Data Base Mappings and Theory of Sketches

In this paper we will present the two basic operations for database schemas used in database mapping systems (separation and Data Federation), and we will explain why the functorial semantics for database mappings needed a new base category instead of usual Set category. Successively, it is presented a definition of the graph G for a schema database mapping system, and the definition of its sketch category Sch(G). Based on this framework we presented functorial semantics for database mapping systems with the new base category DB.Comment: 21 page

Data Base Mappings and Monads: (Co)Induction

In this paper we presented the semantics of database mappings in the relational DB category based on the power-view monad T and monadic algebras. The objects in this category are the database-instances (a database-instance is a set of n-ary relations, i.e., a set of relational tables as in standard RDBs). The morphisms in DB category are used in order to express the semantics of view-based Global and Local as View (GLAV) mappings between relational databases, for example those used in Data Integration Systems. Such morphisms in this DB category are not functions but have the complex tree structures based on a set of complex query computations between two database-instances. Thus DB category, as a base category for the semantics of databases and mappings between them, is different from the Set category used dominantly for such issues, and needs the full investigation of its properties. In this paper we presented another contributions for an intensive exploration of properties and semantics of this category, based on the power-view monad T and the Kleisli category for databases. Here we stressed some Universal algebra considerations based on monads and relationships between this DB category and the standard Set category. Finally, we investigated the general algebraic and induction properties for databases in this category, and we defined the initial monadic algebras for database instances.Comment: 31 page

DB Category: Denotational Semantics for View-based Database Mappings

We present a categorical denotational semantics for a database mapping, based on views, in the most general framework of a database integration/exchange. Developed database category DB, for databases (objects) and view-based mappings (morphisms) between them, is different from Set category: the morphisms (based on a set of complex query computations) are not functions, while the objects are database instances (sets of relations). The logic based schema mappings between databases, usually written in a highly expressive logical language (ex. LAV, GAV, GLAV mappings, or tuple generating dependency) may be functorially translated into this "computation" category DB. A new approach is adopted, based on the behavioral point of view for databases, and behavioral equivalences for databases and their mappings are established. By introduction of view-based observations for databases, which are computations without side-effects, we define a fundamental (Universal algebra) monad with a power-view endofunctor T. The resulting 2-category DB is symmetric, so that any mapping can be represented as an object (database instance) as well, where a higher-level mapping between mappings is a 2-cell morphism. Database category DB has the following properties: it is equal to its dual, complete and cocomplete. Special attention is devoted to practical examples: a query definition, a query rewriting in GAV Database-integration environment, and the fixpoint solution of a canonical data integration model.Comment: 40 page

Matching, Merging and Structural Properties of Data Base Category

Main contribution of this paper is an investigation of expressive power of the database category DB. An object in this category is a database-instance (set of n-ary relations). Morphisms are not functions but have complex tree structures based on a set of complex query computations. They express the semantics of view-based mappings between databases. The higher (logical) level scheme mappings between databases, usually written in some high expressive logical language, may be functorially translated into this base "computation" DB category . The behavioral point of view for databases is assumed, with behavioural equivalence of databases corresponding to isomorphism of objects in DB category. The introduced observations, which are view-based computations without side-effects, are based (from Universal algebra) on monad endofunctor T, which is the closure operator for objects and for morphisms also. It was shown that DB is symmetric (with a bijection between arrows and objects) 2-category, equal to its dual, complete and cocomplete. In this paper we demonstrate that DB is concrete, locally small and finitely presentable. Moreover, it is enriched over itself monoidal symmetric category with a tensor products for matching, and has a parameterized merging database operation. We show that it is an algebraic lattice and we define a database metric space and a subobject classifier: thus, DB category is a monoidal elementary topos.Comment: 27 page