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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.)

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  • $\begingroup$ I doubt that Question 2 has an affirmative answer in general. In my head, the kingdom of Benabou cosmoi $(V, \otimes, I)$ is broken into four phyla, according to the answers to the following two questions: a.) Is $(V,\otimes, I)$ cartesian? and b.) Is $Hom(I,-): V \to Set$ conservative? If the answer to (b) is negative (e.g. when $V = (sSet, \times, \Delta[0])$ or $V = ([0,\infty], +, 0)$), then I think that the answer to your Question (2) is probably negative. $\endgroup$ – Tim Campion Jul 27 at 21:12
  • $\begingroup$ @TimCampion Is your second $V$ presentable? $\endgroup$ – Rune Haugseng Jul 28 at 10:37
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    $\begingroup$ @RuneHaugseng Any idempotent complete small category is accessible, and $[0,\infty]$ is cocomplete, so it's locally presentable. $\endgroup$ – Kevin Arlin Jul 28 at 21:46
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    $\begingroup$ @RuneHaugseng No, it's really true. Try letting $\kappa$ be the cardinality of the set of morphisms. $\endgroup$ – Kevin Arlin Jul 28 at 23:14
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    $\begingroup$ It's something like that. If $J\to \mathcal C$ is $\kappa-$filtered where $|\mathcal C|<\kappa$ then there's a subgraph $J'\to J$ with $|J'|<\kappa$ such that the image of $J'$ coincides with the image of $J$, then a cocone over $J'$ is a cocone over $J$, and any diagram containing a cocone over itself has a retract of that cocone as a colimit. $\endgroup$ – Kevin Arlin Jul 29 at 1:35
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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.

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    $\begingroup$ Regarding to Q2, I'm not sure you are fully correct: My understanding is that while there is indeed a mismatch between locally $\lambda$-presentable in the enriched and unenriched case, it is still is the case that if $V$ itself is locally presentable presentable, then local presentability in the enriched case and unenriched case are equivalent, just for possibly different value of $\lambda$. The point is that if $M$ is $V$ enriched and locally presentable, then the tensor $V \otimes M \to M$ is accessible, hence preserve $\lambda$-presentable object for some $\lambda$. $\endgroup$ – Simon Henry yesterday

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