Notions of Cauchy completeness for normed categories

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.