# Closure of presentable objects under finite limits

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.

• 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. – Mike Shulman Mar 5 '19 at 8:11