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Decomposing the Univalence Axiom [article]

Ian Orton, Andrew M. Pitts
2018 arXiv   pre-print
We first present some existing work, collected together from various published and unpublished sources; we then present a new decomposition of the univalence axiom into simpler axioms.  ...  This paper investigates Voevodsky's univalence axiom in intensional Martin-L\"of type theory. In particular, it looks at how univalence can be derived from simpler axioms.  ...  Acknowledgements We would like to thank the anonymous referees for their insightful comments which materially improved the quality of the paper.  ... 
arXiv:1712.04890v3 fatcat:ib7kw4i7qjhitfirujvdvwrgx4

Decomposing the Univalence Axiom

Ian Orton, Andrew M. Pitts, Michael Wagner
2018 Types for Proofs and Programs  
We first present some existing work, collected together from various published and unpublished sources; we then present a new decomposition of the univalence axiom into simpler axioms.  ...  This paper investigates Voevodsky's univalence axiom in intensional Martin-Löf type theory. In particular, it looks at how univalence can be derived from simpler axioms.  ...  T Y P E S 2 0 1 7 6:12 Decomposing the Univalence Axiom Proof.  ... 
doi:10.4230/lipics.types.2017.6 dblp:conf/types/OrtonP17 fatcat:ll2jlz2j7jhsfoi5rmi3f624zi

Decomposing the Univalence Axiom [article]

Ian Orton, Andrew Pitts, Apollo-University Of Cambridge Repository, Apollo-University Of Cambridge Repository
2018
We first present some existing work, collected together from various published and unpublished sources; we then present a new decomposition of the univalence axiom into simpler axioms.  ...  This paper investigates Voevodsky's univalence axiom in intensional Martin-Löf type theory. In particular, it looks at how univalence can be derived from simpler axioms.  ...  Acknowledgements We would like to thank the anonymous referees for their insightful comments which materially improved the quality of the paper.  ... 
doi:10.17863/cam.27200 fatcat:urpz3wmlxncsjaflvodl4ajcgi

Voevodsky's Univalence Axiom in homotopy type theory [article]

Steve Awodey, Álvaro Pelayo, Michael A. Warren
2013 arXiv   pre-print
This interpretation has given rise to the univalent foundations program, which is the topic of the current special year at the Institute for Advanced Study.  ...  In this short note we give a glimpse of homotopy type theory, a new field of mathematics at the intersection of algebraic topology and mathematical logic, and we explain Vladimir Voevodsky's univalent  ...  We thank the Institute for Advanced Study for the excellent resources which have been made available to the authors during the preparation of this article.  ... 
arXiv:1302.4731v1 fatcat:jklrherhkne57io4lpjiqmm6d4

Voevodsky's Univalence Axiom in Homotopy Type Theory

Steve Awodey, Álvaro Pelayo, Michael A. Warren
2013 Notices of the American Mathematical Society  
Acknowledgments We thank the Institute for Advanced Study for the excellent resources which have been made available to the authors during the preparation of this article.  ...  Awodey is partly supported by NSF Grant DMS-1001191 and AFOSR Grant 11NL035, and was supported by the Friends of the Institute for Advanced Study and the Charles Simonyi Endowment.  ...  The most pressing of them is perhaps the "constructivity" of the Univalence Axiom itself, conjectured by Voevodsky in [23] .  ... 
doi:10.1090/noti1043 fatcat:s6keb2tyrvapnctl2opc3tpohi

Equivalences for free: univalent parametricity for effective transport

Nicolas Tabareau, Éric Tanter, Matthieu Sozeau
2018 Proceedings of the ACM on Programming Languages  
However, existing proof assistants based on type theory treat this principle as an axiom, and it is not yet clear how to extend them to handle univalence internally.  ...  Homotopy Type Theory promises a unification of the concepts of equality and equivalence in Type Theory, through the introduction of the univalence principle.  ...  ACKNOWLEDGMENTS We thank Pierre-Évariste Dagand, Eric Finster and the anonymous reviewers for useful feedback, as well as Simon Boulier and Gaëtan Gilbert for contributing parts of the Coq development.  ... 
doi:10.1145/3236787 dblp:journals/pacmpl/TabareauTS18 fatcat:5v7ewz4nxrahrbwpu6k3zr37yy

The Marriage of Univalence and Parametricity [article]

Nicolas Tabareau, Éric Tanter, Matthieu Sozeau
2020 arXiv   pre-print
The resulting concept, univalent parametricity, is an heterogeneous extension of parametricity strengthened with univalence that fully realizes programming and proving modulo equivalences.  ...  We illustrate the use of univalent parametricity on several examples, including a recent integration of native integers in Coq.  ...  ACKNOWLEDGMENTS We thank the reviewers of the original ICFP 2018 publication, as well as the ICFP 2018 participants for asking several questions that forced us to better understand the limits of our initial  ... 
arXiv:1909.05027v3 fatcat:bhzydmmuc5euxd2wz4cdnrmymu

The Marriage of Univalence and Parametricity

Nicolas Tabareau, Éric Tanter, Matthieu Sozeau
2021 Journal of the ACM  
In addition to the theory of univalent parametricity, we present a lightweight framework implemented in the Coq proof assistant that allows the user to transparently transfer definitions and theorems for  ...  The resulting concept, called univalent parametricity , is an extension of parametricity strengthened with univalence that fully realizes programming and proving modulo equivalences.  ...  We are thankful as well to Assia Mahboubi for suggesting concrete applications for validating the use of our approach, which likewise pushed us to extend and enhance this work on univalent parametricity  ... 
doi:10.1145/3429979 fatcat:xrcsi5wjnbdunbm4djsqxfdbwq

Parametricity, automorphisms of the universe, and excluded middle [article]

Auke Bart Booij, Martín Hötzel Escardó, Peter LeFanu Lumsdaine, Michael Shulman
2017 arXiv   pre-print
univalence axiom.  ...  We also address the interaction between classical axioms and the existence of automorphisms of a type universe.  ...  extensionality, whereas the type expressing "having a two-sided inverse" may in general have more than one inhabitant, in particular a ecting the consistency of the univalence axiom.  ... 
arXiv:1701.05617v2 fatcat:xsyj6rgrlfhuxkpk3r44ntrlhy

Parametricity, Automorphisms of the Universe, and Excluded Middle

Auke B. Booij, Martín H. Escardó, Peter LeFanu Lumsdaine, Michael Shulman, Michael Wagner
2018 Types for Proofs and Programs  
univalence axiom.  ...  We also address the interaction between classical axioms and the existence of automorphisms of a type universe.  ...  extensionality, whereas the type expressing "having a two-sided inverse" may in general have more than one inhabitant, in particular affecting the consistency of the univalence axiom.  ... 
doi:10.4230/lipics.types.2016.7 dblp:conf/types/BooijELS16 fatcat:cnzbmqba5vek3lybem53ly4t5y

The Univalence Principle [article]

Benedikt Ahrens and Paige Randall North and Michael Shulman and Dimitris Tsementzis
2022 arXiv   pre-print
The Univalence Principle is the statement that equivalent mathematical structures are indistinguishable.  ...  Our work builds on Makkai's First-Order Logic with Dependent Sorts, but is expressed in Voevodsky's Univalent Foundations (UF), extending previous work on the Structure Identity Principle and univalent  ...  We furthermore thank the anonymous referees of the LICS version, and of the present AMS Memoir, for their helpful suggestions.  ... 
arXiv:2102.06275v3 fatcat:owvhzqeqt5bmznwo2ttjfi5pkm

Categorical structures for type theory in univalent foundations [article]

Benedikt Ahrens and Peter LeFanu Lumsdaine and Vladimir Voevodsky
2017 arXiv   pre-print
Specifically, we construct maps between the various types of structures, and show that assuming the Univalence axiom, some of the comparisons are equivalences.  ...  We work throughout in (intensional) dependent type theory; some results, but not all, assume the univalence axiom. All the material of this paper has been formalized in Coq, over the UniMath library.  ...  and the univalence of Set and PreShv(C) rely additionally on the univalence axiom.  ... 
arXiv:1705.04310v1 fatcat:fc3zxn7gsjd2dgbwlchv7il7b4

Categorical Structures for Type Theory in Univalent Foundations

Benedikt Ahrens, Peter LeFanu Lumsdaine, Vladimir Voevodsky, Marc Herbstritt
2017 Annual Conference for Computer Science Logic  
We work throughout in (intensional) dependent type theory; some results, but not all, assume the univalence axiom. All the material of this paper has been formalized in Coq, over the UniMath library.  ...  We study these in the setting of univalent foundations, where the relationships between them can be stated more transparently.  ...  and the univalence of Set and PreShv(C) rely additionally on the univalence axiom.  ... 
doi:10.4230/lipics.csl.2017.8 dblp:conf/csl/AhrensLV17 fatcat:al4c2vv3wfhsnnjspyws6jnjba

Homotopy type theory and Voevodsky's univalent foundations [article]

Álvaro Pelayo, Michael A. Warren
2012 arXiv   pre-print
He has subsequently advocated a program, which he calls univalent foundations, of developing mathematics in the setting of type theory with the Univalence Axiom and possibly other additional axioms motivated  ...  In this direction, Vladimir Voevodsky observed that it is possible to model type theory using simplicial sets and that this model satisfies an additional property, called the Univalence Axiom, which has  ...  We thank Pierre-Louis Curien, Marcelo Fiore, Helmut Hofer and Vladimir Voevodsky for useful discussions on the topic of this paper.  ... 
arXiv:1210.5658v1 fatcat:qb2kukj5cnccxao7xb4alkfav4

The Definitional Side of the Forcing

Guilhem Jaber, Gabriel Lewertowski, Pierre-Marie Pédrot, Matthieu Sozeau, Nicolas Tabareau
2016 Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science - LICS '16  
Being implemented as a Coq plugin, this work gives the possibility to formalize easily consistency results, for instance the consistency of the negation of Voevodsky's univalence axiom.  ...  Focusing on the call-by-name translation, we adapt it to the dependent case and prove that it is compatible with the definitional equality of our system, thus avoiding coherence problems.  ...  univalence axiom.  ... 
doi:10.1145/2933575.2935320 dblp:conf/lics/JaberLPST16 fatcat:nnumvblstngtrbfjrc5t74laxu
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