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.