Hostname: page-component-7d8f8d645b-xs5cw Total loading time: 0 Render date: 2023-05-28T04:24:35.989Z Has data issue: false Feature Flags: { "useRatesEcommerce": true } hasContentIssue false

The biequivalence of locally cartesian closed categories and Martin-Löf type theories

Published online by Cambridge University Press:  29 April 2014

CNRS, ENS de Lyon, Inria, UCBL, Université de Lyon, Laboratoire LIP, 46 Allée d'Italie, 69364 Lyon, France Email:
Department of Computer Science and Engineering, Chalmers University of Technology, S-412 96 Göteborg, Sweden Email:


Seely's paper Locally cartesian closed categories and type theory (Seely 1984) contains a well-known result in categorical type theory: that the category of locally cartesian closed categories is equivalent to the category of Martin-Löf type theories with Π, Σ and extensional identity types. However, Seely's proof relies on the problematic assumption that substitution in types can be interpreted by pullbacks. Here we prove a corrected version of Seely's theorem: that the Bénabou–Hofmann interpretation of Martin-Löf type theory in locally cartesian closed categories yields a biequivalence of 2-categories. To facilitate the technical development, we employ categories with families as a substitute for syntactic Martin-Löf type theories. As a second result, we prove that if we remove Π-types, the resulting categories with families with only Σ and extensional identity types are biequivalent to left exact categories.

Copyright © Cambridge University Press 2014 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)


Abadi, M., Cardelli, L., Curien, P.-L. and Lévy, J.-J. (1990) Explicit substitutions. In: ACM Conference on Principles of Programming Languages, San Francisco.Google Scholar
Bénabou, J. (1985) Fibred categories and the foundation of naive category theory. Journal of Symbolic Logic 50 1037.CrossRefGoogle Scholar
Cartmell, J. (1986) Generalized algebraic theories and contextual categories. Annals of Pure and Applied Logic 32 209243.CrossRefGoogle Scholar
Curien, P.-L. (1993) Substitution up to isomorphism. Fundamenta Informaticae 19 (1–2)5186.Google Scholar
Dybjer, P. (1996) Internal type theory. In: TYPES '95, Types for Proofs and Programs. Springer-Verlag Lecture Notes in Computer Science 1158 120134.CrossRefGoogle Scholar
Hofmann, M. (1994) On the interpretation of type theory in locally cartesian closed categories. In: Pacholski, L. and Tiuryn, J. (eds.) Computer Science Logic – 8th Workshop, CSL. Springer-Verlag Lecture Notes in Computer Science 933 427441.CrossRefGoogle Scholar
Hofmann, M. (1996) Syntax and semantics of dependent types. In: Pitts, A. and Dybjer, P. (eds.) Semantics and Logics of Computation, Cambridge University Press 79130.Google Scholar
Lawvere, F. W. (1970) Equality in hyperdoctrines and comprehension schema as an adjoint functor. In: Heller, A. (ed.) Applications of Categorical Algebra, Proceedings of Symposia in Pure Mathematics, AMS.Google Scholar
Leinster, T. (1999) Basic bicategories. arXiv:math/9810017v1.Google Scholar
Martin-Löf, P. (1975) An intuitionistic theory of types: Predicative part. In: Rose, H. E. and Shepherdson, J. C. (eds.) Logic Colloquium '73, North Holland 73118.Google Scholar
Martin-Löf, P. (1982) Constructive mathematics and computer programming. In: Logic, Methodology and Philosophy of Science, VI, 1979, North-Holland 153175.Google Scholar
Martin-Löf, P. (1984) Intuitionistic Type Theory, Bibliopolis.Google Scholar
Martin-Löf, P. (1986) Amendment to intuitionistic type theory. Notes from a lecture given in Göteborg.Google Scholar
Martin-Löf, P. (1992) Substitution calculus. Notes from a lecture given in Göteborg.Google Scholar
Mimram, S. (2004) Decidability of equality in categories with families. Report, Magistère d'Informatique et Modelisation, École Normale Superieure de Lyon. Available at Scholar
Pitts, A. M. (2000) Categorical logic. In: Abramsky, S., Gabbay, D. M. and Maibaum, T. S. E. (eds.) Handbook of Logic in Computer Science, Volume 5 – Algebraic and Logical Structures, Oxford University Press, Chapter 2, 39128.Google Scholar
Seely, R. (1984) Locally cartesian closed categories and type theory. Mathematical Proceedings of the Cambridge Philosophical Society 95 (1)3348.CrossRefGoogle Scholar
Tasistro, A. (1993) Formulation of Martin-Löf's theory of types with explicit substitutions. Licentiate Thesis. Technical report, Department of Computer Sciences, Chalmers University of Technology and University of Göteborg.Google Scholar
Taylor, P. (1999) Practical Foundations of Mathematics, Cambridge University Press.Google Scholar