On the Semantics of Coinductive Types in Martin-Löf Type Theory

  • Federico De Marchi
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3629)


There are several approaches to the problem of giving a categorical semantics to Martin-Löf type theory with dependent sums and products and extensional equality types. The most established one relies on the notion of a type-category (or category with attributes) with \({\it \Sigma}\) and \({\it \Pi}\) types. We extend such a semantics by introducing coinductive types both on the syntactic level and in a type-category. Soundness of the semantics is preserved.

As an example of such a category, we prove that the type-category built over a locally cartesian closed category \({\mathcal C}\) admits coinductive types whenever \({\mathcal C}\) has final coalgebras for all polynomial functors.


Type Theory Categorical Semantic Canonical Projection Syntactic Category Coherence Condition 
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  1. 1.
    Abbott, M., Altenkirch, T., Ghani, N.: Representing strictly positive types. Presented at APPSEM annual meeting, invited for submission to Theoretical Computer Science (2004)Google Scholar
  2. 2.
    Aczel, P.: Non-Well-Founded Sets. In: Center for the Study of Language and Information. CSLI Lecture Notes, vol. 14. Stanford University, Stanford (1988)Google Scholar
  3. 3.
    Aczel, P., Adámek, J., Milius, S., Velebil, J.: Infinite trees and completely iterative theories: a coalgebraic view. Theoretical Computer Science 300, 1–45 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Altenkirch, T.: Extensional equality in intensional type theory. In: 14th Symposium on Logic in Computer Science (LICS 1999), pp. 412–421. IEEE, Los Alamitos (1999)Google Scholar
  5. 5.
    Barr, M.: Terminal coalgebras for endofunctors on sets (1999), Available from
  6. 6.
    Bénabou, J.: Fibered categories and the foundations of naive category theory. J. Symbolic Logic 50(1), 10–37 (1985)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Cartmell, J.: Generalised algebraic theories and contextual categories. Ann. Pure Appl. Logic 32(3), 209–243 (1986)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Coquand, T.: Infinite objects in type theory. In: Barendregt, H., Nipkow, T. (eds.) TYPES 1993. LNCS, vol. 806, pp. 62–78. Springer, Heidelberg (1994)Google Scholar
  9. 9.
    Gaspes, V.: Infinite objects in type theory (1997)Google Scholar
  10. 10.
    Hallnäs, L.: On the syntax of infinite objects: an extension of Martin-Löf’s theory of expressions. In: Martin-Löf, P., Mints, G. (eds.) COLOG 1988. LNCS, vol. 417, pp. 94–104. Springer, Heidelberg (1990)Google Scholar
  11. 11.
    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
  12. 12.
    Jacobs, B.: Categorical logic and type theory. Studies in Logic and the Foundations of Mathematics, vol. 141. North-Holland Publishing Co., Amsterdam (1999)zbMATHGoogle Scholar
  13. 13.
    Jacobs, B., Rutten, J.: A tutorial on (co)algebras and (co)induction. Bulletin of the EATCS 62, 222–259 (1996)Google Scholar
  14. 14.
    Lindström, I.: A construction of non-well-founded sets within Martin-Löf’s type theory. Journal of Symbolic Logic 54(1), 57–64 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Martin-Löf, P.: Intuitionistic type theory. Studies in Proof Theory. Lecture Notes, vol. 1. Bibliopolis, Naples (1984)zbMATHGoogle Scholar
  16. 16.
    Martin-Löf, P.: Mathematics of infinity. In: Martin-Löf, P., Mints, G. (eds.) COLOG 1988. LNCS, vol. 417, pp. 146–197. Springer, Heidelberg (1990)Google Scholar
  17. 17.
    Mendler, N.P., Panangaden, P., Constable, R.L.: Infinite objects in type theory. In: Symposium on Logic in Computer Science (LICS 1986), pp. 249–257. IEEE Computer Society Press, Los Alamitos (1986)Google Scholar
  18. 18.
    Nordström, B., Petersson, K., Smith, J.M.: Programming in Martin-Löf’s type theory. International Series of Monographs on Computer Science, vol. 7. The Clarendon Press/Oxford University Press (1990)Google Scholar
  19. 19.
    Pitts, A.M.: Categorical logic. Handbook of logic in computer science, vol. 5, pp. 39–128. Oxford Sci. Publ./Oxford Univ. Press (2000)Google Scholar
  20. 20.
    Seely, R.A.G.: Locally cartesian closed categories and type theory. Math. Proc. Cambridge Philos. Soc. 95(1), 33–48 (1984)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Streicher, T.: Semantics of type theory. Progress in Theoretical Computer Science. Birkhäuser (1991); Correctness, completeness and independence resultsGoogle Scholar
  22. 22.
    Turi, D., Rutten, J.: On the foundations of final coalgebra semantics: non-well-founded sets, partial orders, metric spaces. Mathematical Structures in Computer Science 8(5), 481–540 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    van den Berg, B., De Marchi, F.: Non-well-founded trees in categories, Available online at (submitted)

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© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Federico De Marchi
    • 1
  1. 1.Department of MathematicsUniversity of UtrechtUtrechtThe Netherlands

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