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Let $\mathcal C$ be a locally presentable category. Then by definition, $\mathcal C$ has all small colimits. Nontrivially, we also have

Theorem 1: (Gabriel and Ulmer?) $\mathcal C$ also has all small limits.

I'm wondering if there exists some kind of "relative" version of this theorem. I think this should take the form of some kind of simplified criterion for checking that a functor between locally presentable categories preserves all small limits. But I don't know what the statement of such a criterion should be.

Question: Let $F: \mathcal C \to \mathcal D$ be a functor between locally presentable categories. Does there exist a criterion for checking that $F$ preserves small limits? (perhaps under additional hypotheses on $F$ -- e.g. maybe as a baseline we should assume that $F$ is accessible)

  • Such a criterion should be specific to the locally presentable setting, so "$F$ has a left adjoint" or "$F$ preserves products and equalizers" don't really fit the bill.

  • It might be argued that there is a relative version of Theorem 1 not along the above lines, namely the special adjoint functor theorem for locally presentable categories (which says that $F$ is a right adjoint iff it is accessible and preserves small limits). But I think I might be justified in hoping for some other statement besides this one to be available.

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    $\begingroup$ Have you seen this: arxiv.org/abs/2110.14192? $\endgroup$
    – Ivan Di Liberti
    Commented Nov 5, 2022 at 19:00
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    $\begingroup$ I wonder why the relative version should be about preservation of limits? The existence of limits (of a given shape) is a special case of a right adjoint functor. So the existence of right adjoints would be more natural, right? $\endgroup$
    – Martin Brandenburg
    Commented Nov 5, 2022 at 23:18
  • $\begingroup$ @IvanDiLiberti I had not seen that, thanks! That result sounds like maybe the strongest thing I could imagine being true along these lines, so it's probably the best answer I could hope for... unless there's some sort of simplification if $F$ is assumed to preserve colimits or something like that... $\endgroup$
    – Tim Campion
    Commented Nov 6, 2022 at 16:56
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    $\begingroup$ @MartinBrandenburg You may be right in the end (and I did mention this possibility in my second bullet above -- but you're right to point out that SAFT is directly used in the proof of Theorem 1, making this perspective all the more compelling). Maybe it's a linguistic trap to say "well, this is a statement about limit existence; its 'relative version' should say something about limit-preservation" $\endgroup$
    – Tim Campion
    Commented Nov 6, 2022 at 16:59

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