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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!
Type theory and homotopy.Type theory and homotopy.?Steve Awodey, 2010. (To appear.) PDF
Homotopy type theory and Voevodsky’s univalent foundations.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.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.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 mathematiciansAn introduction to univalent foundations for mathematicians?. Dan Grayson, arxiv
A self-contained, brief and complete formulation of Voevodsky’s Univalence AxiomA self-contained, brief and complete formulation of Voevodsky's Univalence Axiom?. Martín Escardó, web, arxiv
A proposition is the (homotopy) type of its proofsA proposition is the (homotopy) type of its proofs?. Steve Awodey. arxiv, 2017
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 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 theoryMichael 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
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
Univalence in locally cartesian closed -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.Inductive Types in Homotopy Type Theory.? S. Awodey, N. Gambino, K. Sojakova. To appear in LICS 2012.S. Awodey, N. Gambino, K. Sojakova. To appear in LICS 2012. arXiv
W-types in homotopy type theoryW-types in homotopy type theory? . Benno van den Berg and Ieke Moerdijk,Benno van den Berg and Ieke Moerdijk?, arXiv
The General Universal Property of the Propositional TruncationThe General Universal Property of the Propositional Truncation? . Nicolai Kraus,Nicolai Kraus, arXiv
Constructing the Propositional Truncation using Non-recursive HITsConstructing the Propositional Truncation using Non-recursive HITs? . Floris van Doorn,Floris van Doorn, arXiV
Constructions with non-recursive higher inductive typesConstructions with non-recursive higher inductive types? . Nicolai Kraus, LiCS 2016,Nicolai Kraus, LiCS 2016, pdf
A Descent Property for the Univalent FoundationsA Descent Property for the Univalent Foundations? , Egbert Rijke,Egbert Rijke, doi
Impredicative Encodings of (Higher) Inductive TypesImpredicative Encodings of (Higher) Inductive Types? . Steve Awodey, Jonas Frey, and Sam Speight.Steve Awodey, Jonas Frey?, and Sam Speight?. arxiv, 2018
W-Types with Reductions and the Small Object ArgumentW-Types with Reductions and the Small Object Argument? , Andrew Swan,Andrew Swan?, arxiv
Formalizations
An experimental library of formalized Mathematics based on the univalent foundations?
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. arXivjournal
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
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
Guarded Cubical Type Theory: Path Equality for Guarded Recursion , Lars Birkedal, Ale? Bizjak, Ranald Clouston, Hans Bugge Grathwohl, Bas Spitters, Andrea Vezzosi,Lars Birkedal?, Ale? Bizjak?, Ranald Clouston?, Hans Bugge Grathwohl?, Bas Spitters, Andrea Vezzosi, arXiv
A syntax for cubical type theoryA syntax for cubical type theory?. Thorsten Altenkirch and Ambrus Kaposi?, PDF
Implementation of Univalence in Cubical SetsImplementation of Univalence in Cubical Sets?, github
A Note on the Uniform Kan Condition in Nominal Cubical SetsA Note on the Uniform Kan Condition in Nominal Cubical Sets?, Robert Harper and Kuen-Bang Hou. arXiv
The Frobenius Condition, Right Properness, and Uniform FibrationsThe 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
A Cubical Model of Homotopy Type TheoryA Cubical Model of Homotopy Type Theory?. Steve Awodey. arxiv, 2016
Cartesian Cubical Computational Type TheoryCartesian Cubical Computational Type Theory?, Carlo Angiuli, [[Favonia?Carlo Angiuli, Favonia, Robert Harper. pdf
Weak ∞-Categories from Intensional Type Theory.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.Higher Categories from Type Theories.?Peter LeFanu Lumsdaine, PhD Thesis: Carnegie Mellon University, 2010. PDF
Model Structures from Higher Inductive Types.Model Structures from Higher Inductive Types.?P. LeFanu Lumsdaine. Unpublished note, Dec. 2011. PDF
A category-theoretic version of the identity type weak factorization systemA category-theoretic version of the identity type weak factorization system?. Jacopo Emmenegger?, arXiv
Locally cartesian closed quasicategories from type theoryLocally cartesian closed quasicategories from type theory?. Chris Kapulkin, arXiv.
Note on the construction of globular weak omega-groupoids from types, topological spaces etcNote on the construction of globular weak omega-groupoids from types, topological spaces etc?. John Bourke?, arXiv
2-Dimensional Directed Dependent 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
Quantum gauge field theory in cohesive homotopy type theoryQuantum gauge field theory in cohesive homotopy type theory?. Urs Schreiber and Michael Shulman. arxiv
Brouwer’s fixed-point theorem in real-cohesive homotopy type theoryBrouwer's fixed-point theorem in real-cohesive homotopy type theory?. Michael Shulman. To appear in MSCS; arxiv
Canonicity for 2-Dimensional Type Theory.Canonicity for 2-Dimensional Type Theory.?Dan Licata and Robert Harper. POPL 2012. PDF
Philosophy: Philosophy
Structuralism, Invariance, and UnivalenceStructuralism, 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 A synthetic approach to higher equalities?. Michael Shulman. To appear in Categories for the working philosopher; arXiv
Univalent Foundations as Structuralist FoundationsUnivalent Foundations as Structuralist Foundations?. Dimitris Tsementzis. Forthcoming in Synthese; Pitt-PhilSci
Homotopy type theory: the logic of space the logic of space?. Michael Shulman. To appear in New Spaces in Mathematics and Physics: arxiv
Other: Other
Martin-Löf Complexes.Martin-Löf Complexes?. S. Awodey, P. Hofstra and M.A. Warren, 2013, Annals of Pure and Applied Logic, 164(10), pp. 928-956.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)Space-Valued Diagrams?, Type-Theoretically (Extended Abstract) . Nicolai Kraus and Christian Sattler.Nicolai Kraus and Christian Sattler?. arXiv