Foundations of Physics Letters

, Volume 12, Issue 1, pp 29–49 | Cite as

Operational resolutions and state transitions in a categorical setting

Article

Abstruct

We define a category with as objects operational resolutions and with as morphisms — not necessarily deterministic — state transitions. We study connections with closure spaces and join-complete lattices and sketch physical applications related to evolution and compoundness. An appendix with preliminaries on quantaloids is included.

Key words

state and property transitions closure space complete lattice quantales and quantaloids 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    J. Adamek, H. Herrlich, and G. E. Strecker,Abstract and Concrete Categories (Wiley, New York, 1990).MATHGoogle Scholar
  2. 2.
    D. Aerts,Found. Phys. 24, 1227 (1982).CrossRefADSMathSciNetGoogle Scholar
  3. 3.
    H. Amira, B. Coecke and I. Stubbe,Helvetica Phys. Acta 71, 554 (1998).MATHMathSciNetGoogle Scholar
  4. 4.
    G. Birkhoff,Lattice Theory, AMS Coll. Publ. (1940).Google Scholar
  5. 5.
    F. Borceux,Handbook of Categorical Algebra Part 1 and 2 (Cambridge University Press, Cambridge, 1994).Google Scholar
  6. 6.
    B. Coecke,Helvetica Phys. Acta 68, 394 (1995).MathSciNetGoogle Scholar
  7. 7.
    B. Coecke,Found. Phys. 28, 1109 (1998).CrossRefMathSciNetGoogle Scholar
  8. 8.
    B. Coecke,Found. Phys. 28, 1347 (1998).CrossRefMathSciNetGoogle Scholar
  9. 9.
    B. Coecke, “Structural characterization of compoundness,” submitted toInt. J. Theor. Phys., for the proceedings of IQSA-Liptovski Jan 1998 (n.d.).Google Scholar
  10. 10.
    B. Coecke and S. Smets, “A logical description for perfect measurements,” submitted toInt. J. Theor. Phys., for the proceedings of IQSA-Liptovski Jan 1998 (n.d.).Google Scholar
  11. 11.
    B. Coecke and I. Stubbe, “On a duality of quantales emerging from an operational resolution,” to appear inInt. J. Theor. Phys., for the proceedings of IQSA, Atlanta, 1997 (1999).Google Scholar
  12. 12.
    W. Daniel,Helvetica Phys. Ada 62, 941 (1989).MathSciNetGoogle Scholar
  13. 13.
    D. J. Foulis,Proc. AMS 11,6488 (I960).CrossRefMathSciNetGoogle Scholar
  14. 14.
    Cl.-A. Faure and A. Frölicher,Geom. Dedicata 47, 25 (1993).MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Cl.-A. Faure, D. J. Moore and C. Piron,Helvetica Phys. Acta 68, 150 (1995).MATHMathSciNetGoogle Scholar
  16. 16.
    J.Y. Girard,Theor. Comp. Sc. 50 (1987).Google Scholar
  17. 17.
    P. T. Johnstone,Stone Spaces (Cambridge University Press, Cambridge, 1982).MATHGoogle Scholar
  18. 18.
    A. Joyal and M. Tierney,Mem. AMS 51, No.309 (1984).Google Scholar
  19. 19.
    S. MacLane,Categories for the Working Mathematician (Springer, New York, 1971/1997).Google Scholar
  20. 20.
    D. J. Moore,Helvetica Phys. Acta 68, 658 (1995).MATHGoogle Scholar
  21. 21.
    D. J. Moore,Int. J. Theor. Phys. 36, 2211 (1997).CrossRefGoogle Scholar
  22. 22.
    C. J. Mulvey,Rend. Circ. Math. Palermo 12, 99 (1986).MathSciNetGoogle Scholar
  23. 23.
    C. J. Mulvey and J. Wick-Pelletier,Canadian Math. Soc. Conf. Proc. 13, 345 (1992).Google Scholar
  24. 24.
    J. Paseka, “Simple Quantales,” P. Simon, ed.,Proceedings, 8th Prague Topology Symposium, p. 314 (1996).Google Scholar
  25. 25.
    C. Piron,Helvetica Phys. Acta 37, 439 (1964).MATHMathSciNetGoogle Scholar
  26. 26.
    C. Piron,Foundations of Quantum Physics (Benjamin, New York, 1976).MATHGoogle Scholar
  27. 27.
    C. Piron,J. Phyl. Logic 6, 481 (1977).CrossRefGoogle Scholar
  28. 28.
    A. M. Pitts,Proc. London Math. Soc. 57, 433 (1988).MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    J. C. T. Pool,Comm. Math. Phys. 9, 118 (1968).MATHCrossRefADSMathSciNetGoogle Scholar
  30. 30.
    K. I. Rosenthal,Quantales and Their Applications, Pitmann Research Notes in Math. 234,Longmann Sc. & Tech. Publ. (1990).Google Scholar
  31. 31.
    K. I. Rosenthal, J. Pure Appl. Alg.77, 67 (1991).CrossRefGoogle Scholar
  32. 32.
    D. N. Yetter,J. Symb. Logic 55, 41 (1990).MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Plenum Publishing Corporation 1999

Authors and Affiliations

  1. 1.FUND-DWISFree University of Brussels Pleinlaan 2BrusselsBelgium
  2. 2.AGEL-MAPAUniversité Catholique de Louvain Ch. du Cyclotron 2Belgium

Personalised recommendations