Enriched Cauchy completions and underlying categories

Question 1

The ordinary Cauchy completion $\overline{C}$ of a small category $C$ satisfies a number of conditions: Every idempotent in $\overline{C}$ splits, there’s an equivalence of categories $[C^{op}, Set] \simeq [\overline{C}^{op}, Set]$ , etc…

There’s also a notion of Cauchy completion for enriched categories, my questions are about it:

1 – Let $X$ be a $V$ -enriched category (where $V$ is a closed symmetric monoidal category with all limits and colimits), what properties does its enriched Cauchy completion $\overline{X}$ satisfy? Like is there an equivalence $[X^{op}, V] \simeq [\overline{X}^{op}, V]$ , etc?

2 – What can be said about the underlying categories of $X$ and $\overline{X}$ ( $X_0$ and $\overline{X}_0$ )?? Is $\overline{X}_0$ the ordinary Cauchy completion of $X$ ? Do we have $[X_0^{op}, Set] \simeq [\overline{X}_0^{op}, Set]$ , etc?

I just wanted to ask before I go about trying to answer this myself.

Question 2

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 $[X^{op}, V] \simeq [\overline{X}^{op}, V]$ . 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 $[0, \infty]$ with monoidal structure $\langle 0, {+}\rangle$ . On the other hand, the underlying category of $X$ is always Cauchy complete in the sense of ordinary categories.

Enriched Cauchy completions and underlying categories

Answer

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