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Idea

A family of types indexed by another type.

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: