Revisiting the categorical interpretation of dependent type theory

We show that Hofmann's and Curien's interpretations of Martin-Lof's type theory, which were both designed to cure a mismatch between syntax and semantics in Seely's original interpretation in locally cartesian closed categories, are related via a natural isomorphism. As an outcome, we obtain a new proof of the coherence theorem needed to show the soundness after all of Seely's interpretation.

[1]  Bart Jacobs.  Comprehension Categories and the Semantics of Type Dependency , 1993, Theor. Comput. Sci..

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

[3]  G. M. Kelly,et al.  Two-dimensional monad theory , 1989 .

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

[5]  Peter Dybjer,et al.  Internal Type Theory , 1995, TYPES.

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

[7]  E. Van Gestel,et al.  Programming in Martin-Löf's Type Theory: an Introduction : Bengt Nordström, Kent Petersson and Jan M. Smith Intl. Series of Monographs on Computer Science, Vol. 7, Oxford Science Publications, Oxford, 1990, 231 pages , 1991 .

[8]  S. Lane.  Categories for the Working Mathematician , 1971 .

[9]  Bart Jacobs,et al.  Categorical Logic and Type Theory , 2001, Studies in logic and the foundations of mathematics.

[10]  Peter Dybjer,et al.  The Biequivalence of Locally Cartesian Closed Categories and Martin-Löf Type Theories , 2014, Math. Struct. Comput. Sci..

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

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

[13]  Shuanhong Wang,et al.  BRAIDED MONOIDAL CATEGORIES ASSOCIATED TO YETTER-DRINFELD CATEGORIES , 2002 .

[14]  Richard Garner,et al.  The identity type weak factorisation system , 2008, Theor. Comput. Sci..

[15]  Tom Leinster.  Higher Operads, Higher Categories , 2003 .

[16]  M. Warren.  Homotopy Theoretic Aspects of Constructive Type Theory , 2008 .

[17]  Jean Giraud,et al.  Cohomologie non abélienne , 1971 .

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

[19]  Peter Freyd,et al.  Aspects of topoi , 1972, Bulletin of the Australian Mathematical Society.

[20]  Vladimir Voevodsky,et al.  Notes on type systems , 2009 .

[21]  Bart Jacobs,et al.  Categorical type theory , 1991 .

[22]  S. Awodey,et al.  Homotopy theoretic models of identity types , 2007, Mathematical Proceedings of the Cambridge Philosophical Society.

[23]  Ross Street,et al.  Braided Tensor Categories , 1993 .

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