Homotopy Type Theory
References (Rev #58, changes)

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A roughly taxonomised listing of some of the papers on Homotopy Type Theory. Titles link to more details, bibdata, etc. Currently very incomplete; please add!

Surveys

  • Type theory and homotopy. Steve Awodey, 2010. (To appear.) PDF
  • Homotopy type theory and Voevodsky’s univalent foundations. Álvaro Pelayo and Michael A. Warren, 2012. (Bulletin of the AMS, forthcoming) arXiv
  • Voevodsky’s Univalence Axiom in homotopy type theory. Steve Awodey, Álvaro Pelayo, and Michael A. Warren, October 2013, Notices of the American Mathematical Society 60(08), pp.1164-1167. arXiv
  • Homotopy Type Theory: A synthetic approach to higher equalities. Michael Shulman. To appear in Categories for the working philosopher; arXiv
  • Univalent Foundations and the UniMath library. Anthony Bordg, 2017. PDF
  • Homotopy type theory: the logic of space. Michael Shulman. To appear in New Spaces in Mathematics and Physics: arxiv
  • An introduction to univalent foundations for mathematicians. Dan Grayson, arxiv
  • A self-contained, brief and complete formulation of Voevodsky’s Univalence Axiom. Martín Escardó, web, arxiv
  • A proposition is the (homotopy) type of its proofs. Steve Awodey. arxiv, 2017

See also Philosophy, below.

General models

  • The groupoid interpretation of type theory. Thomas Streicher and Martin Hofmann, in Sambin (ed.) et al., Twenty-five years of constructive type theory. Proceedings of a congress, Venice, Italy, October 19?21, 1995. Oxford: Clarendon Press. Oxf. Logic Guides. 36, 83-111 (1998). PostScript
  • Homotopy theoretic models of identity types. Steve Awodey and Michael Warren, Mathematical Proceedings of the Cambridge Philosophical Society, 2009. PDF
  • Homotopy theoretic aspects of constructive type theory. Michael A. Warren, Ph.D. thesis: Carnegie Mellon University, 2008. PDF
  • Two-dimensional models of type theory, Richard Garner, Mathematical Structures in Computer Science 19 (2009), no. 4, pages 687–736. RG’s website
  • Topological and simplicial models of identity types. Richard Garner and Benno van den Berg, to appear in ACM Transactions on Computational Logic (TOCL). PDF
  • The strict ∞-groupoid interpretation of type theory Michael Warren, in Models, Logics and Higher-Dimensional Categories: A Tribute to the Work of Mihály Makkai, AMS/CRM, 2011. PDF
  • Homotopy-Theoretic Models of Type Theory. Peter Arndt and Chris Kapulkin. In Typed Lambda Calculi and Applications, volume 6690 of Lecture Notes in Computer Science, pages 45?60. arXiv
  • Combinatorial realizability models of type theory, Pieter Hofstra and Michael A. Warren, 2013, Annals of Pure and Applied Logic 164(10), pp. 957-988. arXiv
  • Natural models of homotopy type theory, Steve Awodey, 2015. arXiv
  • Subsystems and regular quotients of C-systems, Vladimir Voevodsky, 2014. arXiv
  • C-system of a module over a monad on sets, Vladimir Voevodsky, 2014. arXiv
  • A C-system defined by a universe category, Vladimir Voevodsky, 2014. arXiv
  • B-systems, Vladimir Voevodsky, 2014. arXiv
  • The local universes model: an overlooked coherence construction for dependent type theories, Peter LeFanu Lumsdaine, Michael A. Warren, to appear in ACM Transactions on Computational Logic, 2014. arXiv
  • Products of families of types in the C-systems defined by a universe category, Vladimir Voevodsky, 2015. arXiv
  • Martin-Lof identity types in the C-systems defined by a universe category, Vladimir Voevodsky, 2015. arXiv
  • The Frobenius Condition, Right Properness, and Uniform Fibrations, Nicola Gambino, Christian Sattler. arXiv
  • Fibred fibration categories, Taichi Uemura, 2016, arXiv
  • A homotopy-theoretic model of function extensionality in the effective topos, Daniil Frumin, Benno van den Berg, arxiv
  • Polynomial pseudomonads and dependent type theory, Steve Awodey, Clive Newstead, 2018, arxiv
  • Towards a Topological Model of Homotopy Type Theory, Paige North, doi
  • The Equivalence Extension Property and Model Structures, Christian Sattler, arxiv
  • Univalent polymorphism, Benno van den Berg, arxiv

Univalence

  • The equivalence axiom and univalent models of type theory (talk at CMU, February 4th, 2010), Vladimir Voevodsky. arXiv
  • Univalence in simplicial sets. Chris Kapulkin, Peter LeFanu Lumsdaine, Vladimir Voevodsky. arXiv
  • Univalence for inverse diagrams and homotopy canonicity. Michael Shulman. arXiv
  • Fiber bundles and univalence. Lecture by Ieke Moerdijk at the Lorentz Center, Leiden, December 2011. Lecture notes by Chris Kapulkin
  • A model of type theory in simplicial sets: A brief introduction to Voevodsky?s homotopy type theory. Thomas Streicher, 2011. PDF
  • Univalence and Function Extensionality. Lecture by Nicola Gambino at Oberwohlfach, February 2011. Lecture notes by Chris Kapulkin and Peter Lumsdaine
  • The Simplicial Model of Univalent Foundations. Chris Kapulkin and Peter LeFanu Lumsdaine and Vladimir Voevodsky, 2012. arXiv
  • The univalence axiom for elegant Reedy presheaves. Michael Shulman, arXiv
  • Univalent universes for elegant models of homotopy types. Denis-Charles Cisinski, arXiv
  • A univalent universe in finite order arithmetic. Colin McLarty, arXiv
  • Constructive Simplicial Homotopy. Wouter Pieter Stekelenburg, arXiv
  • Higher Homotopies in a Hierarchy of Univalent Universes. Nicolai Kraus and Christian Sattler, arXiv, DOI
  • Univalence for inverse EI diagrams. Michael shulman, arXiv
  • Univalent completion. Benno van den Berg, Ieke Moerdijk, arXiv
  • On lifting univalence to the equivariant setting. Anthony Bordg, arXiv
  • Univalence in locally cartesian closed \infty-categories. David Gepner and Joachim Kock. Forum Math. 29 (2017), no. 3, 617–652

Inductive and higher-inductive types

  • Inductive Types in Homotopy Type Theory. S. Awodey, N. Gambino, K. Sojakova. To appear in LICS 2012. arXiv
  • W-types in homotopy type theory. Benno van den Berg and Ieke Moerdijk, arXiv
  • Homotopy-initial algebras in type theory Steve Awodey, Nicola Gambino, Kristina Sojakova. arXiv, Coq code
  • The General Universal Property of the Propositional Truncation. Nicolai Kraus, arXiv
  • Non-wellfounded trees in Homotopy Type Theory. Benedikt Ahrens, Paolo Capriotti, Régis Spadotti. TLCA 2015, doi:10.4230/LIPIcs.TLCA.2015.17, arXiv
  • Constructing the Propositional Truncation using Non-recursive HITs. Floris van Doorn, arXiV
  • Constructions with non-recursive higher inductive types. Nicolai Kraus, LiCS 2016, pdf
  • The join construction. Egbert Rijke, arXiv
  • Semantics of higher inductive types. Michael Shulman and Peter LeFanu Lumsdaine, arXiv
  • A Descent Property for the Univalent Foundations, Egbert Rijke, doi
  • Impredicative Encodings of (Higher) Inductive Types. Steve Awodey, Jonas Frey, and Sam Speight. arxiv, 2018
  • W-Types with Reductions and the Small Object Argument, Andrew Swan, arxiv

Formalizations

  • An experimental library of formalized Mathematics based on the univalent foundations, Vladimir Voevodsky, Math. Structures Comput. Sci. 25 (2015), no. 5, pp 1278-1294, 2015. arXiv journal

  • A preliminary univalent formalization of the p-adic numbers. Álvaro Pelayo, Vladimir Voevodsky, Michael A. Warren, 2012. arXiv

  • Univalent categories and the Rezk completion. Benedikt Ahrens, Chris Kapulkin, Michael Shulman, Math. Structures Comput. Sci. 25 (2015), no. 5, 1010?1039. arXiv:1303.0584 (on internal categories in HoTT)

  • The HoTT Library: A formalization of homotopy type theory in Coq, Andrej Bauer, Jason Gross, Peter LeFanu Lumsdaine, Mike Shulman, Matthieu Sozeau, Bas Spitters, 2016 arxiv

Synthetic homotopy theory

  • Calculating the fundamental group of the circle in homotopy type theory. Dan Licata and Michael Shulman, LICS 2013, available here and on arXiv
  • π n(S n)\pi_n(S^n) in Homotopy Type Theory, Dan Licata and Guillaume Brunerie, Invited Paper, CPP 2013, PDF
  • Homotopy limits in type theory. Jeremy Avigad, Chris Kapulkin, Peter LeFanu Lumsdaine, Math. Structures Comput. Sci. 25 (2015), no. 5, 1040?1070. arXiv
  • Eilenberg-MacLane Spaces in Homotopy Type Theory. Dan Licata and Eric Finster, LICS 2014, PDF and code
  • A Cubical Approach to Synthetic Homotopy Theory. Dan Licata and Guillaume Brunerie, LICS 2015, PDF
  • Synthetic Cohomology in Homotopy Type Theory, Evan Cavallo, PDF
  • A mechanization of the Blakers-Massey connectivity theorem in Homotopy Type Theory, LICS 2016 Kuen-Bang Hou (Favonia), Eric Finster, Dan Licata, Peter LeFanu Lumsdaine, arXiv
  • The Seifert-van Kampen Theorem in Homotopy Type Theory, Kuen-Bang Hou and Michael Shulman, PDF
  • On the homotopy groups of spheres in homotopy type theory, Guillaume Brunerie, Ph.D. Thesis, 2016, arxiv
  • The real projective spaces in homotopy type theory, Ulrik Buchholtz and Egbert Rijke, LICS 2017, arxiv
  • Covering Spaces in Homotopy Type Theory, Kuen-Bang Hou (Favonia), doi
  • Higher Groups in Homotopy Type Theory, Ulrik Buchholtz, Floris van Doorn, Egbert Rijke, arXiv:1802.04315

Higher category theory

  • Univalent categories and the Rezk completion. Benedikt Ahrens, Chris Kapulkin, Michael Shulman, Math. Structures Comput. Sci. 25 (2015), no. 5, 1010?1039. arXiv:1303.0584 (on internal categories in HoTT)
  • A type theory for synthetic \infty-categories. Emily Riehl, Michael Shulman. arxiv, 2017
  • Univalent Higher Categories via Complete Semi-Segal Types. Paolo Capriotti, Nicolai Kraus, arxiv, 2017

Homotopical ideas and truncations in type theory

  • Generalizations of Hedberg?s Theorem. Nicolai Kraus, Martín Escardó, Thierry Coquand, and Thorsten Altenkirch.TLCA 2013, pdf
  • Notions of anonymous existence in Martin-Lof type theory. Nicolai Kraus, Martín Escardó, Thierry Coquand, and Thorsten Altenkirch. pdf
  • Idempotents in intensional type theory. Michael Shulman, arXiv
  • Functions out of Higher Truncations. Paolo Capriotti, Nicolai Kraus, and Andrea Vezzosi. CSL 2015 arxiv
  • Truncation levels in homotopy type theory. Nicolai Kraus, PhD Thesis: University of Nottingham, 2015. pdf
  • Parametricity, automorphisms of the universe, and excluded middle. Auke Bart Booij, Martín Hötzel Escardó, Peter LeFanu Lumsdaine, Michael Shulman. arxiv

Applications to computing

  • Homotopical patch theory. Carlo Angiuli, Ed Morehouse, Dan Licata, Robert Harper, PDF
  • Guarded Cubical Type Theory: Path Equality for Guarded Recursion, Lars Birkedal, Ale? Bizjak, Ranald Clouston, Hans Bugge Grathwohl, Bas Spitters, Andrea Vezzosi, arXiv

Cubical models and cubical type theory

  • A Cubical Approach to Synthetic Homotopy Theory. Dan Licata and Guillaume Brunerie, LICS 2015, PDF
  • A syntax for cubical type theory. Thorsten Altenkirch and Ambrus Kaposi, PDF
  • Implementation of Univalence in Cubical Sets, github
  • A Note on the Uniform Kan Condition in Nominal Cubical Sets, Robert Harper and Kuen-Bang Hou. arXiv
  • The Frobenius Condition, Right Properness, and Uniform Fibrations, Nicola Gambino, Christian Sattler. (Note: this is a duplicate of an entry in the section “General Models” above; accident?) arXiv
  • Cubical Type Theory: a constructive interpretation of the univalence axiom, Cyril Cohen, Thierry Coquand, Simon Huber, and Anders Mortberg, arxiv and github implementation
  • Canonicity for cubical type theory, Simon Huber, arxiv.
  • Nominal Presentation of Cubical Sets Models of Type Theory, Andrew M. Pitts, pdf
  • Axioms for Modelling Cubical Type Theory in a Topos, Ian Orton and Andrew M. Pitts, pdf Agda code
  • Computational Higher Type Theory I: Abstract Cubical Realizability, Carlo Angiuli, Robert Harper, Todd Wilson, arxiv, 2016
  • Computational Higher Type Theory II: Dependent Cubical Realizability, Carlo Angiuli, Robert Harper, arxiv, 2016
  • The univalence axiom in cubical sets. Marc Bezem, Thierry Coquand, Simon Huber. arxiv, 2017
  • A Cubical Model of Homotopy Type Theory. Steve Awodey. arxiv, 2016
  • Cartesian Cubical Computational Type Theory, Carlo Angiuli, Favonia, Robert Harper. pdf

Syntax of type theory:

  • The identity type weak factorisation system. Nicola Gambino and Richard Garner, Theoretical Computer Science 409 (2008), no. 3, pages 94?109. RG?s website
  • Types are weak ∞-groupoids. Richard Garner and Benno van den Berg, to appear. RG?s website
  • Weak ∞-Categories from Intensional Type Theory. Peter LeFanu Lumsdaine, TLCA 2009, Brasília, Logical Methods in Computer Science, Vol. 6, issue 23, paper 24. PDF
  • Higher Categories from Type Theories. Peter LeFanu Lumsdaine, PhD Thesis: Carnegie Mellon University, 2010. PDF
  • A coherence theorem for Martin-Löf?s type theory. Michael Hedberg, Journal of Functional Programming 8 (4): 413?436, July 1998.
  • Model Structures from Higher Inductive Types. P. LeFanu Lumsdaine. Unpublished note, Dec. 2011. PDF
  • A category-theoretic version of the identity type weak factorization system. Jacopo Emmenegger, arXiv
  • Locally cartesian closed quasicategories from type theory. Chris Kapulkin, arXiv.
  • Note on the construction of globular weak omega-groupoids from types, topological spaces etc. John Bourke, arXiv
  • The Homotopy Theory of Type Theories. Chris Kapulkin and Peter LeFanu Lumsdaine, arXiv.
  • Polynomial Pseudomonads and Dependent Type Theory. Steve Awodey and Clive Newstead. arxiv, 2018

Strict equality types

  • Homotopy Type System, Vladimir Voevodsky
  • Extending Homotopy Type Theory with Strict Equality, Thorsten Altenkirch, Paolo Capriotti, Nicolai Kraus, CSL 2016, arXiv
  • Two-Level Type Theory and Applications, Danil Annenkov, Paolo Capriotti, Nicolai Kraus, arxiv, 2017

Directed type theory

  • 2-Dimensional Directed Dependent Type Theory. Dan Licata and Robert Harper. MFPS 2011. See also Chapters 7 and 8 of Dan?s thesis. PDF
  • A type theory for synthetic \infty-categories. Emily Riehl, Michael Shulman. arxiv

Cohesion and modalities

  • Quantum gauge field theory in cohesive homotopy type theory. Urs Schreiber and Michael Shulman. arxiv
  • Brouwer’s fixed-point theorem in real-cohesive homotopy type theory. Michael Shulman. To appear in MSCS; arxiv
  • Modalities in homotopy type theory. Egbert Rijke, Michael Shulman, Bas Spitters. arxiv

Theories and models

  • Homotopy Model Theory I: Syntax and Semantics, Dimitris Tsementzis, arXiv

Computational interpretation:

  • Canonicity for 2-Dimensional Type Theory. Dan Licata and Robert Harper. POPL 2012. PDF

Philosophy:

  • Structuralism, Invariance, and Univalence. Steve Awodey. Philosophia Mathematica (2014) 22 (1): 1-11. online
  • Identity in Homotopy Type Theory, Part I: The Justification of Path Induction. James Ladyman and Stuart Presnell. Philosophia Mathematica (2015), online
  • Homotopy Type Theory: A synthetic approach to higher equalities. Michael Shulman. To appear in Categories for the working philosopher; arXiv
  • Univalent Foundations as Structuralist Foundations. Dimitris Tsementzis. Forthcoming in Synthese; Pitt-PhilSci
  • Homotopy type theory: the logic of space. Michael Shulman. To appear in New Spaces in Mathematics and Physics: arxiv

Other:

  • Martin-Löf Complexes. S. Awodey, P. Hofstra and M.A. Warren, 2013, Annals of Pure and Applied Logic, 164(10), pp. 928-956. PDF, arXiv
  • Space-Valued Diagrams, Type-Theoretically (Extended Abstract). Nicolai Kraus and Christian Sattler. arXiv

Revision on September 28, 2018 at 11:34:25 by Richard Williamson?. See the history of this page for a list of all contributions to it.