### 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 Freyd-category into a closed Freyd-category. The universal property characterises the embedding as the free completion of the Freyd-category as ca [→, 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.

Original language | English |
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Title of host publication | Typed Lambda Calculi and Applications 6th International Conference, TLCA 2003 Valencia, Spain, June 10–12, 2003 Proceedings |

Place of Publication | Berlin |

Publisher | Springer |

Pages | 301-315 |

Number of pages | 15 |

Volume | 2701 |

DOIs | |

Publication status | Published - 2003 |

### Publication series

Name | Lecture Notes in Computer Science |
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Publisher | Springer |

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## Cite this

Power, J. (2003). A universal embedding for the higher order structure of computational effects. In

*Typed Lambda Calculi and Applications 6th International Conference, TLCA 2003 Valencia, Spain, June 10–12, 2003 Proceedings*(Vol. 2701, pp. 301-315). (Lecture Notes in Computer Science). Springer. https://doi.org/10.1007/3-540-44904-3_21