Freely adding one layer of quantifiers to a Boolean doctrine

Abstract: We show how to freely add the first layer of quantifier alternation depth to a Boolean doctrine over a small base category. This amounts to a doctrinal version of Herbrand's theorem for formulas with quantifier alternation depth at most one modulo a universal theory. To achieve this, 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 $M$.