Weighted colimits and formal balls in generalized metric spaces

1. mits of Cauchy sequences in a (possibly nonsymmetric) metric space are shown to be weighted colimits (a notion introduced by Borceux and Kelly, 1975). As a consequence, further insights from enriched category theory are applicable to the theory of metric spaces, thus continuing Lawvereʹs (1973) approach. Many of the recently proposed definitions of generalized limit turn out to be theorems from enriched category theory. e dual of the space of metrical predicates (‘fuzzy subsets’) of a metric space is shown to contain the collection F of formal balls (Weihrauch and Schreiber, 1981; Edalat and Heckmann, 1996) as a quasi-metric subspace. Formal balls are related to ordinary closed balls by means of the Isbell conjugation. For an ordinary metric space X, the subspace of minimal elements of F is isometric to X by the co-Yoneda embedding.