TITLE: Enriched vs ordinary filtered colimits
QUESTION [19 upvotes]: Filtered categories can be defined as those categories $\mathbf{C}$ such that $\mathbf{C}$-indexed colimits in $\mathrm{Set}$ commute with finite limits.
Similarly, for categories enriched in $\mathbf{V}$ (where the appropriate notion of colimits is colimits weighted by enriched presheaves) one can define a presheaf $W \colon \mathbf{C}^{\mathrm{op}} \rightarrow \mathbf{V}$ to be ($\kappa$-)flat if $W$-weighted colimits in $\mathbf{V}$ commute with finite ($\kappa$-small) limits in $\mathbf{V}$ (for some regular cardinal $\kappa$). Borceux, Quinteiro, and Rosický take this as a starting point to develop a theory of accessible and presentable $\mathbf{V}$-categories in their paper "A theory of enriched sketches".
BQR show that in some ways flat weighted colimits are closely related to ordinary (conical) filtered colimits. For example, they show that if $\mathbf{C}$ has finite ($\kappa$-small) weighted limits, then a presheaf on $\mathbf{C}$ is ($\kappa$-)flat if and only if it is a ($\kappa$-)filtered ordinary colimit of representable presheaves. However, they give a counterexample that shows this need not be true for arbitrary $\mathbf{C}$ - but in this example it is still true that flat presheaves are filtered colimits of absolute colimits of representables.
Question 1: A $\kappa$-filtered ordinary colimit of absolute colimits of representables is always a $\kappa$-flat presheaf. Is anything further known (or expected) about the other direction, i.e. whether every $\kappa$-flat presheaf can be decomposed as such a colimit (or some variant involving two cardinals)?
Let me add a second closely related question that indicates why one might care about the first one. BQR prove that if $\mathbf{M}$ is a presentable $\mathbf{V}$-category then its underlying ordinary category is also presentable.
Question 2: Suppose $\mathbf{M}$ is a cocomplete $\mathbf{V}$-category whose underlying category is presentable. Does this imply that $\mathbf{M}$ is a presentable $\mathbf{V}$-category?
(This would be the case if the two classes of presheaves in the first question coincide.)

REPLY [5 votes]: We deal with Question 1 in this paper, joint with Steve Lack.
In general it is not true that every $\kappa$-flat presheaf lies in the closure of the resperesntables under $\kappa$-filtered colimits and absolute colimits. That's the case for example when $\mathcal V=\mathbf{Set}^G$, for a non-trivial finite group $G$ (see Section 5 of the paper).
However, there are many bases of enrichments for which the equality holds. In fact, whenever $\mathcal V$ is locally $\kappa$-presentable as a closed category and $\mathcal V(I,-)$ is (weakly) cocontinuous and (weakly) strong monoidal, then every $\kappa$-flat presheaf is a $\kappa$-filtered colimit of representables. (In this case absolute colimits reduce to the usual splittings of idempotents.) Therefore the existence and preservation of $\kappa$-flat colimits is equivalent to that of $\kappa$-filtered colimits in any $\mathcal V$-category (for $\mathcal V$ as above). Examples of such bases are the cartesian closed categories $\mathbf{Set}$ of sets, $\mathbf{Cat}$ of small categories, $\mathbf{SSet}$ of simplicial sets, $\mathbf{2}$ of the free-living arrow, $\mathbf{Pos}$ of partially ordered set, etc.
We also investigate the case when $\kappa$-flat colimits are generated by $\kappa$-filtered colimits plus other absolute colimits (such as finite direct sums and copowers by dualizable objects). Examples of $\mathcal V$ for this setting include the monoidal categories $\mathbf{CMon}$ of commutative monoids, $\mathbf{Ab}$ of abelian groups, $R\textit{-}\mathbf{Mod}$ of $R$-modules for a commutativering $R$, and $\mathbf{GAb}$ of graded abelian groups.