2025/10/01 by Hofmann, Dirk, Tholen, Walter
#18B35 #18D20 #18D60 #54A20 #Category Theory (math.CT) #FOS: Mathematics
paper · doi:10.48550/arxiv.2510.00912
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 \(V\), we extend these two characterisations of Lawvere-style completeness to \(V\)-normed categories, thus replacing \([0,∞]\) and \(Set\) more generally by the category \(Set/ /V\) of \(V\)-normed sets. We also establish improvements of recent results regarding the normed convergence of Cauchy sequences in two important \(V\)-normed categories.