📝 SummarXiv

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.