Advertisement

Wellfounded Trees and Dependent Polynomial Functors

  • Nicola Gambino
  • Martin Hyland
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3085)

Abstract

We set out to study the consequences of the assumption of types of wellfounded trees in dependent type theories. We do so by investigating the categorical notion of wellfounded tree introduced in [16]. Our main result shows that wellfounded trees allow us to define initial algebras for a wide class of endofunctors on locally cartesian closed categories.

Keywords

Type Theory Natural Transformation Monoidal Category Left Adjoint Forgetful Functor 
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. 1.
    Abbott, M.: Categories of Containers. Ph.D. thesis, University of Leicester (2003)Google Scholar
  2. 2.
    Abbott, M., Altenkirch, T., Ghani, N.: In: Gordon, A.D. (ed.) FOSSACS 2003. LNCS, vol. 2620, pp. 23–38. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  3. 3.
    Aczel, P.: The Type Theoretic Interpretation of Constructive Set Theory: Inductive Definitions. In: Barcan Marcus, R.B., et al. (eds.) Logic, Methodology and Philosophy of Science, VII, North-Holland, Amsterdam (1986)Google Scholar
  4. 4.
    Barr, M., Wells, C.: Toposes, triples and theories. Springer, Heidelberg (1985)zbMATHGoogle Scholar
  5. 5.
    Dybjer, P.: Representing inductively defined sets by wellorderings in Martin-Löf’s type theory. Theoretical Computer Science 176, 329–335 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Ehrhard, T., Regnier, L.: The differential lambda-calculus. Theoretical Computer Science 309, 1–41 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Freyd, P.: Aspects of topoi. Bull. Austral. Math. Soc. 7, 1–76 (1972)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Griffor, E., Rathjen, M.: The Strength of Some Martin-Löf’s Type Theories. Archiv for Mathematical Logic 33, 347–385 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Hofmann, M.: On the interpretation of type theory in locally cartesian closed categories. In: Pacholski, L., Tiuryn, J. (eds.) CSL 1994. LNCS, vol. 933, pp. 427–441. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  10. 10.
    Kelly, G.M., Street, R.: Review of the elements of 2-categories. In: Proc. Sydney Category Theory Seminar 1972/73 LNM 420, pp. 75–103. Springer, Heidelberg (1974)Google Scholar
  11. 11.
    Kelly, G.M.: A unified treatment of transfinite constructions for free algebras, free monoids, colimit, associated sheaves and so on. Bull. Austral. Math. Soc. 22, 1–83 (1980)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Kelly, G.M.: Basic Concepts of Enriched Category Theory. Cambridge University Press, Cambridge (1982)zbMATHGoogle Scholar
  13. 13.
    Mac Lane, S.: Categories for the working mathematician. Springer, Heidelberg (1998)zbMATHGoogle Scholar
  14. 14.
    Maietti, M.E.: The type theory of categorical universes. Ph.D. thesis, Università di Padova (1998)Google Scholar
  15. 15.
    Martin-Löf, P.: Intuitionistic Type Theory. Bibliopolis (1984)Google Scholar
  16. 16.
    Moerdijk, I., Palmgren, E.: Wellfounded trees in categories. Annals of Pure and Applied Logic 104, 189–218 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Nordström, B., Petersson, K., Smith, J.: Programming in Martin-Löf Type Theory. Oxford University Press, Oxford (1990)zbMATHGoogle Scholar
  18. 18.
    Seely, R.A.G.: Locally cartesian closed categories and type theory. Math. Proc. Camb. Phil. Soc. 95, 33–48 (1984)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Street, R.: The formal theory of monads. J. of Pure and Appl. Algebra 2, 149–168 (1972)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Nicola Gambino
    • 1
  • Martin Hyland
    • 1
  1. 1.Department of Pure Mathematics and Mathematical StatisticsUniversity of Cambridge 

Personalised recommendations