A Lawvere metric space is Cauchy complete iff it is Cauchy complete regarded as a $[0,\infty]$-enriched category. I have two questions about this observation:
- Is Cauchy completeness (in the setting of enriched category theory) equivalent to the existence of some type of (weighted) limits of colimits?
- Which Lawvere metric spaces are complete (resp. cocomplete) as enriched categories?