Functional Kan Simplicial Sets: Non-Constructivity of Exponentiation

Author Erik Parmann

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Erik Parmann

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Erik Parmann. Functional Kan Simplicial Sets: Non-Constructivity of Exponentiation. In 21st International Conference on Types for Proofs and Programs (TYPES 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 69, pp. 8:1-8:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


Functional Kan simplicial sets are simplicial sets in which the horn-fillers required by the Kan extension condition are given explicitly by functions. We show the non-constructivity of the following basic result: if B and A are functional Kan simplicial sets, then A^B is a Kan simplicial set. This strengthens a similar result for the case of non-functional Kan simplicial sets shown by Bezem, Coquand and Parmann [TLCA 2015, v. 38 of LIPIcs]. Our result shows that-from a constructive point of view-functional Kan simplicial sets are, as it stands, unsatisfactory as a model of even simply typed lambda calculus. Our proof is based on a rather involved Kripke countermodel which has been encoded and verified in the Coq proof assistant.
  • constructive logic
  • simplicial sets
  • semantics of simple types


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