9
$\begingroup$

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.

$\endgroup$

1 Answer 1

10
$\begingroup$

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.

$\endgroup$
5
  • 2
    $\begingroup$ Thanks! Would you be able to supply a few more details? $\endgroup$ Mar 4, 2019 at 18:47
  • 1
    $\begingroup$ One needs that the pullback functor preserves directed colimits. So, my answer is valid for locally finitely presentable categories only. $\endgroup$ Mar 4, 2019 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. $\endgroup$ Mar 4, 2019 at 22:13
  • 1
    $\begingroup$ Pullbacks preserve directed colimits also in localizations of locally finitely presentable categories. But, in general, I would expect a negative answer. $\endgroup$ Mar 5, 2019 at 7:19
  • 2
    $\begingroup$ 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. $\endgroup$ Mar 5, 2019 at 8:11

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.