Infinitary generalizations of Deligne's completeness theorem

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Given a regular cardinal κ such that κ<κ=κ, we study a class of toposes with enough points, the κ-separable toposes. These are equivalent to sheaf toposes over a site with κ-small limits that has at most κ many objects and morphisms, the (basis for the) topology being generated by at most κ many covering families, and that satisfy a further exactness property T. We prove that these toposes have enough κ-points, that is, points whose inverse image preserve all κ-small limits. This generalizes the separable toposes of Makkai and Reyes, that are a particular case when κ=ω, when property T is trivially satisfied. This result is essentially a completeness theorem for a certain infinitary logic that we call κ-geometric, where conjunctions of less than κ formulas and existential quantification on less than κ many variables is allowed. We prove that κ-geometric theories have a κ-classifying topos having property T, the universal property being that models of the theory in a Grothendieck topos with property T correspond to κ-geometric morphisms (geometric morphisms the inverse image of which preserves all κ-small limits) into that topos. Moreover, we prove that κ-separable toposes occur as the κ-classifying toposes of κ-geometric theories of at most κ many axioms in canonical form, and that every such κ-classifying topos is κ-separable. Finally, we consider the case when κ is weakly compact and study the κ-classifying topos of a κ-coherent theory (with at most κ many axioms), that is, a theory where only disjunction of less than κ formulas are allowed, obtaining a version of Deligne's theorem for κ-coherent toposes.