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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.

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  • $\begingroup$ I should also ask that the chain be 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. $\endgroup$
    – Tim Campion
    Jan 27 at 20:28
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The last Remark in my joint paper gives a positive answer to the Question.

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  • $\begingroup$ Oh I see. I missed that it said "well $\lambda^+$-filtrable". $\endgroup$
    – Tim Campion
    Aug 31 '20 at 17:08
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    $\begingroup$ To spell it out for myself, Remark 8.10 talks about eliminating retracts, but the point is that Corollary 8.9(2) implies a positive answer to the Question. To flesh out the proof a bit, it is deduced from Theorem 8.8(1) with $\mu = \aleph_0$, since $\aleph_0 \triangleleft \lambda$ for all $\lambda$ (even singular $\lambda$ under the definitions in the paper). If $\mathcal C$ is locally $\theta$-presentable, then it is $(\aleph_0, \theta)$-accessible. So we get that $\mathcal C$ is almost $\lambda^+$-filtrable for all $\lambda \geq \theta$. $\endgroup$
    – Tim Campion
    Aug 31 '20 at 17:19
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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.

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    $\begingroup$ Can this be simplified further by using Remark 1.30 in Adamek-Rosicky, that if $\mu\ge\lambda$ then every $\mu$-presentable object is a $\mu$-small colimit of $\lambda$-presentable objects? It seems that one could then express that $\mu$-small diagram as a union of subdiagrams to obtain a suitable chain. In particular, this would seem to eliminate the retract. (Although, the elimination of the retract in AR 1.30 depends on a highly technical argument in Makkai-Pare that I've never really understood.) $\endgroup$ Sep 2 '20 at 0:47
  • $\begingroup$ It just occurred to me that in order to be really useful, one should also ask that the chain be smooth, i.e. that the canonical morphism $\varinjlim_{\beta<\alpha} C_\beta \to C_\alpha$ is an isomorphism for all limit ordinals $\alpha < \kappa^+$. This is indeed the case in the construction discussed here, so long as we take $I_\alpha = \cup_{\beta < \alpha} I_\beta$ for limit ordinals $\alpha < \kappa^+$ (as we may). $\endgroup$
    – Tim Campion
    Jan 27 at 20:21

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