TITLE: Classification of absolute 2-limits?
QUESTION [8 upvotes]: Let $\mathcal V$ be a good enriching category. Recall that an enriched limit weight $\phi: D \to \mathcal V$ is called absolute if $\phi$-weighted limits are preserved by any $\mathcal V$-enriched functor whatsoever. If this is a familiar concept, please jump to my question at the end. But otherwise, I will provide some context. For instance,

Idempotent splitting is absolute for $Set$-enrichment and (hence) for any enriching category;
Finite products are absolute for enrichment in abelian groups, commutative monoids, etc;
Limits of Cauchy sequences are absolute for enrichment in $([0,\infty],\geq,+)$;
$\infty$-categorically, finite limits are absolute for enrichment in spectra;
Eilenberg-Moore objects for idempotent monads are absolute for enrichment in $Cat$.

There are a number of general things to say here:

(Street) A $\mathcal V$-weight $\phi: D \to \mathcal V$ is absolute if and only if $\phi$ has a left adjoint $\bar \phi: D^{op} \to \mathcal V$ in the bicategory of $\mathcal V$-categories and $\mathcal V$-profunctors.
Following Lawvere and the third example above, a $\mathcal V$-category with all absolute $\mathcal V$-limits is called Cauchy complete. The Cauchy completion of a $\mathcal V$-category $C$ is obtained by freely adjoining absolute $\mathcal V$-limits to $C$, and may be constructed as the $\mathcal V$-category of absolute weights on $C$. This construction exhibits the Cauchy complete $\mathcal V$-categories as a reflective subcategory of $\mathcal V\text{-}Cat$.
Thus a $\mathcal V$-weight $\phi: D \to \mathcal V$ is absolute if and only if it lies in the Cauchy completion of $D$.

These general facts are useful in working out explicit characterizations of the absolute weights for particular $\mathcal V$:

An ordinary ($Set$-enriched) category $C$ is Cauchy-complete iff $C$ has all split idempotents; a $Set$-enriched weight $\phi: D \to \mathcal V$ is absolute iff it is a retract of a representable (and in particular an ordinary conical limit is absolute iff the indexing diagram has a cofinal idempotent);
An $Ab$-enriched category $C$ is Cauchy-complete iff $C$ has split idempotents and finite products;
A generalized metric space is Cauchy-complete iff it is Cauchy-complete in the usual sense;
A spectrally-enriched $\infty$-category is Cauchy-complete iff it is stable with split idempotents;

But I don't know an analogous statement for $Cat$-enrichment.

Question: When does a 2-category $C$ have all absolute weighted 2-limits?

Clearly $C$ must have split idempotents and Eilenberg-Moore objects for idempotent monads -- do these suffice? Or are there more absolute 2-limit weights out there which can't be built from these?

REPLY [4 votes]: A 2-category is Cauchy complete (in the sense you describe) if and only if idempotents splits, which is, if and only if its underlying ordinary category is Cauchy complete.
This holds more generally in this context: the base of enrichement $\mathcal V=(\mathcal V_0,\otimes,I)$ is symmetryc monoidal closed and locally presentable, and the functor $\mathcal V_0(I,-):\mathcal V_0\to\mathbf{Set}$ is (weakly) cocontinuous and (weakly) strong monoidal. (weakly here means that the induced comparison maps need not be bijections, but just surjections.) In this case a $\mathcal V$-category is Cauchy complete if and only if it has splittings of idempotents. This is proven for example in Corollary 3.16 here.
The idea is that, given an absolute $\mathcal V$-weight $\phi\colon\mathcal D\to\mathcal V$, then the category of elements $\mathcal E$ of $\mathcal V_0(I,\phi_0-)\colon \mathcal D_0\to\mathbf{Set}$ is absolute (in the ordinary sense, i.e. $\mathcal E$-colimits are preserved by any functor) and that $\phi$ can be written as a conical colimit
$$ \phi\cong \textit{colim}(\mathcal{E}_{\mathcal V}\to\mathcal D^{op}\to[\mathcal D,\mathcal V] )$$
indexed on $\mathcal E$.
Examples of such bases are of course $\mathbf{Cat}$ and $\mathbf{Set}$, but also $\mathbf{SSet}$, $\mathbf{Pos}$, and $\mathcal V\textit{-}\mathbf{Cat}$ whenever $\mathcal V$ is locally presentable.