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Let $\mathcal{C}$ and $\mathcal{D}$ be pretopoi, and let $f: \mathcal{C} \rightarrow \mathcal{D}$ be a pretopos functor (that is, a functor which preserves finite coproducts, finite limits, and epimorphisms). Let $M( \mathcal{C} )$ be the category of models of $\mathcal{C}$ (that is, pretopos functors from $\mathcal{C}$ to the category of sets) and define $M( \mathcal{D} )$-similarly.

The conceptual completeness theorem of Makkai-Reyes asserts that if $f$ induces an equivalence of categories $M(f): M( \mathcal{D} ) \rightarrow M( \mathcal{C} )$, then $f$ is itself an equivalence of categories.

I am wondering about the following more general situation. Suppose that the functor $M(f)$ is an op-fibration in sets (in other words, that the category $M( \mathcal{D} )$ can be obtained by applying the Grothendieck construction to a functor from $M( \mathcal{C} )$ to the category of sets). I would like to conclude that $\mathcal{D}$ can be obtained as a filtered colimit of pretopoi of the form $\mathcal{C}_{ / C}$ (in other words, that $\mathcal{D}$ is the pretopos associated to a Pro-object of $\mathcal{C}$).

Is something like this true, and/or available in the categorical logic literature?

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    $\begingroup$ Might Breiner's 'Logical schemes' approach (andrew.cmu.edu/user/awodey/students/breiner.pdf) be helpful. He considers conceptual completeness in section 4.4. "We reframe Makkai & Reyes’ conceptual completeness theorem as a theorem about schemes…The theorem follows immediately from our scheme construction." (p. 9). $\endgroup$ – David Corfield Jun 30 '16 at 8:23
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I believe the answer is:

"Yes" for pretopoi associated to classical first-order theories.

"No, but close" in general. (Close means: you cannot conclude that $\mathcal{D}$ is pro-etale over $\mathcal{C}$, but you can "cover" $\mathcal{D}$ by objects which are pro-etale over both.)

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    $\begingroup$ Since you are developing an $\infty$-categorical version of Makkai-Reyes conceptual completeness in A.9 of Spectral Algebraic Geometry, what can be said of the analogue of your question and answer? $\endgroup$ – David Corfield Jun 30 '16 at 11:40
  • $\begingroup$ @David: You can ask the same question for infty-pretopoi. The answer is "yes" when C is initial (which motivated the question). That is, a bounded coherent infty-topos X is profinite iff all maps between geometric morphisms f_*: S -> X are invertible. The answer is "no" in general (the 1-categorical counterexample gives an infty-categorical counterexample), but I think the idea can be salvaged by using a more liberal notion of pro-etale (a similar phenomenon appears in the work of Bhatt-Scholze on pro-etale maps in algebraic geometry). $\endgroup$ – Jacob Lurie Jul 3 '16 at 6:46
  • $\begingroup$ Thanks! As to whether there's a categorical logician who might be able to speak to the original question (and perhaps the infty-analogue), Steve Awodey (supervisor of the Breiner I mentioned above) should be well-placed. There's a 4 page summary of their scheme-theoretic approach here: andrew.cmu.edu/user/sbreiner/documents/…, including suggestions towards non-affine logical schemes and pseudoelementary classes. $\endgroup$ – David Corfield Jul 4 '16 at 13:42
  • $\begingroup$ I'd be curious to see this "more liberal notion of pro-etale map". $\endgroup$ – David Roberts Jul 27 '16 at 14:44

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