Family of types in type theory
Consider the example of the dependent type of number sequences of length $n$. It might be defined like this in Coq:

  Inductive Sequence : nat -&gt; Type :=
  | nil : Sequence O
  | cons : forall (n : nat), nat -&gt; Sequence n -&gt; Sequence (S n).


For every n : nat we have a type Sequence n. We say that Sequence is indexed by nat or that Sequence depends on nat. We could also say that Sequence n depends on the index n.

In any case, we are just discussing terminology here. The following are equiavalent ways of saying the same thing:


Type $B$ depends on type $A$
$B$ is a type family indexed by type $A$
$B$ is a dependent type indexed by type $A$
$B : A \to \mathsf{Type}$
for $x : A$, $B(x)$ is a type
$x : A \vdash B(x) \ \mathsf{type}$


The reason we use the word family is that in mathematics a collection of sets indexed by a set, $\lbrace A_i \rbrace_{i \in I}$, is called a family of sets. It corresponds to a dependent type $A$ indexed by a type $I$.