References (Rev #65, 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
- Synthetic homotopy theory
- Higher category theory
- Homotopical ideas and truncations in type theory
- General models
- Univalence
- Inductive, coinductive, and higher-inductive types
- Formalizations
- Applications to computing
- Cubical models and cubical type theory
- Syntax of type theory
- Strict equality types
- Directed type theory
- Cohesion and modalities
- Theories and models
- Computational interpretation
- Philosophy
- Other

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

See also Philosophy, below.

*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*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*On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory*, Floris van Doorn, Ph.D. Thesis 2018 arXiv:1808.10690*Localization in Homotopy Type Theory*, J. Daniel Christensen, Morgan Opie, Egbert Rijke, Luis Scoccola, arXiv:1807.04155*Cellular Cohomology in Homotopy Type Theory*, Ulrik Buchholtz, Kuen-Bang Hou (Favonia), arXiv:1802.02191*The join construction*. Egbert Rijke, 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*)*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

*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

*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

*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

S. Awodey, N. Gambino, K. Sojakova. To appear in LICS 2012. arXiv~~Inductive Types in Homotopy Type Theory.?~~Inductive Types in Homotopy Type Theory.. Benno van den Berg and Ieke Moerdijk?, arXiv~~W-types in homotopy type theory?~~W-types in homotopy type theorySteve Awodey, Nicola Gambino, Kristina Sojakova. arXiv, Coq code~~Homotopy-initial algebras in type theory?~~Homotopy-initial algebras in type theory. Nicolai Kraus, arXiv~~The General Universal Property of the Propositional Truncation?~~The General Universal Property of the Propositional Truncation. Benedikt Ahrens, Paolo Capriotti, Régis Spadotti?. TLCA 2015, doi:10.4230/LIPIcs.TLCA.2015.17, arXiv~~Non-wellfounded trees in Homotopy Type Theory?~~Non-wellfounded trees in Homotopy Type Theory. Floris van Doorn, arXiV~~Constructing the Propositional Truncation using Non-recursive HITs?~~Constructing the Propositional Truncation using Non-recursive HITs. Nicolai Kraus, LiCS 2016, pdf~~Constructions with non-recursive higher inductive types?~~Constructions with non-recursive higher inductive types.~~The join construction~~Semantics of higher inductive types]]~~Egbert Rijke~~Michael Shulman and [[Peter LeFanu Lumsdaine?, arXiv~~Semantics of higher inductive types?~~A Descent Property for the Univalent Foundations~~.~~,~~Michael~~Egbert~~Shulman~~Rijke~~and~~~~Peter~~doi~~LeFanu~~~~Lumsdaine~~~~,~~~~arXiv~~~~A Descent Property for the Univalent Foundations?~~Impredicative Encodings of (Higher) Inductive Types~~,~~.~~Egbert~~Steve~~Rijke~~Awodey,~~doi~~Jonas Frey?, and Sam Speight?. arxiv, 2018~~Impredicative Encodings of (Higher) Inductive Types?~~W-Types with Reductions and the Small Object Argument~~.~~,~~Steve Awodey~~Andrew Swan?,~~Jonas Frey?~~arxiv~~, and~~~~Sam Speight?~~~~.~~~~arxiv~~~~, 2018~~, Rasmus Ejlers Møgelberg, Niccolò Veltri,~~W-Types with Reductions and the Small Object Argument?~~Bisimulation as path type for guarded recursive types~~Andrew Swan?~~arxiv~~,~~~~arxiv~~,~~Bisimulation~~Signatures~~as~~and~~path~~Induction~~type~~Principles for~~guarded~~Higher~~recursive~~Inductive-Inductive~~types~~Types?~~Rasmus~~~~Ejlers~~~~Møgelberg,~~~~Niccolò~~~~Veltri,~~~~arxiv~~Ambrus Kaposi?, András Kovács? arXiv:1902.00297

*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

*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

. Dan Licata and Guillaume Brunerie, LICS 2015, PDF~~A Cubical Approach to Synthetic Homotopy Theory~~A Cubical Approach to Synthetic Homotopy Theory. Thorsten Altenkirch and Ambrus Kaposi?, PDF~~A syntax for cubical type theory?~~A syntax for cubical type theory, github~~Implementation of Univalence in Cubical Sets?~~Implementation of Univalence in Cubical Sets, Robert Harper and Kuen-Bang Hou. arXiv~~A Note on the Uniform Kan Condition in Nominal Cubical Sets?~~A Note on the Uniform Kan Condition in Nominal Cubical Sets, Nicola Gambino, Christian Sattler?. (Note: this is a duplicate of an entry in the section “General Models” above; accident?) arXiv~~The Frobenius Condition, Right Properness, and Uniform Fibrations?~~The Frobenius Condition, Right Properness, and Uniform Fibrations*Cubical Type Theory: a constructive interpretation of the univalence axiom*, Cyril Cohen?, Thierry Coquand, Simon Huber, and Anders Mortberg, arxiv and github implementation, Simon Huber, arxiv.~~Canonicity for cubical type theory?~~Canonicity for cubical type theory, Andrew M. Pitts, pdf~~Nominal Presentation of Cubical Sets Models of Type Theory?~~Nominal Presentation of Cubical Sets Models of Type Theory]], Ian Orton and Andrew M. Pitts, pdf Agda code~~Axioms for Modelling Cubical Type Theory in a Topos?~~Axioms for Modelling Cubical Type Theory in a Topos*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*Computational Higher Type Theory III: Univalent Universes and Exact Equality*, Carlo Angiuli, Kuen-Bang Hou, Robert Harper, arxiv, 2017*Computational Higher Type Theory IV: Inductive Types*, Evan Cavallo, Robert Harper, arxiv, 2018. Marc Bezem?, Thierry Coquand, Simon Huber. arxiv, 2017~~The univalence axiom in cubical sets?~~The univalence axiom in cubical sets. Steve Awodey. arxiv, 2016~~A Cubical Model of Homotopy Type Theory?~~A Cubical Model of Homotopy Type Theory, Carlo Angiuli, Favonia, Robert Harper. pdf~~Cartesian Cubical Computational Type Theory?~~Cartesian Cubical Computational Type Theory~~On Higher Inductive Types in Cubical Type Theory?~~On Higher Inductive Types in Cubical Type Theory,~~,~~Thierry Coquand, Simon Huber, Andes Mortberg?, arxiv, 2018

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

- 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

*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

*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

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

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

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

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

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Revision on February 14, 2019 at 10:24:12 by Ali Caglayan. See the history of this page for a list of all contributions to it.