Weighted colimits and formal balls in generalized metric spaces

(a) Limits of Cauchy sequences in a (possibly non-symmetric) 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 Lawvere''s (1973) approach. Many of the recently proposed definitions of generalized limit turn out to be theorems from enriched category theory.