Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-23T22:10:19.850Z Has data issue: false hasContentIssue false
  • English
  • Français

The parametric continuation monad †

Published online by Cambridge University Press:  24 August 2015

PAUL-ANDRÉ MELLIÈS
PAUL-ANDRÉ MELLIÈS*
Affiliation:
CNRS, Laboratoire PPS, UMR 7126, Université Paris Diderot, Sorbonne Paris Cité, F-75205 Paris, France Email: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Every dialogue category comes equipped with a continuation monad defined by applying the negation functor twice. In this paper, we advocate that this double negation monad should be understood as part of a larger parametric monad (or a lax action) with parameter taken in the opposite of the dialogue category. This alternative point of view has one main conceptual benefit: it reveals that the strength of the continuation monad is the fragment of a more fundamental and symmetric structure – provided by a distributivity law between the parametric continuation monad and the canonical action of the dialogue category over itself. The purpose of this work is to describe the formal properties of this parametric continuation monad and of its distributivity law.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 27 , Special Issue 5: Computing with Lambda-terms. A Special Issue Dedicated to Corrado Böhm for his 90th Birthday , June 2017 , pp. 651 - 680
Copyright
Copyright © Cambridge University Press 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

†

Dedicated to Corrado Böhm, on the occasion of his 90th birthday.

References

Atkey, R. (2009). Parameterised notions of computation. Journal of Functional Programming 19 (3 & 4) 335–376.Google Scholar
Beck, J. (1969). Distributive laws. Lecture Notes in Mathematics 80 119–140.Google Scholar
Bénabou, J. (1963). Catégories avec multiplication. C. R. Académie des Sciences, Paris 256 1887–1890.Google Scholar
Blute, R., Cockett, R., Seely, R. and Trimble, T. (1996). Natural deduction and coherence for weakly distributive categories. Journal of Pure and Applied Algebra 113 229–296.Google Scholar
Cockett, R. and Seely, R. (1997). Weakly distributive categories. Journal of Pure and Applied Algebra 114 (2) 133–173.Google Scholar
Girard, J-Y. (1987) Linear logic. Theoretical Computer Science, London Mathematical 50 (1) 1–102.Google Scholar
Girard, J-Y. (1995). Linear logic, its syntax and semantics. In: Girard, J.-Y., Lafont, Y. and Regnier, L. (eds.) Advances in Linear Logic, London Mathematical Society Lecture Notes Series volume 222, Cambridge University Press.Google Scholar
Hu, H. and Joyal, A. (1999) Coherence completions of categories. Theoretical Computer Science 01/1999; 27 (2) 153–184.Google Scholar
Kelly, M. and Lack, S. (2004). Monoidal functors generated by adjunctions, with applications to transport of structure. Fields Institute Communications 43 319–340.Google Scholar
Kelly, M. and Street, R. (1974). Review of the elements of 2-categories. In: Proceedings of the Sydney Category Theory Seminar 1972/73. LNM 420, Springer 75–103.Google Scholar
Kock, A. (1970). On double dualization monads. Mathematica Scandinavica 27 151–165.Google Scholar
Leinster, T. (2004). Higher Operads, Higher Categories, London Mathematical Society Lecture Note Series volume 298, Cambridge University Press.Google Scholar
Moggi, E. (1991). Notions of computation and monads. Information and Computation 93 (1) 55–92.Google Scholar
Melliès, P.-A. (2006). A transfer theorem along left adjoints. Unpublished manuscript.Google Scholar
Melliès, P.-A. (2009). Categorical semantics of linear logic. Published in Interactive models of computation and program behaviour. Panoramas et Synthèses 27, Société Mathématique de France.Google Scholar
Melliès, P.-A. (2012). Game semantics in string diagrams. In: Proceedings of the Annual Symposium on Logic in Computer Science.Google Scholar
Melliès, P.-A. (2013a). A micrological study of negation. Submitted. Manuscript available on the author's webpage.Google Scholar
Melliès, P.-A. (2013b). A micrological study of helical negation. Submitted. Manuscript available on the author's webpage.Google Scholar
Melliès, P.-A. (2015). Dialogue categories and chiralities. In: Publications of the Research Institute in Mathematical Sciences. To appear.Google Scholar
Melliès, P.-A. and Tabareau, N. (2009). Resource modalities in tensorial logic. Annals of Pure and Applied Logic 161 (5) 632–653.Google Scholar
Uustalu, T. (2003) Generalizing substitution. Theoretical Informatics and Applications 37 (4) 315–336.Google Scholar