Towards a Topological Model of Homotopy Type Theory

  • Paige North
Conference paper
Part of the Trends in Mathematics book series (TM)


The model of homotopy type theory in simplicial sets [7] has proven to be a grounding and motivating influence in the development of homotopy type theory. The classical theory of topological spaces has also proven to be motivational to the subject. Though the Quillen equivalence between simplicial sets and topological spaces provides, in some weak sense, a model in topological spaces, we explore the extent to which the category of topological spaces may be a more direct and strict model of homotopy type theory. We define a notion of model of homotopy type theory, and show that the category of topological spaces fully embeds into such a model.


  1. 1.
    T. Barthel, E. Riehl, On the construction of functorial factorizations for model categories. Algebr. Geom. Topol. 13, 1089–1124 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    P.I. Booth, R. Brown, Spaces of partial maps, fibered mapping spaces and the compact-open topology. Gen. Topol. Appl. 8, 181–195 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    B. Day, Limit Spaces and Closed Span Categories. Lecture Notes in Mathematics, vol. 420 (Springer, Berlin, 1974), pp. 65–74.Google Scholar
  4. 4.
    N. Gambino, R. Garner, The identity type weak factorization system. Theor. Comput. Sci. 409, 94–109 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    J.M.E. Hyland, Filter spaces and continuous functionals. Ann. Math. Logic 16, 101–143 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    A. Joyal, A categorical description of homotopy type theory, in Conference talk, Conference on Type Theory, Homotopy Theory and Univalent Foundations, Barcelona. CRM, 24 Sept 2013Google Scholar
  7. 7.
    K. Kapulkin, P.L. Lumsdaine, V. Voevodsky, The simplicial model of univalent foundations (2012, preprint). arXiv:1211.2851Google Scholar
  8. 8.
    A. Strøm, Note on cofibrations II. Math. Scand. 11, 130–142 (1968)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.DPMMSUniversity of CambridgeCambridgeUK

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