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.
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References
A. Asperti, The internal model of polymorphic lambda calculus, Preprint, Carnegie Mellon 1988.
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.
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.
P. Freyd, Aspects of topoi, Bull. Austral. Math. Soc. 7 (1972), 1–76, 467–480.
J.Y. Girard, Interprétation fonctionelle et élimination des coupures dans l'arithmétique d'ordre supérieur, Thèse de Doctorat d'État, Paris 1972.
—, The system F of variable types fifteen years later, Theoretical Computer Science 45 (1986), 159–192.
J.W. Gray, The integration of logical and algebraic types, Preprint, University of Illinois at Urbana-Champagne 1989.
F. Lamarche, Modelling polymorphisms with categories, Ph.D. Thesis, McGill University 1988.
J. Lambek, A fixpoint theorem for complete categories, Math. Zeitschr. 103 (1968), 151–161.
—, Subequalizers, Can. Math. Bull. 13 (1970), 337–349.
........ and P.J. Scott, Introduction to higher order intuitionistic logic, Cambridge studies in advanced mathematics 7, Cambridge University Press 1986.
G. Longo, Some aspects of impredicativity, Notes on Weyl's philosophy of mathematics and today's type theory, Preprint, Carnegie Mellon 1988.
— 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.
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.
A. Prouté, Catégories et lambda-calcul, Preprint, Université de Paris VI, 1988.
J.C. Reynolds, Towards a theory of type structure, Colloque sur la programmation, B. Robinet (ed.), Springer LNCS 19 (1974), 408–425.
— 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.
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.
R.A.G. Seely, Categorical semantics for higher order polymorphic lambda calculus, J. Symbolic Logic 52 (1987), 969–989.
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© 1989 Springer-Verlag Berlin Heidelberg
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Lambek, J. (1989). Fixpoints revisited. In: Meyer, A.R., Taitslin, M.A. (eds) Logic at Botik '89. Logic at Botik 1989. Lecture Notes in Computer Science, vol 363. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-51237-3_17
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DOI: https://doi.org/10.1007/3-540-51237-3_17
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