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Monad compositions II: Kleisli strength

Published online by Cambridge University Press:  01 June 2008

ERNIE MANES  and
PHILIP MULRY
ERNIE MANES
Affiliation:
Department of Mathematics and Statistics, University of Massachusetts at Amherst, U.S.A.
PHILIP MULRY
Affiliation:
Department of Computer Science, Colgate University, Hamilton, New York, 13346, U.S.A. Email: pmulry@mail.colgate.edu
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Abstract

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In this paper we introduce the concept of Kleisli strength for monads in an arbitrary symmetric monoidal category. This generalises the notion of commutative monad and gives us new examples, even in the cartesian-closed category of sets. We exploit the presence of Kleisli strength to derive methods for generating distributive laws. We also introduce linear equations to extend the results to certain quotient monads. Mechanisms are described for finding strengths that produce a large collection of new distributive laws, and consequently monad compositions, including the composition of monadic data types such as lists, trees, exceptions and state.

Information

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 18 , Issue 3 , June 2008 , pp. 613 - 643
Copyright
Copyright © Cambridge University Press 2008

References

Ádamek, J. and Lawvere, F. W. (2001) How Algebraic is algebra? Theory and Applications of Categories 8 253–283.Google Scholar
Barr, M. (1970) Coequalizers and free triples. Mathematische Zeitschrift 116 307–322.Google Scholar
Beck, J. (1969) Distributive laws. Springer-Verlag Lecture Notes in Mathematics 80 119–140.CrossRefGoogle Scholar
Eilenberg, S. and Kelly, G.M. (1966) Closed categories. Proceedings of the Conference on Categorical Algebra, La Jolla, Springer-Verlag 421–562.CrossRefGoogle Scholar
King, D. and Wadler, P. (1993) Combining monads. In: Launchbury, J. and Sansom, P. M. (eds.) Proceedings of the 1992 Glasgow Workshop on Functional Programming, Springer-Verlag 134–143.CrossRefGoogle Scholar
Kock, A. (1970) Monads on symmetric monoidal closed categories. Archiv der Mathematik 21 1–10.CrossRefGoogle Scholar
Koslowski, J. (2005) A monadic approach to polycategories. Theory and Applications of Categories 14 125–156.Google Scholar
Linton, F. E. J. (1966) Autonomous equational categories. Journal of Mathematics and Mechanics 15 637–642.Google Scholar
Lüth, C. and Ghani, N. (2002) Composing Monads Using Coproducts. International Conference on Functional Programming ICFP'02, ACM Press 133–144.Google Scholar
Mac Lane, S. (1971) Categories for the Working Mathematician, Springer-Verlag.CrossRefGoogle Scholar
Marmolejo, F., Rosebrugh, R. and Wood, R. J. (2002) A basic distributive law. Journal of Pure and Applied Algebra 168 209–226.Google Scholar
Manes, E. G. and Mulry, P. S. (2007) Monad Compositions I: General Constructions and Recursive Distributive laws. Theory and Applications of Categories 18 172–208.Google Scholar
Manes, E. G. and Mulry, P. S. (2008) Monad Compositions III. (To appear.)Google Scholar
Moggi, E. (1991) Notions of computations and monads. Information and Computation 93 55–92.Google Scholar
Mulry, P. S. (1994) Lifting theorems for Kleisli categories. Springer-Verlag Lecture Notes in Computer Science 802 304–319.CrossRefGoogle Scholar
Mulry, P. S. (1998) Monads in semantics. Electronic Notes in Theoretical Computer Science 12 275–286.CrossRefGoogle Scholar
Scott, D. (1970) Outline of a mathematical theory of computation. Technical Monograph PRG-2, Oxford University Computing Laboratory.Google Scholar