Enriched vs ordinary filtered colimits

Question

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.)

Answer 1

For Q1: something related is dealt with in a context more general than the classical one by Adamek, Borceux, Lack and Rosicky in their paper "A classification of accessible categories". They replace finite or $\kappa$-small limits with an arbitrary class of limits $\mathbb{D}$, and consider a condition which they call soundness, one of whose consequences is a decomposition of every $\mathbb{D}$-flat weight as a suitably "$\mathbb{D}$-filtered" colimit of representables.

This is all in the unenriched context, which is not what you want, but the point is that they make axiomatic assumptions which are more or less exactly what is needed to force the answer to your question 1 to be true. Make of that what you will, but it at least suggests that it's not automatic, and will probably require a bespoke argument in each situation.

For Q2: No. I guess the classical reference is Kelly's "Structures defined by finite limits in the enriched context." If $\mathcal V$ is a symmetric monoidal closed category which is locally $\kappa$-presentable as a closed category (i.e., it is locally $\kappa$-presentable and the $\kappa$-presentable objects are closed under the monoidal structure), then there is a good notion of locally $\lambda$-presentable $\mathcal V$-category: they are precisely the cocomplete $\mathcal V$-categories, whose underlying ordinary categories are locally $\lambda$-presentable, and whose $\lambda$-presentable objects are closed under tensors (=copowers) with $\lambda$-presentable objects of $\mathcal V$. Without this last condition, there is a gap through which to thread a negative answer to your question.

EDIT

I agree with Simon that if $\mathcal{C}$ is a cocomplete $\mathcal{V}$-category whose underlying category is locally presentable, then one can always find some $\kappa$ such that $\mathcal{C}$ is locally presentable as a $\mathcal{V}$-category, meaning that $\mathcal{C}_0$ is locally $\kappa$-presentable and the $\kappa$-presentable objects are closed under tensors by $\kappa$-presentable objects of $\mathcal{V}$.

Here, by saying that $X \in \mathcal C$ is $\kappa$-presentable, I just mean that $\mathcal C(X,\text{-}) \colon \mathcal{C} \rightarrow \mathcal V$ preserves conical filtered colimits (this is Kelly's definition). As Rune says, one could also talk of $X \in \mathcal C$ being $\kappa$-compact, meaning that $\mathcal C(X,\text{-})$ preserves $\kappa$-flat colimits. Since there is in no reason to believe that every $\kappa$-flat weight is a $\kappa$-filtered conical colimit of representables, these two notions will in general be distinct.

However, they coincide when $\mathcal C$ is locally $\kappa$-presentable as a $\mathcal V$-category: so all the $\kappa$-presentable objects are $\kappa$-compact in the enriched sense. This is actually in the BQR paper you cite (Lemma 6.5) and follows from the following fact. Let us write $\mathcal A$ for the essentially small full subcategory of $\kappa$-presentable objects. Clearly $\mathcal A$ has $\kappa$-small colimits, and $\mathcal C$ is the free completion $\kappa\text-\mathbf{Filt}(\mathcal A)$ of $\mathcal A$ under conical $\kappa$-filtered colimits. But in fact, $\mathcal C$ is also the free completion $\kappa\text-\mathbf{Flat}(\mathcal A)$ of $\mathcal A$ under $\kappa$-flat colimits. Given this, a functor out of $\mathcal C$ preserves conical $\kappa$-filtered colimits iff it is the left Kan extension of its own restriction to $\mathcal A$, iff it preserves $\kappa$-flat colimits: in particular, $\kappa$-presentability and $\kappa$-compactness in $\mathcal C$ will coincide.

That $\kappa\text-\mathbf{Flat}(\mathcal A) = \kappa\text-\mathbf{Filt}(\mathcal A)$ is proven in Theorem 6.11 of Kelly's "Structures defined by...", or equally by Prop 4.5 of BQR (as you mention in your question.)

Answer 2

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.