Weak Monoidal DB Topos

Abstract

Chapter 9 considers the topological properties of the DB category: the database metric space, its Subobject classifier, and we demonstrate that DB is a weak monoidal topos. It is shown that DB is monoidal biclosed, finitely complete and cocomplete, locally small and locally finitely presentable category with hom-objects (“exponentiations”) and a subobject classifier. It is well known that the intuitionistic logic is a logic of an elementary (standard) topos. However, we obtain that DB is not an elementary, but a weak monoidal topos. We obtain that in the specific case when the universe of database values is a finite set this logic corresponds to the standard propositional logic. This is the case when the database-mapping system is completely specified by the FOL. However, in the case when we deal with incomplete information and hence we obtain the SOtgds with existentially quantified Skolem functions and our universe must include the infinite set of distinct Skolem constants (for recursive schema-mapping or schema integrity constraints), our logic is then an intermediate or superintuitionistic logic in which the weak excluded middle formula ¬ϕ∨¬¬ϕ is valid. Thus, this weak monoidal topos of DB has more theorems than intuitionistic logic but less than the standard propositional logic.