Notions of Cauchy completeness for normed categories

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 $[0,\mathrm{\infty }]$ and $\mathsf{S}\mathsf{e}\mathsf{t}$ more generally by the category $\mathsf{S}\mathsf{e}\mathsf{t}\mathrm{/}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\mathrm{/}\mathcal{V}$ 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.