Abstract
Abstract We show how the display-map category of finite (symmetric) simplicial complexes can be seen as representing the totality of database schemas and instances in a single mathematical structure. We give a sound interpretation of a certain dependent type theory in this model and show how it allows for the syntactic specification of schemas and instances and the manipulation of the same with the usual type-theoretic operations.
Highlights
Databases being, essentially, collections of tables of data, a foundational question is how to best represent such collections of tables mathematically in order to study their properties and ways of manipulating them
Areas exist in which the relational model is less adequate than in others
We show that (S, D) is a model of a certain dependent type theory including the usual type-forming operations and with context constants
Summary
Collections of (possibly interrelated) tables of data, a foundational question is how to best represent such collections of tables mathematically in order to study their properties and ways of manipulating them. The interesting aspect is that this category can in a natural way be seen as a category of tables; collecting in a single mathematical structure—an indexed or fibered category— the totality of schemas and instances. This representation can be introduced as follows. Future work includes exploiting the more geometric perspective on tables that this models offers (see (Spivak 2009)), and the modeling of schemas with multiple relation variables over the same attributes and instances with multiple keys representing the same data. Knowledge of the very basic notions of category theory, such as category, functor, and natural transformation, is assumed
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