Theory and Applications of Type-Theoretic Universes

Compositional presentations of models of GATs

We now assume that the theory of signatures has $U$ -small dependent products and identity types

Proposition. Let

A : T \to U

be a small type in the context of a model of

U

and let

P

be the initial model of

(M : T) \times A (M)

. Then

P . M

is a model of

T

. There is a classical notion of finite presentation (a finite colimit of representable models) that coincides with the models which can be represented in this way.

Example.

TwoCycle TwoCycle FigureEight FigureEight : {open G : ReflGraph} \to U :\equiv {∣ x : V, y : V, f : E x y, g : E y x ∣} : {open G : ReflGraph} \to U :\equiv {∣ c_{1} : TwoCycle, c_{2} : TwoCycle, e : c_{1} . y = c_{2} . x ∣}