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](https://arxiv.org/abs/1902.06777), 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.



In a locally presentable category, is every object (a retract of) the colimit of a chain of smaller objects?
The last Remark in my joint paper gives a positive answer to the Question.


