Sheaf-like categories and applications

In this ongoing work we explore the  notion of category of sheaves over a complete Boolean algebra H, Sh(H), though the equivalent description of category of H-valued sets, H-Sets (≃ Sh(H)):  this equivalence is known to extend immediately to complete Heyting algebras. Here we provide further extensions to the  notion of H-valued set to more general algebras (associated to logics) --MV-algebras and BCK-algebras -- and we start to explore some possibilities of applications of this new setting, mainly to Functional Analysis and Continuous Logic