A family of types is a function $P$ from a type $A$ to a universe. Every term of $A$ corresponds to a type $P(A)$.

Definition

A type family is a map $P:X \to \mathcal{U}$.

Fibrations

Type families can be thought of as fibrations in classical homotopy theory. The base space is $X$, the total space is $\sum_{(x:X)}P(x)$ and the fiber $P(\star_X)$. This gives the fibration: