A Model of Type Theory in Cubical Sets

We present a model of type theory with dependent product, sum, and identity, in cubical sets. We describe a universe and explain how to transform an equivalence between two types into an equality. We also explain how to model propositional truncation and the circle. While not expressed internally in type theory, the model is expressed in a constructive metalogic. Thus it is a step towards a computational interpretation of Voevodsky's Univalence Axiom.

[1]  Andrew M. Pitts.  An Equivalent Presentation of the Bezem-Coquand-Huber Category of Cubical Sets , 2014, ArXiv.

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

[3]  Hongwei Xi,et al.  Proceedings of the 2007 workshop on Programming languages meets program verification , 2007, ICFP 2007.

[4]  Alonzo Church,et al.  A formulation of the simple theory of types , 1940, Journal of Symbolic Logic.

[5]  Alley Stoughton,et al.  Substitution Revisited , 1988, Theor. Comput. Sci..

[6]  W Hurewicz,et al.  ON THE CONCEPT OF FIBER SPACE. , 1955, Proceedings of the National Academy of Sciences of the United States of America.

[7]  Fahd Ali Al-Agl,et al.  Theory and Applications of Categories , 1993 .

[8]  Ronald Brown.  Moore hyperrectangles on a space form a strict cubical omega-category with connections , 2009 .

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

[10]  Robin O. Gandy,et al.  On the axiom of extensionality – Part I , 1956, Journal of Symbolic Logic.

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

[12]  A. Kock.  Monads on symmetric monoidal closed categories , 1970 .

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

[14]  Alexander Katovsky,et al.  Category Theory , 2010, Arch. Formal Proofs.

[15]  J. Lambek,et al.  Introduction to higher order categorical logic , 1986 .

[16]  M. Hofmann,et al.  The groupoid interpretation of type theory , 1998 .

[17]  Vladimir Voevodsky,et al.  Notes on homotopy λ-calculus , 2006 .

[18]  Ross Street,et al.  Cosmoi of internal categories , 1980 .

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

[20]  Bengt Nordström,et al.  Programming in Martin-Löf's Type Theory , 1990 .

[21]  Jean-Pierre Serre,et al.  Homologie Singuliere Des Espaces Fibres , 1951 .

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

[23]  Thierry Coquand,et al.  A Generalization of Takeuti-Gandy Interpretation , 2013 .

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

[25]  Thierry Coquand,et al.  Isomorphism is equality , 2013 .

[26]  ANTONIO A. R. MONTEIRO.  COMBINATORIAL HOMOTOPY THEORY , 2006 .

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

[28]  Thomas Streicher,et al.  A model of type theory in simplicial sets: A brief introduction to Voevodsky's homotopy type theory , 2014, J. Appl. Log..

[29]  G. B. M..  Principia Mathematica , 1911, Nature.

[30]  Thierry Coquand,et al.  A Kripke model for simplicial sets , 2015, Theor. Comput. Sci..

[31]  P. Lumsdaine,et al.  THE SIMPLICIAL MODEL OF UNIVALENT FOUNDATIONS , 2014 .

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

[33]  Jeremy Avigad,et al.  A Machine-Checked Proof of the Odd Order Theorem , 2013, ITP.

[34]  Jon P. May.  Simplicial objects in algebraic topology , 1993 .

[35]  Thomas Nikolaus,et al.  Algebraic models for higher categories , 2010, 1003.1342.

[36]  Steven Awodey,et al.  Homotopy Type Theory , 2015, ICLA.

[37]  Steve Awodey.  Identity Types - Topological and Categorical Structure , .

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

[39]  D. W. Anderson,et al.  Fibrations and geometric realizations , 1978 .

[40]  D. M. Kan,et al.  ABSTRACT HOMOTOPY. III. , 1956 .

[41]  J. Meigs,et al.  WHO Technical Report , 1954, The Yale Journal of Biology and Medicine.

[42]  Marco Grandis,et al.  CUBICAL SETS AND THEIR SITE , 2003 .

[43]  Alexander Grothendieck,et al.  Pursuing Stacks , 2021, 2111.01000.

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

[45]  P. Johnstone.  Sketches of an Elephant: A Topos Theory Compendium Volume 1 , 2002 .

[46]  Andrej Bauer,et al.  Homotopy Type Theory: Univalent Foundations of Mathematics , 2013, ArXiv.

[47]  Andrej Bauer,et al.  Equilogical spaces , 2004, Theor. Comput. Sci..

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

[49]  S. I. Gelʹfand,et al.  Methods of Homological Algebra , 1996 .

[50]  Sally Popkorn,et al.  A Handbook of Categorical Algebra , 2009 .

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

[52]  Andrew M. Pitts,et al.  Categorical logic , 2001, LICS 2001.

[53]  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.

[54]  Richard Garner,et al.  Topological and Simplicial Models of Identity Types , 2012, TOCL.

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

[56]  Christine Paulin-Mohring,et al.  Inductive Definitions in the system Coq - Rules and Properties , 1993, TLCA.

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

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

[59]  Paul G. Goerss,et al.  Simplicial Homotopy Theory , 2009, Modern Birkhäuser Classics.

[60]  Peter T. Johnstone.  Cartesian monads on toposes , 1997 .

[61]  Simon Docherty.  A Model Of Type Theory In Cubical Sets With Connections , 2014 .

[62]  Georges Gonthier,et al.  Formal Proof—The Four- Color Theorem , 2008 .

[63]  E. Bishop.  Foundations of Constructive Analysis , 2012 .

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

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