TITLE: How to compute (enriched) Cauchy completions?
QUESTION [8 upvotes]: Lawvere famously explained that the following three constructions are all secretly "the same" construction:

Completing an ordinary category by including splittings of all idempotents.
Completing a linear category by including all direct sums and splittings of idempotents.
Completing a metric space by including limits of all Cauchy sequences.

Indeed, for any reasonable type of enriched category, there is an "enriched Cauchy completion". The above examples correspond to categories enriched in $\mathrm{Set}$, in $\mathrm{AbGp}$, and in $\mathbb R_{\geq 0}$.
See Karoubi envelope and Cauchy completion in the nLab.
Suppose I hand you some interesting world in which to enrich categories. What techniques are there to work out the meaning of "Cauchy completion" in that world? By "work out the meaning of," I mean for example the statement that an $\mathrm{AbGp}$-enriched category is (enriched) Cauchy complete iff it contains direct sums and splittings of idempotents.
In my case, I have some known (enriched) absolute limits, and I'm mostly trying to prove that my list is complete — that if I have limits for every entry on my list, then I have all absolute limits.

REPLY [4 votes]: Another approach is to use the fact that absolute weights are small-projective in the enriched functor category, i.e. mapping out of them preserves all colimits.  Now take a weight $W$, assumed to be absolute and hence small-projective, and express it as a canonical weighted colimit of representables $W = \mathrm{colim}^W Y$, with $Y$ the Yoneda embedding.  Mapping out of $W$ preserves this colimit, so $$\hom(W,W) \cong \hom(W,\mathrm{colim}^W Y) \cong \mathrm{colim}^W \hom(W,Y-).$$
In particular, the identity map of $W$ corresponds to a map $I \to \mathrm{colim}^W \hom(W,Y-)$.  If you have an explicit construction of colimits in your enriching category, you can work out what this means, and then use some naturality to express $W$ in terms of your basic colimits.
For instance, in the case $V=\mathrm{Set}$, colimits are quotients of disjoint unions, so this map gives us a particular object $x$ and a map $W \to Y x$, which naturality implies is a section of the coprojection $Y x \to W$.  Hence $W$ is a retract of a representable, so all absolute $\mathrm{Set}$-colimits are generated by splitting idempotents.  Similarly, in the case $V=\mathrm{Ab}$, colimits are quotients of direct sums, so this map gives us a finite number of objects $x_1,\dots,x_n$ and a map $W \to Y x_1 \oplus \cdots\oplus Y x_n$, exhibiting $W$ as a retract of this direct sum.  Hence all absolute $\mathrm{Ab}$-colimits are generated by finite direct sums and splitting idempotents.  And so on.