Categorical pairs and the indicative shift

The paper introduces the notion of a categorical pair, a pair of categories (C, C′) such that every morphism in C is an object in C′. Arrows in C′ can express relationships between the morphisms of C. In particular we show that by using a model of the linguistic process of naming, we can ensure that morphisms F in C can have an indirect self-reference of the form a → Fa where this arrow occurs in the category C′. This result is shown to complement and clarify known fixed point theorems in logic and categories, and is applied to Gödel’s Incompleteness Theorem, the Cantor Diagonal Process and the Lawvere Fixed Point Theorem.