Advertisement

Applied Categorical Structures

, Volume 17, Issue 5, pp 445–466 | Cite as

A Convenient Category of Locally Preordered Spaces

  • Sanjeevi Krishnan
Article

Abstract

As a practical foundation for a homotopy theory of abstract spacetime, we extend a category of certain compact partially ordered spaces to a convenient category of “locally preordered” spaces. We show that our new category is Cartesian closed that the forgetful functor to the category of compactly generated spaces creates all limits and colimits.

Keywords

Preordered space Ditopology Compactly generated space 

Mathematics Subject Classifications (2000)

54E99 54F05 18F99 18D15 68Q85 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Borceux, F.: Handbook of categorical algebra 2: categories and structures. In: Encyclopedia of Mathematics and its Applications, vol. 51, pp. xviii+443. Cambridge University Press, Cambridge (1994)Google Scholar
  2. 2.
    Brown, K.: The geometry of rewriting systems: a proof of the Anick-Groves-Squier theorem, algorithms and classification in combinatorial group theory, (Berekely, CA, 1989). Math. Sci. Res. Inst. Publ. 23, 137–163 (1992)Google Scholar
  3. 3.
    Bubenik, P., Worytkiewicz, K.: A model category for local pospaces. Homology Homotopy Appl. 8(1), 263–292 (2006)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Fahrenberg, U.: Directed homology. In: Proc. GETCO&CMCIM 2003, Electronic Notes in Theoretical Computer Science, vol. 100. Elsevier, Amsterdam (2004)Google Scholar
  5. 5.
    Fajstrup, L., Goubault, E., Haucourt, E., Raussen, M.: Components of the fundamental category. Appl. Categ. Structures 12(1), 84–108 (2004)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Fajstrup, L., Goubault, E., Raussen, M.: Algebraic topology and concurrency. Theoret. Comput. Sci. 357(1–3), 241–278 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Gaucher, P.: A model category for the homotopy theory of concurrency. Homology Homotopy Appl. 5(1), 549–599 (2003)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Gierz, G., Hoffman, K.H., Keimel, K., Lawsonj, J.D., Mislove, M., Scott, D.S.: Continuous lattices and domains. In: Encyclopedia of Mathematics and Applications, vol. 63. Cambridge University Press, Cambridge (2003)Google Scholar
  9. 9.
    Goubault, E.: Geometry and concurrency: a user’s guide. Math. Structures Comput. Sci. 10(4), 411–425 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Goubault, E., Goubault-Larrecq, J.: On the geometry of intutionistic S4 proofs. Homology Homotopy Appl. 5(2), 137–209 (2003)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Goubault, E., Haucourt, E.: Components of the fundamental category II. Appl. Categ. Structures 15(4), 387–414 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Grandis, M.: Directed homotopy theory. I.. Cahiers Topologie Géom. Differentielle Catég. 44(4), 281–316 (2003)zbMATHMathSciNetGoogle Scholar
  13. 13.
    Grandis, M.: Directed homotopy theory. II. Homotopy constructs. Theory Appl. Categ. 10(14), 369–391 (2002)zbMATHMathSciNetGoogle Scholar
  14. 14.
    Grandis, M.: Inequilogical spaces, directed homology and noncommutative geometry. Homology Homotopy Appl. 6(1), 413–437 (2004)zbMATHMathSciNetGoogle Scholar
  15. 15.
    Grandis, M.: Ordinary and directed combinatorial homotopy, applied to image analysis and concurrency. Homology Homotopy Appl. 5(2), 211–231 (2003)zbMATHMathSciNetGoogle Scholar
  16. 16.
    Haucourt, E.: Comparing topological models for concurrency. In: GETCO 2005 proceedings, San Francisco, 23–26 August 2005Google Scholar
  17. 17.
    Kobayashi, Y.: Complete rewriting systems and homology of monoid algebras. J. Pure Appl. Algebra 65(3), 263–275 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Kobayashi, Y., Otto, F., Squier, C.: A finiteness condition for rewriting systems. Theoret. Comput. Sci. 131, 271–294 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Krishnan, S.: A homotopy theory of locally preordered spaces. Ph.D. thesis, University of Chicago, Chicago IL (2006)Google Scholar
  20. 20.
    Lafont, Y.: A new finiteness condition for monoids presented by complete rewriting systems (after Craig C. Squier). J. Pure Appl. Algebra 98, 229–244 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Nachbin, L.: Topology and order, translated from the Portuguese by Lulu Bechtolsheim. Van Nostrand Math. Stud. 4, 122 (1965)MathSciNetGoogle Scholar
  22. 22.
    Patchkoria, A.: Homology and cohomology monoids of presimplicial semimodules. Bulletin of the Georgian Academy of Sciences 162(1), 9–12 (2000)zbMATHMathSciNetGoogle Scholar
  23. 23.
    Pratt, V.: Modelling concurrency with geometry. In: Proc. 18th ACM Symp. on Principles of Programming Languages, pp. 311–322. New York, ACM (1991)Google Scholar
  24. 24.
    Squier, C.C.: Word problems and a homological finiteness condition for monoids. J. Pure Appl. Algebra 49, 201–217 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Steenrod, N.: A convenient category of spaces. Michigan Math. J. 14, 133–152 (1967)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Modélisation et Analyse de Systèmes en InteractionCEA LIST & Ecole PolytechniqueParisFrance

Personalised recommendations