Normalization and the Yoneda embedding

  • Authors:

  • Djordje Čubrić;Peter Dybjer;Philip Scott

  • Affiliations:

  • DPMMS, University of Cambridge, 16 Mill Lane, Cambridge CB2 1SB, UK. Email: cubric@triples.math.mcgill.ca;Department of Computing Science, Chalmers University of Technology, S-412 96 Göteborg, Sweden. Email: peterd@cs.chalmers.se;Department of Mathematics, University of Ottawa, 585 King Edward, Ottawa, Ontario K1N 6N5, Canada. Email: phil@mathstat.uottawa.ca

  • Venue:
  • Mathematical Structures in Computer Science
  • Year:
  • 1998

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Abstract

We show how to solve the word problem for simply typed λβη-calculus by using a few well-known facts about categories of presheaves and the Yoneda embedding. The formal setting for these results is 𝒫-category theory, a version of ordinary category theory where each hom-set is equipped with a partial equivalence relation. The part of 𝒫-category theory we develop here is constructive and thus permits extraction of programs from proofs. It is important to stress that in our method we make no use of traditional proof-theoretic or rewriting techniques. To show the robustness of our method, we give an extended treatment for more general λ-theories in the Appendix.