Relating models of impredicative type theories

  • Bart Jacobs
  • Eugenio Moggi
  • Thomas Streicher
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 530)


The object of study of this paper is the categorical semantics of three impredicative type theories, viz. Higher Order λ-calculus , the Calculus of Constructions and Higher Order ML. The latter is particularly interesting because it is a two-level type theory with type dependency at both levels. Having described appropriate categorical structures for these calculi, we establish translations back and forth between all of them. Most of the research in the paper concerns the theory of fibrations and comprehension categories.


Type Theory Natural Transformation Categorical Semantic Category Theory Generic Object 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Cartmell, J. [1978] Generalised Algebraic Theories and Contextual Categories, Ph.D. thesis, Univ. Oxford.Google Scholar
  2. Coquand, T. and Huet G. [1988] The Calculus of Constructions, in: Information and Computation (1988), vol. 73, num. 2/3.Google Scholar
  3. Ehrhard, Th. [1988] A Categorical Semantics of Constructions, in: Logic in Computer Science (Computer Society Press, Washington, 1988) 264–273.Google Scholar
  4. Hyland, J.M.E. and Pitts, A.M. [1989] The Theory of Constructions: categorical semantics and topos-theoretic Models, in: Gray, J. and Scedrov, A., (eds.), Categories in Computer Science and Logic (Contemp. Math. 92, AMS, Providence, 1989).Google Scholar
  5. Jacobs, B.P.F. [1990] Comprehension Categories and the Semantics of Type Dependency, manuscript.Google Scholar
  6. [1991] Categorical Type Theory, Ph.D. thesis, Univ. Nijmegen.Google Scholar
  7. Moggi, E. [1991] A Category Theoretic Account of Program Modules, Math. Struct. in Comp. Sc. (1991), vol. 1.Google Scholar
  8. Pavlović, D. [1990] Predicates and Fibrations, Ph.D. thesis, Univ. Utrecht.Google Scholar
  9. Pitts, A.M. [1989] Categorical Semantics of Dependent Types, Notes of a talk given at SRI Menlo Park and at the Logic Colloquium in Berlin.Google Scholar
  10. Seely, R.A.G. [1984] Locally cartesian closed Categories and Type Theory, Math. Proc. Camb. Phil. Soc.95 33–48.Google Scholar
  11. [1987] Categorical Semantics for higher order Polymorphic Lambda Calculus, J. Symb. Log. 52 969–989.Google Scholar
  12. Streicher, Th. [1989] Correctness and Completeness of a Categorical Semantics of the Calculus of Constructions, Ph.D. thesis, Univ. Passau.Google Scholar
  13. Taylor, P. [1987] Recursive Domains, indexed Categories and Polymorphism, Ph.D. thesis, Univ. Cambridge.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1991

Authors and Affiliations

  • Bart Jacobs
    • 1
  • Eugenio Moggi
    • 2
  • Thomas Streicher
    • 3
  1. 1.Dep. Comp. Sci.NijmegenNL.
  2. 2.Dip. InformaticaGenovaI.
  3. 3.Fak. Math. & Inform.PassauD.

Personalised recommendations