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arXiv · 2506.22196

Scott's Representation Theorem and the Univalent Karoubi Envelope

Arnoud van der Leer·Kobe Wullaert·Benedikt Ahrens
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Abstract

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 $\lambda$-terms.

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Arnoud van der Leer, Kobe Wullaert, Benedikt Ahrens. 2025-06-27. Scott's Representation Theorem and the Univalent Karoubi Envelope. https://arxiv.org/abs/2506.22196

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