Data-loading Functors

Definition 1.1. A database schema is a small category C, and a morphism of schemas from C to C′ is a functor F : C → C′. An object c of C is called a type or a table in C, and an arrow f : c→ c′ is called a column of c valued in c′. A database state on schema C is a functor δ : C → Set. Given an object c ∈ Ob(C), an element x ∈ δ(c) is called a c-representative in δ, or a row of δ(c). We refer to a pair (x, f), where x is a c-representative in δ and f : c → c′ is a column of c, as a cell in δ, and to each cell (x, f) the element δ(f)(x) ∈ δ(c′) is called the value of the (x, f)-cell or the f -value of x. A morphism of database states from δ to δ′ on C is a natural transformation p : δ → δ′. We write C–Set to denote the category of database states on C and sometimes refer to it as the category of C-sets.