Non-Constructivity in Kan Simplicial Sets

Authors Marc Bezem, Thierry Coquand, Erik Parmann

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Marc Bezem
Thierry Coquand
Erik Parmann

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Marc Bezem, Thierry Coquand, and Erik Parmann. Non-Constructivity in Kan Simplicial Sets. In 13th International Conference on Typed Lambda Calculi and Applications (TLCA 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 38, pp. 92-106, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


We give an analysis of the non-constructivity of the following basic result: if X and Y are simplicial sets and Y has the Kan extension property, then Y^X also has the Kan extension property. By means of Kripke countermodels we show that even simple consequences of this basic result, such as edge reversal and edge composition, are not constructively provable. We also show that our unprovability argument will have to be refined if one strengthens the usual formulation of the Kan extension property to one with explicit horn-filler operations.
  • Constructive logic
  • simplicial sets
  • semantics of simple types


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