Homotopy Type Theory
type family


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


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


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

P( X) x:XP(x)XP(\star_X)\to \sum_{x:X}P(x) \to X

See also

universe Synthetic homotopy theory


HoTT Book