar
  21. [21]
    G. Q. Zhang. dI-domains as information systems. In G. Ausiello, M. DezaniCiancaglini, and S. Ronchi Della Rocca, editors, Automata, Languages and Programming, volume 372 of Lecture Notes in Computer Science, pages 773–788, Berlin, 1989. Springer Verlag.Google Scholar

Copyright information

© British Computer Society 1993

Authors and Affiliations

  • Abbas Edalat
    • 1
  1. 1.Department of ComputingImperial CollegeLondonUK

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