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.