Cubical Type Theory: A Constructive Interpretation of the Univalence Axiom

This paper presents a type theory in which it is possible to directly manipulate n-dimensional cubes (points, lines, squares, cubes, etc.) based on an interpretation of dependent type theory in a cubical set model. This enables new ways to reason about identity types, for instance, function extensionality is directly provable in the system. Further, Voevodsky's univalence axiom is provable in this system. We also explain an extension with some higher inductive types like the circle and propositional truncation. Finally we provide semantics for this cubical type theory in a constructive meta-theory.

[1]  Andrew Polonsky.  Extensionality of lambda- , 2014, TYPES.

[2]  Thorsten Altenkirch.  Extensional equality in intensional type theory , 1999, Proceedings. 14th Symposium on Logic in Computer Science (Cat. No. PR00158).

[3]  Nicola Gambino,et al.  Uniform Fibrations and the Frobenius Condition , 2015 .

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

[5]  P. J. Higgins,et al.  Nonabelian Algebraic Topology: Filtered Spaces, Crossed Complexes, Cubical Homotopy Groupoids , 2011 .

[6]  Andrew Wakelin Swan.  An algebraic weak factorisation system on 01-substitution sets: a constructive proof , 2016, J. Log. Anal..

[7]  Per Martin-Löf,et al.  An intuitionistic theory of types , 1972 .

[8]  Jean-Philippe Bernardy,et al.  Type-theory in color , 2013, ICFP.

[9]  J. A. Kalman,et al.  Lattices with involution , 1958 .

[10]  D. Scott,et al.  Sheaves and logic , 1979 .

[11]  Peter Aczel,et al.  On Relating Type Theories and Set Theories , 1998, TYPES.

[12]  Denis-Charles Cisinski,et al.  Univalent universes for elegant models of homotopy types , 2014, 1406.0058.

[13]  Andrew M. Pitts.  Nominal Presentation of Cubical Sets Models of Type Theory , 2014, TYPES.

[14]  Thierry Coquand,et al.  A Model of Type Theory in Cubical Sets , 2013, TYPES.

[15]  Thomas Streicher,et al.  Semantics of type theory - correctness, completeness and independence results , 1991, Progress in theoretical computer science.

[16]  andrew. polonsky.  On Extensionality of λ ∗ , 2015 .

[17]  H. Priestley,et al.  Distributive Lattices , 2004 .

[18]  Vladimir Voevodsky.  The equivalence axiom and univalent models of type theory. (Talk at CMU on February 4, 2010) , 2014 .

[19]  Daniel R. Licata,et al.  A Cubical Approach to Synthetic Homotopy Theory , 2015, 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science.

[20]  D. M. Kan,et al.  ABSTRACT HOMOTOPY. , 1955, Proceedings of the National Academy of Sciences of the United States of America.

[21]  P. Martin-Löf.  An Intuitionistic Theory of Types: Predicative Part , 1975 .

[22]  T. Streicher.  Semantics of Type Theory , 1991, Progress in Theoretical Computer Science.

[23]  Andrew M. Pitts,et al.  Nominal Sets: Names and Symmetry in Computer Science , 2013 .

[24]  Peter LeFanu Lumsdaine,et al.  The simplicial model of Univalent Foundations (after Voevodsky) , 2012, Journal of the European Mathematical Society.

[25]  Thierry Coquand,et al.  A Presheaf Model of Parametric Type Theory , 2015, MFPS.

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

[27]  Jaap van Oosten,et al.  The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. http: //homotopytypetheory.org/book, Institute for Advanced Study, 2013, vii + 583 pp , 2014, Bulletin of Symbolic Logic.

[28]  Giovanni Sambin,et al.  Twenty-five years of constructive type theory. , 1998 .