The ordinary Cauchy completion ¯C of a small category C satisfies a number of conditions: Every idempotent in ¯C splits, there’s an equivalence of categories [Cop,Set]≃[¯Cop,Set], etc…
There’s also a notion of Cauchy completion for enriched categories, my questions are about it:
1 – Let X be a Venriched category (where V is a closed symmetric monoidal category with all limits and colimits), what properties does its enriched Cauchy completion ¯X satisfy? Like is there an equivalence [Xop,V]≃[¯Xop,V], etc?
2 – What can be said about the underlying categories of X and ¯X (X0 and ¯X0)?? Is ¯X0 the ordinary Cauchy completion of X? Do we have [Xop0,Set]≃[¯Xop0,Set], etc?
I just wanted to ask before I go about trying to answer this myself.
Answer
It works the other way around — you should have tried to answer your question by yourself before you posted it here 🙂

Yes, there is an equivalence [Xop,V]≃[¯Xop,V]. You may find more details in “Basic Concepts of Enriched Category Theory” by M. Kelly (Chapter 5.5).

No. The name “Cauchy completion” has been chosen to suggest that it is a generalization of the usual concept of Cauchy completion for metric spaces. Indeed, if you take a generalized Lawvere metric space X, then the usual completion of X under Cauchy sequences coincide with the Cauchy completion of X, when X is thought of as a category enriched over poset [0,∞] with monoidal structure ⟨0,+⟩. On the other hand, the underlying category of X is always Cauchy complete in the sense of ordinary categories.
Attribution
Source : Link , Question Author : Richard Jennings , Answer Author : Michal R. Przybylek