Homotopy Type Theory References > history (Rev #38, 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

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

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

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

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.

  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

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

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

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

  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)

  arXiv
• 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

  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

  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
• $\pi_n(S^n)$ 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

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

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, pdf and github implementation
• 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

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.

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

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
• 2-Dimensional Directed Dependent Type Theory. Dan Licata and Robert Harper. MFPS 2011. See also Chapters 7 and 8 of Dan?s thesis. PDF
• Extending Homotopy Type Theory with Strict Equality, Thorsten Altenkirch, Paolo Capriotti, Nicolai Kraus, CSL 2016, arXiv
• Space-Valued Diagrams, Type-Theoretically (Extended Abstract). Nicolai Kraus and Christian Sattler. arXiv