Family of types in type theory

Can anyone simplify the meaning of families of types index by a type. It looks i get it but quite not understood it. What do you mean by a "family" ?

I understand index by a value (n length sequence) then what you mean by index by a type. Any example how you index by a type ? Thanks.

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

Inductive Sequence : nat -> Type :=
  | nil : Sequence O
  | cons : forall (n : nat), nat -> Sequence n -> 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 $I$.

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