Advertisement

Fixpoints revisited

  • J. Lambek
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 363)

Abstract

This is an attempt to update some old results by the author on least fixpoints of endofunctors of categories. It is now assumed that the categories in question are cartesian or bicartesian closed and that the functors can be expressed as polynomials. Moreover, in place of completeness one now requires only a weak kind of product in addition to joint equalizers of families of pairs of arrows. Nonetheless, the actual construction of least fixpoints has remained essentially the same. To find examples of categories with these properties a general method is described for adjoining equalizers to categories without sacrificing other structure.

Keywords

Initial Object Arbitrary Family Admissible Family Polynomial Family Springer LNCS 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. A. Asperti, The internal model of polymorphic lambda calculus, Preprint, Carnegie Mellon 1988.Google Scholar
  2. E. Bainbridge, P. Freyd, A. Scedrov and P. Scott, Functorial polymorphism, in: Logical foundations of functorial programming, G. Huet (ed.), Addison Wesley, Reading, Mass. 1989, 248–260.Google Scholar
  3. A. Carboni, P. Freyd and A. Scedrov, A categorical approach to realizability of polymorphic types, 3rd ACM Symp. Math. Found. Progr. Lang. Semantics, Springer LNCS 298 (1987), 23–42.Google Scholar
  4. P. Freyd, Aspects of topoi, Bull. Austral. Math. Soc. 7 (1972), 1–76, 467–480.Google Scholar
  5. J.Y. Girard, Interprétation fonctionelle et élimination des coupures dans l'arithmétique d'ordre supérieur, Thèse de Doctorat d'État, Paris 1972.Google Scholar
  6. —, The system F of variable types fifteen years later, Theoretical Computer Science 45 (1986), 159–192.Google Scholar
  7. J.W. Gray, The integration of logical and algebraic types, Preprint, University of Illinois at Urbana-Champagne 1989.Google Scholar
  8. F. Lamarche, Modelling polymorphisms with categories, Ph.D. Thesis, McGill University 1988.Google Scholar
  9. J. Lambek, A fixpoint theorem for complete categories, Math. Zeitschr. 103 (1968), 151–161.Google Scholar
  10. —, Subequalizers, Can. Math. Bull. 13 (1970), 337–349.Google Scholar
  11. ........ and P.J. Scott, Introduction to higher order intuitionistic logic, Cambridge studies in advanced mathematics 7, Cambridge University Press 1986.Google Scholar
  12. G. Longo, Some aspects of impredicativity, Notes on Weyl's philosophy of mathematics and today's type theory, Preprint, Carnegie Mellon 1988.Google Scholar
  13. — and E. Moggi, Constructive natural deduction and its “modest” interpretation, in: Semantics of natural and computer languages, J. Meseguer et al (eds.), MIT Press, Cambridge Mass. 1989.Google Scholar
  14. A.M. Pitts, Polymorphism is set theoretic constructively, in: Category theory and computer science, D. Pitt et al. (eds.), Springer LNCS 283 (1987), 12–35.Google Scholar
  15. A. Prouté, Catégories et lambda-calcul, Preprint, Université de Paris VI, 1988.Google Scholar
  16. J.C. Reynolds, Towards a theory of type structure, Colloque sur la programmation, B. Robinet (ed.), Springer LNCS 19 (1974), 408–425.Google Scholar
  17. — and G.D. Plotkin, On functors expressible in the polymorphic typed lambda calculus, in: G. Huet (ed.), Logical foundations of functional programming, Addison-Wesley, Reading Mass. 1989, 118–140.Google Scholar
  18. P.H. Rodenburg and F.J. van der Linden, Manufacturing a cartesian closed category with exactly two objects out of a C-monoid, Studia Logica (1989), to appear.Google Scholar
  19. R.A.G. Seely, Categorical semantics for higher order polymorphic lambda calculus, J. Symbolic Logic 52 (1987), 969–989.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1989

Authors and Affiliations

  • J. Lambek
    • 1
  1. 1.Mathematics DepartmentMcGill UniversityMontrealCanada

Personalised recommendations