Abstract
This article summarizes mathematical formalisms used to describe object and object-oriented databases. Different approaches were used to describe object databases, object database models respectively.
We focus on algebraic approaches including graphs, monoid comprehensions, monads and semi-monads, and applied type theory. Especially we introduce formal categorical approach to object database modeling.
Several categorical models of object databases are described as well as models of some other formalisms using category theory. We use category theory as an unifying framework for any useful formalism, its model respectively.
Furthermore, category theory is used in order to compare not only different formalisms for object oriented databases but also for different database paradigms comparison, i.e. object, relational, and XML. The vision of developing database framework using formal models and their transformations-based on the MDA concept from OMG-based on category theory is finally suggested.