A roughly taxonomised listing of some of the papers on Homotopy Type Theory. Titles link to more details, bibdata, etc.


  • 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 ω-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]
  • The univalence axiom for inverse diagrams. 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]
  • A preliminary univalent formalization of the p-adic numbers. Álvaro Pelayo, Vladimir Voevodsky, Michael A. Warren, 2012. [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 ω-groupoids. Richard Garner and Benno van den Berg, to appear. RG’s website.
  • Weak ω-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.

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]
  • Inductive Types in Homotopy Type Theory. S. Awodey, N. Gambino, K. Sojakova. To appear in LICS 2012.  [arXiv]
  • Model Structures from Higher Inductive Types.  P. LeFanu Lumsdaine.  Unpublished note, Dec. 2011. [PDF]
  • 2-Dimensional Directed Dependent Type Theory. Dan Licata and Robert Harper. MFPS 2011. See also Chapters 7 and 8 of Dan’s thesis. [PDF]

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