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.

smooth, i.e. that the canonical morphism $\varinjlim_{\beta<\alpha} C_\beta \to C_\alpha$ be an isomorphism for all $\alpha < \lambda$, because the small object argument only interacts well with smooth chains. This is indeed the case in the construction discussed in the answers, with just an iota of extra care (see below). And to clarify from the "Motivation": the small object argument does produce an accessible functor and so in some sense interacts well with highly-filtered colimits; it's just a weaker sense than when it comes to smooth chains. $\endgroup$