Weak ω-Categories from Intensional Type Theory

  • Peter LeFanu Lumsdaine
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5608)


Higher-dimensional categories have recently emerged as a natural context for modelling intensional type theories; this raises the question of what higher-categorical structures the syntax of type theory naturally forms. We show that for any type in Martin-Löf Intensional Type Theory, the system of terms of that type and its higher identity types forms a weak ω-category in the sense of Leinster. Precisely, we construct a contractible globular operad \({P_{\mathit{ML}^{\mathrm{Id}}}}\) of type-theoretically definable composition laws, and give an action of this operad on any type and its identity types.


Type Theory Identity Type Syntactic Category Parallel Pair Weak Factorisation System 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Hofmann, M., Streicher, T.: The groupoid interpretation of type theory. In: Twenty-Five Years of Constructive Type Theory (Venice, 1995). Oxford Logic Guides, vol. 36, pp. 83–111. Oxford Univ. Press, New York (1998)Google Scholar
  2. 2.
    Gambino, N., Garner, R.: The identity type weak factorisation system. Theoretical Computer Science (to appear) (2008) arXiv:0808.2122 Google Scholar
  3. 3.
    Garner, R.: 2-dimensional models of type theory. Mathematical Structures in Computer Science (to appear) (2008) arXiv:0808.2122Google Scholar
  4. 4.
    Awodey, S., Warren, M.A.: Homotopy theoretic models of identity types. Math. Proc. of the Cam. Phil. Soc. (to appear) (2008) arXiv:0709.0248 Google Scholar
  5. 5.
    Leinster, T.: Higher Operads, Higher Categories. London Mathematical Society Lecture Note Series, vol. 298. Cambridge University Press, Cambridge (2004) arXiv:math/0305049 CrossRefzbMATHGoogle Scholar
  6. 6.
    Batanin, M.A.: Monoidal globular categories as a natural environment for the theory of weak n-categories. Adv. Math. 136(1), 39–103 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Lumsdaine, P.Lef.: Weak ω-categories from intensional type theory (extended version) (2008) arXiv:0812.0409 Google Scholar
  8. 8.
    van den Berg, B.: Types as weak ω-categories. Lecture delivered in Uppsala, and unpublished notes (2006)Google Scholar
  9. 9.
    Garner, R., van den Berg, B.: Types are weak ω-groupoids (submitted) (2008) arXiv:0812.0298 Google Scholar
  10. 10.
    Jacobs, B.: Categorical Logic and Type Theory. Studies in Logic and the Foundations of Mathematics, vol. 141. North-Holland Publishing Co., Amsterdam (1999)zbMATHGoogle Scholar
  11. 11.
    Warren, M.A.: Homotopy Theoretic Aspects of Constructive Type Theory. PhD thesis, Carnegie Mellon University (2008)Google Scholar
  12. 12.
    Cartmell, J.: Generalised algebraic theories and contextual categories. Ann. Pure Appl. Logic 32(3), 209–243 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Leinster, T.: A survey of definitions of n-category. Theory Appl. Categ. 10, 1–70 (2002) (electronic) arXiv:math/0107188 MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Street, R.: The petit topos of globular sets. Journal of Pure and Applied Algebra 154, 299–315 (2000)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Peter LeFanu Lumsdaine
    • 1
  1. 1.Carnegie Mellon UniversityUSA

Personalised recommendations