Coproducts of ideal monads
Ghani, Neil ; Uustalu, Tarmo
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 38 (2004), p. 321-342 / Harvested from Numdam

The question of how to combine monads arises naturally in many areas with much recent interest focusing on the coproduct of two monads. In general, the coproduct of arbitrary monads does not always exist. Although a rather general construction was given by Kelly [Bull. Austral. Math. Soc. 22 (1980) 1-83], its generality is reflected in its complexity which limits the applicability of this construction. Following our own research [C. Lüth and N. Ghani, Lect. Notes Artif. Intell. 2309 (2002) 18-32], and that of Hyland, Plotkin and Power [IFIP Conf. Proc. 223 (2002) 474-484], we are looking for specific situations when simpler constructions are available. This paper uses fixed points to give a simple construction of the coproduct of two ideal monads.

Publié le : 2004-01-01
DOI : https://doi.org/10.1051/ita:2004016
Classification:  08B20,  18C15,  18C50,  68Q55
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     author = {Ghani, Neil and Uustalu, Tarmo},
     title = {Coproducts of ideal monads},
     journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
     volume = {38},
     year = {2004},
     pages = {321-342},
     doi = {10.1051/ita:2004016},
     mrnumber = {2098194},
     zbl = {1072.18006},
     language = {en},
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Ghani, Neil; Uustalu, Tarmo. Coproducts of ideal monads. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 38 (2004) pp. 321-342. doi : 10.1051/ita:2004016. http://gdmltest.u-ga.fr/item/ITA_2004__38_4_321_0/

[1] P. Aczel, J. Adámek, S. Milius and J. Velebil, Infinite trees and completely iterative theories: a coalgebraic view. Theor. Comput. Sci. 300 (2003) 1-45. | Zbl 1028.68077

[2] J. Adámek, S. Milius and J. Velebil, Free iterative theories: a coalgebraic view. Math. Struct. Comput. Sci. 13 (2003) 259-320. | Zbl 1030.18004

[3] J. Adámek and J. Rosický, Locally Presentable and Accessible Categories. London Math. Soc. Lect. Note Ser. 189 (1994). | MR 1294136 | Zbl 0795.18007

[4] M. Barr, Coequalizers and free triples. Math. Z. 116 (1970) 307-322. | Zbl 0194.01701

[5] M. Barr and C. Wells, Toposes, Triples and Theories. Grundlehren der mathematischen Wissenschaften 275 (1985). | MR 771116 | Zbl 0567.18001

[6] J. Beck, Distributive laws, in Seminar on Triples and Categorical Homology Theory (ETH, 1966/67), edited by B. Eckmann, Springer-Verlag, Berlin. Lect. Notes Math. 80 (1969) 119-140. | Zbl 0186.02902

[7] M. Fiore, G. Plotkin and D. Turi, Abstract syntax and variable binding, in Proc. of 14th Ann. IEEE Symp. on Logic in Computer Science, LICS'99, Trento, July 1999, IEEE CS Press, Los Alamitos, CA (1999) 193-202.

[8] N. Ghani, C. Lüth, F. De Marchi and J. Power, Dualising initial algebras. Math. Struct. Comput. Sci. 13 (2003) 349-370. | Zbl 1049.18005

[9] N. Ghani, C. Lüth and F. De Marchi, Coalgebraic monads, in Proc. of 5th Wksh. on Coalgebraic Methods in Computer Science, CMCS'02, Grenoble, Apr. 2002, edited by L.S. Moss, Elsevier, Amsterdam. Electr. Notes in Theor. Comput. Sci. 65 (2002).

[10] N. Ghani and T. Uustalu, Explicit substitutions and higher-order syntax, in Proc. of 2nd ACM SIGPLAN Wksh. on Mechanized Reasoning about Languages with Variable Binding, MERLIN'03, Uppsala, Aug. 2003, edited by F. Honsell, M. Miculan, A. Momigliano, ACM Press, New York (2003).

[11] J.A. Goguen, A categorical manifesto. Math. Struct. Comput. Sci. 1 (1991) 49-67. | Zbl 0747.18001

[12] M. Hyland, G. Plotkin and J. Power, Combining computational effects: commutativity and sum, in Proc. of IFIP 17th World Computer Congress, TC1 Stream / 2nd IFIP Int. Conf. on Theoretical Computer Science, TCS 2002, Montreal, Aug. 2002, edited by A. Baeza-Yates, U. Montanari, and N. Santoro, Kluwer Academic Publishers, Dordrecht. IFIP Conf. Proc. 223 (2002) 474-484.

[13] C. Jones and G.D. Plotkin, A probabilistic powerdomain of evaluations, in Proc. of 4th Ann. IEEE Symp. Logic in Computer Science, LICS'89, Pacific Grove, CA, June 1989, IEEE CS Press, Washington, DC (1989) 186-195. | Zbl 0716.06003

[14] M. Jones and L. Duponcheel, Composing monads, Techn. report RR-1004. Dept. of Comp. Sci. Yale Univ. (1993).

[15] G.M. Kelly, A unified treatment of transfinite constructions for free algebras, free monoids, colimits, associated sheaves and so on. Bull. Austral. Math. Soc. 22 (1980) 1-83. | Zbl 0437.18004

[16] G.M. Kelly and J. Power, Adjunctions whose counits are equalizers, and presentations of finitary monads. J. Pure Appl. Algebra 89 (1993) 163-179. | Zbl 0779.18003

[17] F.E.J. Linton, Some aspects of equational categories, in Proc. of Conf. on Categorical Algebra, La Jolla, CA, June 1965, edited by S. Eilenberg, D. K. Harrison, S. Mac Lane, H. Röhrl, Springer-Verlag, Berlin (1966) 84-94. | Zbl 0201.35003

[18] C. Lüth, Categorical Term Rewriting: Monads and Modularity. Ph.D. Thesis, Lab. for Foundations of Comp. Sci., Univ. of Edinburgh (1998).

[19] C. Lüth and N. Ghani, Monads and modularity, in Proc. of 4th Int. Wksh. on Frontiers of Combining Systems, FroCoS 2002, Santa Margherita Ligure, Apr. 2002, edited by A. Armando, Springer-Verlag, Berlin. Lect. Notes Artif. Intell. 2309 (2002) 18-32. | Zbl 1057.68063

[20] C. Lüth and N. Ghani, Monads and modular term rewriting, in Proc. of 7th Int. Conf. on Category Theory in Computer Science, CTCS'97, Santa Margherita Ligure, Sept. 2002, edited by E. Moggi and G. Rosolini, Springer-Verlag, Berlin. Lect. Notes Comput. Sci. 1290 (1997) 69-86. | Zbl 0889.68084

[21] E.G. Manes, Algebraic Theories. Springer-Verlag, Berlin, Grad. Texts in Math. 26 (1976). | MR 419557 | Zbl 0353.18007

[22] R. Matthes and T. Uustalu, Substitution in non-wellfounded syntax with variable binding, in Proc. of 6th Wksh. on Coalgebraic Methods in Computer Science, CMCS'03, Warsaw, Apr. 2003, edited by H. P. Gumm, Elsevier, Amsterdam. Electr. Notes Theor. Comput. Sci. 82 (2003). | Zbl 1071.68063

[23] M. Mislove, Nondeterminism and probabilistic choice: obeying the laws, in Proc. 11 Int. Conf. on Concurrency Theory, CONCUR 2000, University Park, PA, Aug. 2000, edited by C. Palamidessi, Springer-Verlag, Berlin. Lect. Notes comput. Sci. 1877 (2000) 350-364. | Zbl 0999.68147

[24] E. Moggi, Computational lambda-calculus and monads, in Proc. of 4th Ann. IEEE Symp. on Logic in Computer Science, LICS'89, Pacific Grove, CA, June 1989, IEEE CS Press, Washington, DC (1989) 14-23. | Zbl 0716.03007

[25] E. Moggi, An abstract view of programming languages. Techn. report ECS-LFCS-90-113, Lab. for Foundations of Comp. Sci., Univ. of Edinburgh (1990).

[26] L. Moss, Parametric corecursion. Theor. Comput. Sci. 260 (2001) 139-163. | Zbl 0973.68134

[27] R. Tix, Continuous D-cones: convexity and powerdomain constructions. Ph.D. Thesis, Techn. Univ. Darmstadt (1999). | Zbl 0998.68533

[28] D. Varacca, The powerdomain of indexed valuations, in Proc. of 17th Ann. IEEE Symp. on Logic in Computer Science, LICS'02, Copenhagen, July 2002, IEEE CS Press, Los Alamitos, CA (2002) 299-308.