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Calculating Colimits Compositionally

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Concurrency, Graphs and Models

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5065))

Abstract

We show how finite limits and colimits can be calculated compositionally using the algebras of spans and cospans, and give as an application a proof of the Kleene Theorem on regular languages.

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References

  1. Abramsky, S., Coecke, B.: A Categorical Semantics of Quantum Protocols. In: Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science: LICS 2004, pp. 415–425. IEEE Computer Society, Los Alamitos (2004)

    Chapter  Google Scholar 

  2. Bloom, S.L., Esik, Z.: Iteration Theories: the equational logic of iterative processes. EATCS Monographs in Theoretical Computer Science. Springer, Heidelberg (1993)

    MATH  Google Scholar 

  3. Bruni, R., Gadducci, F., Montanari, U.: Normal forms for algebras of connection. Theor. Comput. Sci. 286(2), 247–292 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  4. Carboni, A., Walters, R.F.C.: Cartesian bicategories I. Journal of Pure and Applied Algebra 49, 11–32 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  5. de Francesco Albasini, L., Rosebrugh, R., Sabadini, N., Walters, R.F.C.: Cospans and free symmetric monoidal categories (in preparation)

    Google Scholar 

  6. Elgot, C.C.: Monadic computation and iterative algebraic theories, Logoc Colloquium 1973, Studies in Logic 80, pp. 175–230. North Holland, Amsterdam (1975)

    Google Scholar 

  7. Gadducci, F., Heckel, R.: An inductive view of graph transformation. In: Parisi-Presicce, F. (ed.) WADT 1997. LNCS, vol. 1376, pp. 223–237. Springer, Heidelberg (1998)

    Google Scholar 

  8. Gadducci, F., Heckel, R., Llabrés, M.: A bi-categorical axiomatisation of concurrent graph rewriting. In: Proc. CTCS 1999, Category Theory and Computer Science. Electronic Notes in Theoretical Computer Science, vol. 29, Elsevier Sciences, Amsterdam (1999)

    Google Scholar 

  9. Joyal, A., Street, R.H.: The geometry of tensor calculus I. Advances in Math. 88, 55–112 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  10. Joyal, A., Street, R., Verity, D.: Traced monoidal categories. Mathematical Proceedings of the Cambridge Philosophical Society 119(3), 447–468 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  11. Katis, P., Sabadini, N., Walters, R.F.C.: Bicategories of processes. Journal of Pure and Applied Algebra 115, 141–178 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  12. Katis, P., Sabadini, N., Walters, R.F.C.: Span(Graph): A categorical algebra of transition systems. In: Johnson, M. (ed.) AMAST 1997. LNCS, vol. 1349, pp. 307–321. Springer, Heidelberg (1997)

    Chapter  Google Scholar 

  13. Katis, P., Sabadini, N., Walters, R.F.C.: On the algebra of systems with feedback and boundary. Rendiconti del Circolo Matematico di Palermo Serie II Suppl. 63, 123–156 (2000)

    MathSciNet  Google Scholar 

  14. Katis, P., Sabadini, N., Walters, R.F.C.: A formalisation of the IWIM Model. In: Porto, A., Roman, G.-C. (eds.) COORDINATION 2000. LNCS, vol. 1906, pp. 267–283. Springer, Heidelberg (2000)

    Chapter  Google Scholar 

  15. Katis, P., Sabadini, N., Walters, R.F.C.: Feedback, trace and fixed-point semantics. Theoret. Informatics Appl. 36, 181–194 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  16. Kelly, G.M., Laplaza, M.L.: Coherence for compact closed categories. J. Pure Appl. Algebra 19, 193–213 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  17. Kock, J.: Frobenius algebras and 2D topological Quantum Field Theories. Cambridge University Press, Cambridge (2004)

    MATH  Google Scholar 

  18. Menni, M., Sabadini, N., Walters, R.F.C.: A universal property of the monoidal 2-category of cospans of ordinals and surjections. Theory and Applications of Categories 18(19), 631–653 (2007)

    MATH  MathSciNet  Google Scholar 

  19. Meseguer, J., Montanari, U.: Petri Nets Are Monoids. Information and Computation 88, 105–155 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  20. Penrose, R.: Applications of negative dimensional tensors. In: Combinatorial Mathematics and its Applications, p. 221. Academic Press, London (1971)

    Google Scholar 

  21. Rosebrugh, R., Sabadini, N., Walters, R.F.C.: Minimization and minimal realization in Span(Graph). Mathematical Structures in Computer Science 14, 685–714 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  22. Rosebrugh, R., Sabadini, N., Walters, R.F.C.: Generic commutative separable algebras and cospans of graphs. Theory and Applications of Categories 15(6), 264–277 (2005)

    MathSciNet  Google Scholar 

  23. Rosebrugh, R., Sabadini, N., Walters, R.F.C.: Calculating colimits and limits compositionally. Category Theory 2007, Carvoeiro, Portugal, 18th (June 2007)

    Google Scholar 

  24. Rosebrugh, R., Sabadini, N., Walters, R.F.C.: Calculating colimits and limits compositionally (in preparation)

    Google Scholar 

  25. Walters, R.F.C.: Lecture to the Sydney Category Seminar (January 26, 1983)

    Google Scholar 

  26. Walters, R.F.C.: The tensor product of matrices, Lecture. In: International Conference on Category Theory, Louvain-la-Neuve (1987)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, Mt. Allison University, Sackville, New Brunswick, Canada

    Robert Rosebrugh

  2. Dipartimento di Scienze delle Cultura, Politiche e dell’Informazione, Università dell’Insubria, Italy

    Nicoletta Sabadini & Robert F. C. Walters

Authors
  1. Robert Rosebrugh
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  2. Nicoletta Sabadini
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  3. Robert F. C. Walters
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Editor information

Pierpaolo Degano Rocco De Nicola José Meseguer

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Rosebrugh, R., Sabadini, N., Walters, R.F.C. (2008). Calculating Colimits Compositionally. In: Degano, P., De Nicola, R., Meseguer, J. (eds) Concurrency, Graphs and Models. Lecture Notes in Computer Science, vol 5065. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-68679-8_36

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  • DOI: https://doi.org/10.1007/978-3-540-68679-8_36

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-68676-7

  • Online ISBN: 978-3-540-68679-8

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