Homotopy Type Theory
References (Rev #3)

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. 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.
  • 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


  • 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
  • A univalent universe in finite order arithmetic. Colin McLarty, 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

Formalizations of set-level mathematics

  • A preliminary univalent formalization of the p-adic numbers. Álvaro Pelayo, Vladimir Voevodsky, Michael A. Warren, 2012. 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
  • Eilenberg-MacLane Spaces in Homotopy Type Theory. Dan Licata and Eric Finster, PDF and code

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

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

Computational interpretation:

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


  • 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

Revision on April 23, 2015 at 03:10:18 by David Roberts. See the history of this page for a list of all contributions to it.