Homotopy Type Theory References > history (Rev #18)

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
• Uniform Fibrations and the Frobenius Condition, Nicola Gambino, Christian Sattler, 2015. 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

Formalizations of set-level mathematics

• 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

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
• 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, PDF and code

Homotopical ideas and truncations in type theory

• 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

Applications to computing

• Homotopical patch theory. Carlo Angiuli, Ed Morehouse, Dan Licata, Robert Harper, PDF

Cubical models and cubical type theory

• A Cubical Approach to Synthetic Homotopy Theory. Dan Licata and Guillaume Brunerie, 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
• Uniform Fibrations and the Frobenius Condition, Nicola Gambino, Christian Sattler. arXiv

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

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

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