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Let $\mathbb D$ be a sound limit doctrine. When $\mathbb D$ is the doctrine of $\lambda$-small limits, then for any $\mathbb D$-presentable category $\mathscr C$ and $X \in \mathscr C$, the coslice $X/\mathscr C$ is also $\mathbb D$-presentable (Proposition 1.57 of Adámek–Rosicky's Locally presentable and accessible categories). However, this does not hold for general $\mathbb D$ (Remark 83 of Centazzo's Generalised algebraic models shows that this property fails for $\mathbb D$ the empty doctrine).

Unfortunately, the proof in AR makes use of syntactic arguments, which do not have obvious analogues for other notions of accessibility, and so it is unclear in what cases this property may or may not hold in general.

Are there any convenient sufficient or necessary conditions on $\mathbb D$ for $\mathscr C$ being $\mathbb D$-presentable to imply $\mathscr C/X$ is $\mathbb D$-presentable? (In particular that hold for the doctrines of $\lambda$-small limits and of finite products.)

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    $\begingroup$ What does "$\kappa$-sifted" mean? If $I$-colimits commute with $\kappa$-small products in $Set$ and $\kappa > \omega$, then $I$ is $\kappa$-filtered, if I recall correctly. If that's what "$\kappa$-sifted" means, then "$\kappa$-sifted" is only special when $\kappa = \omega$. $\endgroup$
    – Tim Campion
    Commented Jan 27, 2021 at 23:28
  • $\begingroup$ Assuming that "$\omega$-sifted" just means "sifted", the answer in that case should be "yes", just thinking about it in terms of universal algebra -- add some generators and relations to your variety coming from the map out of $X$. But it would be nice to have a conceptual reason which applies to a clear class of sound doctrines. $\endgroup$
    – Tim Campion
    Commented Jan 27, 2021 at 23:31
  • $\begingroup$ @TimCampion: thanks for pointing that out. I suppose I was looking for a more conceptual categorical reason. Perhaps it would be worth instead asking for a way to see that both locally $\kappa$-presentable and locally strongly presentable categories have this coslice property. $\endgroup$
    – varkor
    Commented Jan 27, 2021 at 23:36
  • $\begingroup$ Okay right -- I learned that uncountable-siftedness is the same as uncountable-filteredness from ABLR. The place it's proved is apparently Adamek,Koubek, and Velebil. It's interesting that Centazzo's counterexample is the empty doctrine. I wonder if it might suffice to assume the doctrine is sound and nonempty? (EDIT: Er -- it would have to be a bit more subtle than that, e.g. the doctrine of idempotents will also be a counterexample) $\endgroup$
    – Tim Campion
    Commented Jan 27, 2021 at 23:39
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    $\begingroup$ It may be delicate -- if you wanted to go further and ask about $\mathbb D$-accessibility, then classically taking a slice or coslice can change the index of accessibility, so you don't stay in the same doctrine. So special features of the locally $\mathbb D$-presentable case must be used. $\endgroup$
    – Tim Campion
    Commented Jan 28, 2021 at 0:01

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