In a locally presentable category, is every object (a retract of) the colimit of a chain of smaller objects?

Question

Let $\mathcal C$ be an accessible category. For $C \in \mathcal C$, define the presentability rank $rk(C)$ of $C$ to be the minimal regular $\kappa$ such that $C$ is $\kappa$-presentable. Following Lieberman, Rosicky, and Vasey, say that $C$ is filtrable if it is the colimit of a chain $C = \varinjlim_{\alpha < \lambda} C_\alpha$ of objects $C_\alpha$ of lower presentability rank $rk(C_\alpha) < rk(C)$, and almost filtrable if it is a retract of the colimit $D$ of such a chain such that $rk(D) = rk(C)$.

Question: Let $\mathcal C$ be an accessible category. Under what conditions can we say that every object $C \in \mathcal C$ of sufficiently large presentability rank is almost filtrable? Does it suffice to assume that $\mathcal C$ is locally presentable?

(Of course, if "chain" is replaced with "highly-filtered colimit", then no conditions are necessary.)

In the above-linked preprint are given various conditions for filtrability dependent on $rk(C)$, but they are not really focused on the locally presentable case. In this case,

• $rk(C)$ is always a successor (unless it's $\aleph_0$ or perhaps if it's smaller than the accessibility rank of $\mathcal C$);

• there's a basic argument which shows that if $rk(C)$ is the successor of a regular cardinal, then $C$ is almost filtrable (and the last Remark in the above-linked paper asserts that the retract can be removed with the fat small object argument).

But I'm not sure how to say anything when $rk(C)$ is the successor of a singular cardinal.

Motivation:

It's important to me to be able to handle all $C \in \mathcal C$ of sufficiently large presentability rank, because this opens up the possibility of a new sort of inductive argument in the theory of locally presentable categories: induction on presentability rank using decomposition by chains. This sort of induction should be particularly well-suited to applications related to the small object argument, which interacts well with chains but not with general highy-filtered colimits.

Answer 1

The last Remark in my joint paper gives a positive answer to the Question.

Answer 2

Since this result is derived by Lieberman, Rosicky, and Vasey as a corollary of some more sophisticated constructions with more sophisticated goals, I think it might be worth "compiling out" the proof here. It turns out to be not so bad. Let $\mathcal C$ be a locally $\lambda$-presentable category, and recall the following fact:

For any $C \in \mathcal C$, if $\mathrm{rk}(C) > \lambda$, then $\mathrm{rk}(C) = \kappa^+$ is a successor. (Proof hint: $\kappa$ is the smallest cardinal such that $C$ is a retract of a $\kappa$-sized colimit of $\lambda$-presentable objects.)

Theorem [Lieberman, Rosicky, and Vasey] Let $\mathcal C$ be a locally $\lambda$-presentable category and let $C \in \mathcal C$ with $\mathrm{rk}(C) = \kappa^+ > \lambda$. Then $C$ is almost filtrable.

Proof: Write $C = \varinjlim_{i \in I} C_i$ as a colimit of $\lambda$-presentable objects. Then $C$ is a retract of the colimit of a $\kappa^+$-small subdiagram, so we may assume without loss of generality that the diagram $I$ is of cardinality $\kappa$. We may write $I = \cup_{\alpha < \mathrm{cf}(\kappa)} I_\alpha$ as the union of a $\mathrm{cf}(\kappa)$-sized increasing chain of subdiagrams of cardinality $|I_\alpha| <\kappa$. Setting $C_\alpha = \varinjlim_{i \in I_\alpha} C_i$, we have $C = \varinjlim_{\alpha < \mathrm{cf}(\kappa)} C_\alpha$, yielding the desired filtration.