Coalgebraic Components in a Many-Sorted Microcosm

  • Ichiro Hasuo
  • Chris Heunen
  • Bart Jacobs
  • Ana Sokolova
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5728)

Abstract

The microcosm principle, advocated by Baez and Dolan and formalized for Lawvere theories lately by three of the authors, has been applied to coalgebras in order to describe compositional behavior systematically. Here we further illustrate the usefulness of the approach by extending it to a many-sorted setting. Then we can show that the coalgebraic component calculi of Barbosa are examples, with compositionality of behavior following from microcosm structure. The algebraic structure on these coalgebraic components corresponds to variants of Hughes’ notion of arrow, introduced to organize computations in functional programming.

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References

  1. 1.
    Baez, J.C., Dolan, J.: Higher dimensional algebra III: n-categories and the algebra of opetopes. Adv. Math. 135, 145–206 (1998)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Barbosa, L.S.: Towards a calculus of state-based software components. Journ. of Universal Comp. Sci. 9(8), 891–909 (2003)Google Scholar
  3. 3.
    Barbosa, L.: Components as Coalgebras. PhD thesis, Univ. Minho (2001)Google Scholar
  4. 4.
    Barr, M., Wells, C.: Toposes, Triples and Theories. Springer, Berlin (1985)CrossRefMATHGoogle Scholar
  5. 5.
    Barr, M., Wells, C.: Category Theory for Computing Science, 3rd edn., Centre de recherches mathématiques, Université de Montréal (1999)Google Scholar
  6. 6.
    Bénabou, J.: Distributors at work. Lecture notes by Thomas Streicher (2000), www.mathematik.tu-darmstadt.de/~streicher/FIBR/DiWo.pdf.gz
  7. 7.
    Blackwell, R., Kelly, G., Power, A.: Two-dimensional monad theory. Journ. of Pure & Appl. Algebra 59, 1–41 (1989)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Bonsangue, M.M., Rutten, J., Silva, A.: Coalgebraic logic and synthesis of Mealy machines. In: Amadio, R.M. (ed.) FOSSACS 2008. LNCS, vol. 4962, pp. 231–245. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  9. 9.
    Borceux, F.: Handbook of Categorical Algebra. Encyclopedia of Mathematics, vol. 50, 51 and 52. Cambridge University Press, Cambridge (1994)CrossRefMATHGoogle Scholar
  10. 10.
    Denecke, K., Wismath, S.L.: Universal Algebra and Applications in Theoretical Computer Science. Chapman and Hall, Boca Raton (2002)MATHGoogle Scholar
  11. 11.
    Fiore, T.M.: Pseudo limits, biadjoints, and pseudo algebras: Categorical foundations of conformal field theory. Memoirs of the AMS 182 (2006)Google Scholar
  12. 12.
    Hasuo, I.: Pseudo functorial semantics (preprint), www.kurims.kyoto-u.ac.jp/~ichiro
  13. 13.
    Hasuo, I.: Tracing Anonymity with Coalgebras. PhD thesis, Radboud University Nijmegen (2008)Google Scholar
  14. 14.
    Hasuo, I., Jacobs, B., Sokolova, A.: The microcosm principle and concurrency in coalgebra. In: Amadio, R.M. (ed.) FOSSACS 2008. LNCS, vol. 4962, pp. 246–260. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  15. 15.
    Hughes, J.: Generalising monads to arrows. Science of Comput. Progr. 37(1–3), 67–111 (2000)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Hyland, M., Power, J.: The category theoretic understanding of universal algebra: Lawvere theories and monads. Elect. Notes in Theor. Comp. Sci. 172, 437–458 (2007)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Jacobs, B.: Semantics of weakening and contraction. Ann. Pure & Appl. Logic 69(1), 73–106 (1994)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Jacobs, B.: Categorical Logic and Type Theory. North-Holland, Amsterdam (1999)MATHGoogle Scholar
  19. 19.
    Jacobs, B., Rutten, J.J.M.M.: A tutorial on (co)algebras and (co)induction. EATCS Bulletin 62, 222–259 (1997)MATHGoogle Scholar
  20. 20.
    Jacobs, B., Heunen, C., Hasuo, I.: Categorical semantics for arrows. Journ. Funct. Progr. (to appear, 2009)Google Scholar
  21. 21.
    Lack, S., Power, J.: Lawvere 2-theories. Presented at CT 2007 (2007), www.mat.uc.pt/~categ/ct2007/slides/lack.pdf
  22. 22.
    Lawvere, F.W.: Functorial Semantics of Algebraic Theories and Some Algebraic Problems in the Context of Functorial Semantics of Algebraic Theories. PhD thesis, Columbia University, 1–121 (1963); Reprints in Theory and Applications of Categories, 5 (2004)Google Scholar
  23. 23.
    Levy, P.B., Power, A.J., Thielecke, H.: Modelling environments in call-by-value programming languages. Inf. & Comp. 185(2), 182–210 (2003)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Mac Lane, S.: Categories for the Working Mathematician, 2nd edn. Springer, Berlin (1998)MATHGoogle Scholar
  25. 25.
    Moggi, E.: Notions of computation and monads. Inf. & Comp. 93(1), 55–92 (1991)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Pattinson, D.: An introduction to the theory of coalgebras. Course notes for NASSLLI (2003), www.indiana.edu/~nasslli
  27. 27.
    Power, J., Robinson, E.: Premonoidal categories and notions of computation. Math. Struct. in Comp. Sci. 7(5), 453–468 (1997)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Rutten, J.J.M.M.: Algebraic specification and coalgebraic synthesis of Mealy automata. Elect. Notes in Theor. Comp. Sci. 160, 305–319 (2006)CrossRefGoogle Scholar
  29. 29.
    Segal, G.: The definition of conformal field theory. In: Tillmann, U. (ed.) Topology, Geometry and Quantum Field Theory. London Math. Soc. Lect. Note Series, vol. 308, pp. 423–577. Cambridge University Press, Cambridge (2004)Google Scholar
  30. 30.
    Uustalu, T., Vene, V.: Comonadic notions of computation. Elect. Notes in Theor. Comp. Sci. 203(5), 263–284 (2008)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Wadler, P.: Monads for functional programming. In: Marktoberdorf Summer School on Program Design Calculi. Springer, Heidelberg (1992)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Ichiro Hasuo
    • 1
    • 4
  • Chris Heunen
    • 2
  • Bart Jacobs
    • 2
  • Ana Sokolova
    • 3
  1. 1.RIMSKyoto UniversityJapan
  2. 2.Radboud University NijmegenThe Netherlands
  3. 3.University of SalzburgAustria
  4. 4.PRESTO Research Promotion ProgramJapan Science and Technology AgencyJapan

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