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?

absolutelyno competence in category theory, but I wolud start from the reasons in logic to say that compactness and completeness are equivalent, and then translate them in topos language. $\endgroup$Heytingfunctor from the syntactic category of the theory to the category of sets. If I recall correctly there is a topos-theoretic formulation involvingopengeometric morphisms. $\endgroup$2more comments