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
Baez, J.C., Dolan, J.: Higher dimensional algebra III: n-categories and the algebra of opetopes. Adv. Math. 135, 145–206 (1998)
Barbosa, L.S.: Towards a calculus of state-based software components. Journ. of Universal Comp. Sci. 9(8), 891–909 (2003)
Barbosa, L.: Components as Coalgebras. PhD thesis, Univ. Minho (2001)
Barr, M., Wells, C.: Toposes, Triples and Theories. Springer, Berlin (1985)
Barr, M., Wells, C.: Category Theory for Computing Science, 3rd edn., Centre de recherches mathématiques, Université de Montréal (1999)
Bénabou, J.: Distributors at work. Lecture notes by Thomas Streicher (2000), www.mathematik.tu-darmstadt.de/~streicher/FIBR/DiWo.pdf.gz
Blackwell, R., Kelly, G., Power, A.: Two-dimensional monad theory. Journ. of Pure & Appl. Algebra 59, 1–41 (1989)
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)
Borceux, F.: Handbook of Categorical Algebra. Encyclopedia of Mathematics, vol. 50, 51 and 52. Cambridge University Press, Cambridge (1994)
Denecke, K., Wismath, S.L.: Universal Algebra and Applications in Theoretical Computer Science. Chapman and Hall, Boca Raton (2002)
Fiore, T.M.: Pseudo limits, biadjoints, and pseudo algebras: Categorical foundations of conformal field theory. Memoirs of the AMS 182 (2006)
Hasuo, I.: Pseudo functorial semantics (preprint), www.kurims.kyoto-u.ac.jp/~ichiro
Hasuo, I.: Tracing Anonymity with Coalgebras. PhD thesis, Radboud University Nijmegen (2008)
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)
Hughes, J.: Generalising monads to arrows. Science of Comput. Progr. 37(1–3), 67–111 (2000)
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)
Jacobs, B.: Semantics of weakening and contraction. Ann. Pure & Appl. Logic 69(1), 73–106 (1994)
Jacobs, B.: Categorical Logic and Type Theory. North-Holland, Amsterdam (1999)
Jacobs, B., Rutten, J.J.M.M.: A tutorial on (co)algebras and (co)induction. EATCS Bulletin 62, 222–259 (1997)
Jacobs, B., Heunen, C., Hasuo, I.: Categorical semantics for arrows. Journ. Funct. Progr. (to appear, 2009)
Lack, S., Power, J.: Lawvere 2-theories. Presented at CT 2007 (2007), www.mat.uc.pt/~categ/ct2007/slides/lack.pdf
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)
Levy, P.B., Power, A.J., Thielecke, H.: Modelling environments in call-by-value programming languages. Inf. & Comp. 185(2), 182–210 (2003)
Mac Lane, S.: Categories for the Working Mathematician, 2nd edn. Springer, Berlin (1998)
Moggi, E.: Notions of computation and monads. Inf. & Comp. 93(1), 55–92 (1991)
Pattinson, D.: An introduction to the theory of coalgebras. Course notes for NASSLLI (2003), www.indiana.edu/~nasslli
Power, J., Robinson, E.: Premonoidal categories and notions of computation. Math. Struct. in Comp. Sci. 7(5), 453–468 (1997)
Rutten, J.J.M.M.: Algebraic specification and coalgebraic synthesis of Mealy automata. Elect. Notes in Theor. Comp. Sci. 160, 305–319 (2006)
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)
Uustalu, T., Vene, V.: Comonadic notions of computation. Elect. Notes in Theor. Comp. Sci. 203(5), 263–284 (2008)
Wadler, P.: Monads for functional programming. In: Marktoberdorf Summer School on Program Design Calculi. Springer, Heidelberg (1992)
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Hasuo, I., Heunen, C., Jacobs, B., Sokolova, A. (2009). Coalgebraic Components in a Many-Sorted Microcosm. In: Kurz, A., Lenisa, M., Tarlecki, A. (eds) Algebra and Coalgebra in Computer Science. CALCO 2009. Lecture Notes in Computer Science, vol 5728. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03741-2_6
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