Freely adding one layer of quantifiers to a Boolean doctrine

Abstract

We describe the layer of quantifier alternation depth at most one of the quantifier completion of a Boolean doctrine over a small category. This amounts to a doctrinal version of Herbrand's theorem for formulas with quantifier alternation depth at most one modulo a universal theory. The resulting construction satisfies a universal property that makes it the free QA-one-step Boolean doctrine. To achieve this version of Herbrand's theorem, we characterize, within the doctrinal setting, the classes $A$ of quantifier-free formulas for which there is a model $M$ such that $A$ is precisely the class of formulas whose universal closure is valid in $M$.