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?
Best Answer
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.)