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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.

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    $\begingroup$ Another good example: completing a dg-category under direct sums, splitting of idempotents, and shifts. An example in the $\infty$-categorical context which I'm not sure has been made precise: completing a spectrally-enriched $\infty$-category under all finite colimits. $\endgroup$ – Tim Campion Dec 22 '17 at 22:47
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Let $V$ be a complete and cocomplete symmetric monoidal closed enriching category. It's a general fact (see Street, Absolute colimits in enriched categories) that the Cauchy completion $\tilde C$ of $C$ is the enriched category of right adjoint bimodules from a point to $C$. That is, an object is an enriched presheaf $R: C^{op} \to V$, an enriched copresheaf $L: C \to V$, a unit $\eta: I \to \int^{c \in C} R(c) \otimes L(c)$, and and counit $\varepsilon: L(c) \otimes R(c') \to C(c,c')$ natural in $c,c'$, satisfying triangle identities.

So the idea is to start with this data, and, assuming that $C$ has all the absolute colimits you know about, show that this adjoint bimodule is representable, i.e. show that there exists $c_\ast \in C$ such that $R \cong C(-,c_\ast)$, $L \cong (c_\ast,-)$, in such a way that $\varepsilon$ becomes composition and $\eta$ becomes the unit map $I \to C(c_\ast,c_\ast)$.

I'm not sure there's anything systematic about how to go about showing this -- it will depend, for example, on the nature of the list of colimits you have to work with -- so maybe it's best just to illustrate with an example. Let's take $V = Ab$. Then, using an explicit description of the coend, we know that $\eta$ must be of the form $\sum_i a_i \otimes b_i$ with $a_i \in R(c_i)$, $b_i \in L(c_i)$ (and the sum is finite). So, assuming we have direct sums, we can just take $\eta$ to live in $R(\oplus_i c_i) \otimes L(\oplus_i c_i)$. Then using $\varepsilon$ we can write down an idempotent on the object $\oplus_i c_i$ which we split to get a representing object for $(R,L,\eta, \varepsilon)$.

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  • $\begingroup$ Another fact that may be useful, especially in the early stages when constructing a list of known absolute colimits, is the fact that if a presheaf $R: C^{op} \to V$ has a left adjoint $L: C \to V$, then $L$ must be the Isbell dual of $R$, and $\varepsilon$ should be the canonical evaluation map. This will cut down the search space substantially; $\eta$, if it exists, will be uniquely determined. $\endgroup$ – Tim Campion Dec 23 '17 at 15:51
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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.

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