The relational data model uses set theory to provide a formal background, thus ensuring a rigorous mathematical data model with support for manipulation. Newer generation database models are based on the object-oriented paradigm, and so fall short of having such a formal background, especially in some of the more complex data manipulation areas. We use category theory to provide a formalism for object databases, in particular the object-relational model. Our model is known as the Product Model. This thesis will describe our formal model for the key aspects of object databases. In particular, we will examine how the Product Model deals with three of the most important problems inherent in object databases, those of queries, closure and views. As well as this, we investigate the more common database concepts, such as keys, relationships and aggregation. We will illustrate the feasibility of this model, by producing a prototype implementation using PIFDM. PIFDM is a semantic data model database system based on the functional model of Shipman, with object-oriented extensions.