Infinitary generalizations of Deligne's completeness theorem

Abstract

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.