Scott's Representation Theorem and the Univalent Karoubi Envelope
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909691171307520 |
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| author | van der Leer, Arnoud Wullaert, Kobe Ahrens, Benedikt |
| author_facet | van der Leer, Arnoud Wullaert, Kobe Ahrens, Benedikt |
| contents | Lambek and Scott constructed a correspondence between simply-typed lambda calculi and Cartesian closed categories. Scott's Representation Theorem is a cousin to this result for untyped lambda calculi. It states that every untyped lambda calculus arises from a reflexive object in some category. We present a formalization of Scott's Representation Theorem in univalent foundations, in the (Rocq-)UniMath library. Specifically, we implement two proofs of that theorem, one by Scott and one by Hyland. We also explain the role of the Karoubi envelope -- a categorical construction -- in the proofs and the impact the chosen foundation has on this construction. Finally, we report on some automation we have implemented for the reduction of $λ$-terms. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_22196 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Scott's Representation Theorem and the Univalent Karoubi Envelope van der Leer, Arnoud Wullaert, Kobe Ahrens, Benedikt Logic in Computer Science Category Theory F.3.2; D.2.4 Lambek and Scott constructed a correspondence between simply-typed lambda calculi and Cartesian closed categories. Scott's Representation Theorem is a cousin to this result for untyped lambda calculi. It states that every untyped lambda calculus arises from a reflexive object in some category. We present a formalization of Scott's Representation Theorem in univalent foundations, in the (Rocq-)UniMath library. Specifically, we implement two proofs of that theorem, one by Scott and one by Hyland. We also explain the role of the Karoubi envelope -- a categorical construction -- in the proofs and the impact the chosen foundation has on this construction. Finally, we report on some automation we have implemented for the reduction of $λ$-terms. |
| title | Scott's Representation Theorem and the Univalent Karoubi Envelope |
| topic | Logic in Computer Science Category Theory F.3.2; D.2.4 |
| url | https://arxiv.org/abs/2506.22196 |