Theory and Applications of Type-Theoretic Universes

Conjunctive queries

From [Spivak 2012Spivak, D.I. 2012. Functorial data migration. Information and Computation 217, 31–51. http://dx.doi.org/10.1016/j.ic.2012.05.001.] we know that we can represent database schema as categories, which are very simple GATs

Corp :\equiv (department : U) \times (person : U) \times (secretary : department \to person)

We can view $Q : (C : Corp) \to U$ as a conjunctive query which returns the type of its results. The initial model of $(C : Corp) \times Q (C)$ is the initial database instance with a result for that conjunctive query.

If we add sum types, we could (maybe) also model DUC queries (disjoint union of conjunctive queries)

Example. We can symmetrize a graph via DUC queries

Symmetrize : Graph \to Graph Symmetrize G :\equiv {∣ V :\equiv G . V, E x y :\equiv G . E x y + G . E y x ∣}