Quantifier completions, choice principles and applications

We give an expositional account of the quantifier completions by using the language of doctrines. This algebraic presentation allows us to exhaustively analyse the behaviour of the existential and universal quantifiers. In particular there are three points we wish to convey: the first is that they preserve the lattice structure of the fibres under opportune hypotheses which turn out to be preserved as well; the second regards the applications, in particular to the dialectica construction; the third is that these free constructions carry on some relevant choice principles.