Compactness theorem and topos theory The theory of classifying topoi due to Makkai, Reyes, Hakim, and Grothendieck supplies a bijection between geometric theories (up to Morita equivalence) and Grothendieck topoi, by assigning to each geometric theory $\mathbb T$ the unique Grothendieck topos $\mathcal E$ such that
$$\hom(\mathcal F, \mathcal E)\cong \text{models of $\mathbb T$ in $\mathcal F$}$$
natural in $\mathcal F$. In particular, the points of the Grothendieck topos associated to $\mathbb T$ are precisely the models of $\mathbb T$ in the category of sets.
This correspondence identifies Deligne's theorem (SGA4, Exposé VI, Section 9), which gives a sufficient condition for a topos to have points, with Gödel's completeness theorem from mathematical logic. A great explanation of that phenomenon has been given by Torsten Ekedahl (MO/68335).
Question: Another fundamental theorem from mathematical logic is the compactness theorem. Can the compactness theorem be phrased as a statement about Grothendieck topoi, similar to how Gödel's completeness theorem can be phrased as a statement about Grothendieck topoi?
 A: The following is an analogy with the compactness theorem. "Compactness" in logic refers to the fact that the first-order theory you're working with is finitary. In topos theory, I believe the closest approximation to this idea is restricting to coherent theories (Johnstone in "Topos Theory" calls them finitary theories). On the topos side, this means restricting to coherent toposes.
The localic coherent toposes are those toposes of the form $\mathbf{Sh}(X)$ for $X$ a spectral space. Take a coherent theory with $\mathbf{Sh}(X)$ as classifying topos. Adding axioms changes the classifying topos, more precisely the new classifying topos is a subtopos $\mathbf{Sh}(Y) \subseteq \mathbf{Sh}(X)$ corresponding to a sublocale $Y \subseteq X$. If you added only finitary/coherent axioms, then $\mathbf{Sh}(Y) \subseteq \mathbf{Sh}(X)$ is a coherent subtopos, and these correspond precisely to the subsets $Y \subseteq X$ that are closed sets for the "patch topology".
So adding a sequence of coherent axioms corresponds to a chain of closed sets
$$X \supseteq Y_1 \supseteq Y_2 \supseteq Y_3 \supseteq \dots$$ for the patch topology. The compactness theorem can then be seen as the statement:
$$\forall i \in I,~ Y_i \neq \varnothing ~\Rightarrow~ \bigcap_{i \in \mathbb{N}} Y_i \neq \varnothing$$
This statement follows from the fact that $X$ is compact for the patch topology (by looking at the complements of these closed sets).
So an analogue of the compactness theorem for toposes would be: if $\mathcal{E}$ is a coherent topos, and $\mathcal{E} \supseteq \mathcal{E}_1 \supseteq \mathcal{E}_2 \supseteq \mathcal{E}_3 \supseteq \dots$ is a sequence of non-empty coherent subtoposes, then the intersection is again non-empty coherent. I don't know a proof of this in the non-localic case.
