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In a locally presentable category $\cal E$, there are arbitrarily large regular cardinals $\lambda$ such that the $\lambda$-presentable (a.k.a. $\lambda$-compact) objects are closed under pullbacks. Namely, the pullback functor ${\cal E}^{(\to\leftarrow)}\to \cal E$ is a right adjoint, hence accesible. Thus it preserves $\lambda$-presentable objects for arbitrarily large $\lambda$, so it's enough to check that the $\lambda$-presentable objects in ${\cal E}^{(\to\leftarrow)}$ are those that are pointwise so in $\cal E$. (A version of this argument is given in this answer in the case of finite products.)

Of course "arbitrarily large" means that for any cardinal $\mu$ there exists a regular cardinal $\lambda>\mu$ with this property. A stronger claim would be that this is true for all sufficiently large regular cardinals $\lambda$, i.e. to reverse the quantifiers and say there exists a $\mu$ such that all regular cardinals $\lambda>\mu$ have this property (that $\lambda$-presentable objects are closed under pullbacks). Is this stronger claim true?

Note that it is certainly not true that all regular cardinals $\lambda$ have this property; counterexamples can be found here.

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The stronger claim is true at least for locally finitely presentable categories; this follows from Proposition 4.3 in https://arxiv.org/pdf/1005.2910.pdf.

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    $\begingroup$ Thanks! Would you be able to supply a few more details? $\endgroup$ – Mike Shulman Mar 4 '19 at 18:47
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    $\begingroup$ One needs that the pullback functor preserves directed colimits. So, my answer is valid for locally finitely presentable categories only. $\endgroup$ – Jiří Rosický Mar 4 '19 at 19:57
  • $\begingroup$ Ah, that makes more sense. That's interesting, but I would really like to know the answer for arbitrary locally presentable categories. $\endgroup$ – Mike Shulman Mar 4 '19 at 22:13
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    $\begingroup$ Pullbacks preserve directed colimits also in localizations of locally finitely presentable categories. But, in general, I would expect a negative answer. $\endgroup$ – Jiří Rosický Mar 5 '19 at 7:19
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    $\begingroup$ I would also expect a negative answer in general. But the first paragraph of the proof of Proposition 6.1.6.7 in Higher Topos Theory appears to claim that it is true even for locally presentable $\infty$-categories, although I don't follow the proof (it seems to have a gap precisely at the place where I would expect a sharp cardinal inequality to occur). So if it does indeed fail in general, it would be nice to have an explicit counterexample. $\endgroup$ – Mike Shulman Mar 5 '19 at 8:11

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