A universal embedding for the higher order structure of computational effects

  • Authors:

  • John Power

  • Affiliations:

  • Laboratory for the Foundations of Computer Science, University of Edinburgh, Edinburgh, Scotland, UK

  • Venue:
  • TLCA'03 Proceedings of the 6th international conference on Typed lambda calculi and applications
  • Year:
  • 2003

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Abstract

We give a universal embedding of the semantics for the first order fragment of the computational λ-calculus into a semantics for the whole calculus. In category theoretic terms, which are the terms of the paper, this means we give a universal embedding of every small Freydcategory into a closed Freyd-category. The universal property characterises the embedding as the free completion of the Freyd-category as a [→, Set]-enriched category under conical colimits. This embedding extends the usual Yoneda embedding of a small category with finite products into its free cocompletion, i.e., the usual category theoretic embedding of a model of the first order fragment of the simply typed λ-calculus into a model for the whole calculus, and similarly for the linear λ-calculus. It agrees with an embedding previously given in an ad hoc way without a universal property, so it shows the definitiveness of that construction.