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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.

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    $\begingroup$ 2. The underlying ordinary category of the enriched Cauchy completion is generally not the ordinary Cauchy completion. For example, $\mathbf{Ab}$-enriched Cauchy completion adds finite direct sums and splits idempotents. $\endgroup$ – Zhen Lin Sep 8 '15 at 21:37
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It works the other way around --- you should have tried to answer your question by yourself before you posted it here :-)

  1. 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).

  2. 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.

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  • $\begingroup$ Thanks Michal, honestly I hadn't revisited this topic in a while :). Don't know if this makes sense but: in the case of Lawvere metric spaces I wonder what happens when the $[0, \infty]$-enriched category $X$ has an added algebraic structure? Like when the objects of $X$ form a ring/algebra/etc? How are $X$ and $\overline{X}$ related in those cases? Does the theory of Lawvere metric spaces (which is a more algebraic approach to metric spaces) tell me anything about them? $\endgroup$ – Richard Jennings Sep 10 '15 at 7:26
  • $\begingroup$ Hi Richard, I am not sure if I understand your question --- of course, the notion of Cauchy completion does not depend on the internal structure of objects of a category. $\endgroup$ – Michal R. Przybylek Sep 10 '15 at 22:02
  • $\begingroup$ Thanks again Michal, maybe it's because I haven't delved into them, but it's just that I see Lawvere metric spaces as an algebraic/categorical interpretation of something that is already known in analysis, not necessarily as something that tells me something new about metric spaces, specially about metric spaces with algebraic structures. $\endgroup$ – Richard Jennings Sep 11 '15 at 21:37

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