# Internal type theory

Conference paper

First Online:

## Abstract

We introduce *categories with families* as a new notion of model for a basic framework of dependent types. This notion is close to ordinary syntax and yet has a clean categorical description. We also present categories with families as a generalized algebraic theory. Then we define categories with families formally in Martin-Löf's intensional intuitionistic type theory. Finally, we discuss the *coherence problem* for these *internal categories with families*.

## Keywords

Type Theory Dependent Type Proof Assistant Coherence Condition Equality Judgement
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## References

- 1.P. Aczel. Galois: a theory development project. A report on work in progress for the Turin meeting on the Representation of Logical Frameworks, 1993.Google Scholar
- 2.T. Altenkirch, V. Gaspes, B. Nordström, and B. von Sydow. A user's guide to ALF. Draft, January 1994.Google Scholar
- 3.J. Bénabou. Fibred categories and the foundation of naive category theory.
*Journal of Symbolic Logic*, 50:10–37, 1985.Google Scholar - 4.I. Beylin and P. Dybjer. Extracting a proof of coherence for monoidal categories from a proof of normalization for monoids. This volume.Google Scholar
- 5.J. Cartmell. Generalized algebraic theories and contextual categories.
*Annals of Pure and Applied Logic*, 32:209–243, 1986.Google Scholar - 6.T. Coquand and P. Dybjer. Intuitionistic model constructions and normalization proofs.
*Mathematical Structures in Computer Science*, 1996. To appear.Google Scholar - 7.P.-L. Curien. Substitution up to isomorphism.
*Fundamenta Informaticae*, 19(1,2):51–86, 1993.Google Scholar - 8.P. Dybjer. Universes and a general notion of simultaneous inductive-recursive definition in type theory. In
*Proceedings of the 1992 Workshop on Types for Proofs and Programs*, 1992.Google Scholar - 9.P. Dybjer and R. Pollack, editors.
*Informal Proceedings of the CLICS-TYPES Workshop on Categories and Type Theory*, Programming Methodology Group, Göteborg University and Chalmers University of Technology, Report 85, 1995.Google Scholar - 10.T. Ehrhard.
*Une sémantique catégorique des types dépendents: Applications au Calcul des Constructions*. PhD thesis, Université Paris VII, 1988.Google Scholar - 11.M. Hofmann. Elimination of extensionality and quotient types in Martin-Löfs type theory. In
*Types for Proofs and Programs, International Workshop TYPES'93, LNCS 806*,1994.Google Scholar - 12.M. Hofmann. Interpretation of type theory in locally cartesian closed categories. In
*Proceedings of CSL*. Springer LNCS, 1994.Google Scholar - 13.M. Hofmann.
*Extensional concepts in intensional type theory*. PhD thesis, University of Edinburgh, 1995.Google Scholar - 14.M. Hofmann. Syntax and semantics of dependent types. In A. Pitts and P. Dybjer, editors,
*Semantics and Logics of Computation*. Cambridge University Press, 1996. To appear.Google Scholar - 15.G. Huet and A. Saibi. Constructive category theory. In
*Proceedings of the Joint CLICS-TYPES Workshop on Categories and Type Theory, Göteborg*, January 1995.Google Scholar - 16.P. Martin-Löf. Constructive mathematics and computer programming. In
*Logic, Methodology and Philosophy of Science, VI, 1979*, pages 153–175. North-Holland, 1982.Google Scholar - 17.P. Martin-Löf. Substitution calculus. Notes from a lecture given in Göteborg, November 1992.Google Scholar
- 18.A. M. Pitts. Categorical logic. In
*Handbook of Logic in Computer Science*. Oxford University Press, 1997. Draft version of article to appear.Google Scholar - 19.R. Pollack.
*The Theory of Lego A Proof Checker for the Extended Calculus of Constructions*. PhD thesis, University of Edinburgh, 1994.Google Scholar - 20.E. Ritter.
*Categorical Abstract Machines for Higher-Order Typed Lambda Calculi*. PhD thesis, Trinity College, Cambridge, September 1992.Google Scholar - 21.R. A. G. Seely. Locally cartesian closed categories and type theory.
*Proceedings of the Cambridge Philosophical Society*, 95:33–48, 1984.Google Scholar

## Copyright information

© Springer-Verlag Berlin Heidelberg 1996