Stone Duality between Queries and Data

The Stone dualities for accessible categories and the subclass of Diers categories provide limit and colimit structuring principles for query languages and the associated database. As a motivating example, we consider relational databases. Relational databases are given by database schemata, the syntax, and the class of sets of relations satisfying the syntactic constraints, called instances. Relational database schemata are shown to be sketched by nite limit sketches. Such sketches include all the usual kinds of dependencies: functional, join and inclusion. The addition of domain constraints results in ((nite limit, countable coproduct)-sketches. Without inclusion dependencies, the accessible category of models of the database sketch consists of all the satisfying instances and homomorphisms between them. With inclusion dependencies, the instances are equivalence classes of the models. Without domain constraints the model category is locally presentable. With domain constraints the model category is locally multipresentable. i.e., a Diers category. The duality theory provides an entirely new means of expressing queries as formal sums of database instances. 1. Introduction A model is a system of sets with relations and functions providing constraints upon the set system. A class of models, all similarly structured, together with the structure preserving maps between them is a category. We consider such categories to be categories of structured data, abstracting the structured data recorded in a computer. There are many examples of such model categories of interest in computer science, both pure and applied. We concentrate on only one such class of examples, relational databases. This enables us to show most of the details for these examples and provides substantive weight to our primary considerations, the duality between model categories of data and the queries on the data. Our interest lies solely in the model categories which are accessible, 20]. Accessible categories may be speciied via a sketch, 20, 9, 4]. We give a brief description of sketches in later sections. Sentences in basic logic and sketches have equivalent model theories. That is, for every sentence in basic logic there is a sketch with the same category of models, and visa versa, 20]. As this reference points out, little of L 11 is lost by considering only basic logic. For these reasons, sketches are called graph-based logic in 8]. For some situations, the sentence, i.e., theory, in basic rst-order logic is the preferred choice for a speciication of computational data and activities. For other situations, a sketch provides the …