Notions of Cauchy completeness for normed categories

Dirk Hofmann; Walter Tholen

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arXiv · 2510.00912

Notions of Cauchy completeness for normed categories

Dirk Hofmann·Walter Tholen

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Abstract

As already mentioned by Lawvere in his 1973 paper, the characterisation of Cauchy completeness of metric spaces in terms of representability of adjoint distributors amounts to the idempotent-split property of an ordinary category when the governing symmetric monoidal-closed category is changed from the extended real half-line to the category of sets. In this paper, for any commutative quantale $\mathcal{V}$, we extend these two characterisations of Lawvere-style completeness to $\mathcal{V}$-normed categories, thus replacing $[0,\infty]$ and $\mathsf{Set}$ more generally by the category $\mathsf{Set}{/\!\!/}\mathcal{V}$ of $\mathcal{V}$-normed sets. We also establish improvements of recent results regarding the normed convergence of Cauchy sequences in two important $\mathcal{V}$-normed categories.

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Dirk Hofmann, Walter Tholen. 2025-10-01. Notions of Cauchy completeness for normed categories. https://arxiv.org/abs/2510.00912

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