Relational foundations for functorial data migration

Abstract

We study the data transformation capabilities associated with schemas that are presented by directed multigraphs and path equations. Unlike most approaches which treat graph-based schemas as abbreviations for relational schemas, we treat graph-based schemas as categories. A schema S is a finitely-presented category, and the collection of all S-instances forms a category, S-Inst. A functor F between schemas S and T , which can be generated from a visual mapping between graphs, induces three adjoint data migration functors, ΣF : S-Inst → T -Inst, ΠF : S-Inst → T -Inst, and ∆F : T -Inst → S-Inst. We present an algebraic query language FQL based on these functors, prove that FQL is closed under composition, prove that FQL can be implemented with the select-projectproduct-union relational algebra (SPCU) extended with a key-generation operation, and prove that SPCU can be implemented with FQL.

Workshop on Topology and Abstract Algebra for Biomedicine

“…As stated before, AAAT has helped shape current tools and methodologies supporting schemas and ontologies. This can be further enhanced by recent work on Category Theory as applied to databases 8 . These can address important issues on data migration, schema changes, data integrity and normalization, and intelligent query strategies.…”

“…These can address important issues on data migration, schema changes, data integrity and normalization, and intelligent query strategies. Most database operations are some combination of three fundamental operations: project, join, union 8,9 .…”

Deconstructing UML, part 1 :

“…Commutative diagrams make these constraints explicit so that they can be built into database systems directly rather than hand-coded in an ad hoc fashion. See [44] for more details on categorical databases. Now that we have a reasonably complete picture of individual UML classes, we are ready to consider associations, UML relationships which allow us to link different classes to one another.…”

Workshop on applied category theory:

“…One approach, pioneered in EASIK 3 [66], uses categories and functors to represent database schemas, states and queries. The Categorical Query Language [67] builds on this idea, using functorial relationships between schemas to implement data migrations and merges. From a rather different perspective, PySheaf provides a rigorous approach to multi-modal data fusion based on an abstract representation of the overlapping information between different data sources.…”