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One of the classical theorems of model theory is the Chang-Łos-Suszko preservation theorem that states that the theories formulated in FOL (first order logic) that are preserved under direct limits (generalized unions) are precisely those theories that are axiomatizable by a collection of $\Pi_2$-sentences, i.e., those of the form $\forall x_1 ...\forall x_m \exists y_1...\exists y_n ~\delta$, where $\delta$ is quantifier-free (see, e.g., these slides of Benno van den Berg).

The above motivates the following question concerning inverse limits (of inverse systems). NB: category theorists commonly refer to inverse limits as limits and direct limits as colimits.

Question. Is there a characterization of FOL theories that are preserved under inverse limits?

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    $\begingroup$ There is a gap between "inverse limit" as commonly understood and "limit" as understood by a category theorist. Do you mean specifically limits of inverse systems, or do you also allow products and equalisers? $\endgroup$
    – Zhen Lin
    Commented Jan 27, 2022 at 3:12
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    $\begingroup$ @ZhenLin I meant specifivally limits of inverse systems. I added a parenthetical comments to clarify (thanks for your question). $\endgroup$
    – Ali Enayat
    Commented Jan 27, 2022 at 3:39
  • $\begingroup$ The link to Benno van den Berg slides doesn't work. $\endgroup$
    – Fedor Pakhomov
    Commented Jan 28, 2022 at 11:07
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    $\begingroup$ I do not know the answer to this question, but the notion of locally multipresentable category/ universal AEC is related to this one (it is stronger) and might be a good starting point: arxiv.org/abs/1707.09005 $\endgroup$
    – Ivan Di Liberti
    Commented Jan 28, 2022 at 18:46
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    $\begingroup$ Nice question; but my off-the-cuff impulse is that I wouldn’t expect such a clean characterisation, since limits of inverse systems (if I’m understanding right that that’s what I’d call directed limits) in Set don’t enjoy such strong well-behavedness properties as direct limits of direct systems (i.e. filtered colimits), since Set is locally finitely presentable but not dually so. Or to put it another way: I’d guess other classes of limits might give more natural answers consider than “limits over inverse systems”. $\endgroup$
    – Peter LeFanu Lumsdaine
    Commented Feb 1, 2022 at 0:33

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