Normalization and the Yoneda Embedding

Received We show how to solve the word problem for simply typed-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, in the Appendix we give an extended treatment for more general-theories.