Constructive version of Boolean algebra

The notion of overlap algebra introduced by G. Sambin provides a constructive version of complete Boolean algebra. Here we show that his notion of overlap morphism corresponds classically to that of map preserving arbitrary joins. Moreover we prove that the power collection of a set is the free overlap algebra join-generated from the set. Then, we generalize the concept of overlap algebra and overlap morphism in various ways to provide constructive versions of the category of Boolean algebras with maps preserving arbitrary existing joins.