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