Theory and Applications of Type-Theoretic Universes

Theory of Signatures

Definition. A model of the theory of signatures consists of a cwf with a universe

U

, such that

This is just enough to be a module system for a dependently typed language. For instance, working internally and eliding $El$

ReflGraph :\equiv (V : U) \times (E : V \to V \to U) \times (refl : (v : V) \to E v v)

Proposition. Closed types in the theory of signatures are generalized algebraic theories.