Internal Type Theory

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-Lof's intensional intuitionistic type theory. Finally, we discuss the coherence problem for these internal categories with families.

[1]  M. Hofmann.  Extensional concepts in intensional type theory , 1995 .

[2]  R. Pollack.  The Theory of LEGO A Proof Checker for the Extended Calculus of Constructions , 1994 .

[3]  Eike Ritter,et al.  Categorical Abstract Machines for Higher-Order Typed lambda-Calculi , 1994, Theor. Comput. Sci..

[4]  Peter Dybjer,et al.  Extracting a proof of coherence for monoidal categories from a formal proof of normalization for monoids , 1996 .

[5]  Per Martin-Löf,et al.  Constructive mathematics and computer programming , 1984 .

[6]  Pierre-Louis Curien.  Substitution up to Isomorphism , 1993, Fundam. Informaticae.

[7]  Martin Hofmann,et al.  Elimination of Extensionality in Martin-Löf Type Theory , 1994, TYPES.

[8]  Thorsten Altenkirch,et al.  A user's guide to {ALF , 1994 .

[9]  Martin Hofmann,et al.  Syntax and semantics of dependent types , 1997 .

[10]  Jean Benabou,et al.  Fibered categories and the foundations of naive category theory , 1985, Journal of Symbolic Logic.

[11]  Peter Dybjer,et al.  Extracting a Proof of Coherence for Monoidal Categories from a Proof of Normalization for Monoids , 1995, TYPES.

[12]  Viggo Stoltenberg-hansen,et al.  In: Handbook of Logic in Computer Science , 1995 .

[13]  R. Seely,et al.  Locally cartesian closed categories and type theory , 1984, Mathematical Proceedings of the Cambridge Philosophical Society.

[14]  John Cartmell,et al.  Generalised algebraic theories and contextual categories , 1986, Ann. Pure Appl. Log..

[15]  Martin Hofmann,et al.  On the Interpretation of Type Theory in Locally Cartesian Closed Categories , 1994, CSL.

[16]  Thierry Coquand,et al.  Intuitionistic model constructions and normalization proofs , 1997, Mathematical Structures in Computer Science.