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In set theory

A family of sets consists of an index set $I$ and, for each element $k$ of $I$, a set $S_k$.

Given $I$, a set $F$ and a function $p\colon F \to I$, we get a family of sets by defining $S_k$ to be the preimage $p^*(k)$.

Conversely, given a family of sets, let $F$ be the disjoint union

and let $f(k,x)$ be $k$.

(We should talk about ways to formalise this concept in various forms of set theory and when the latter construction above requires the axiom of collection.)

In category theory

A family of sets consists of an discrete groupoid $I$ called the index set, and a functor $F:I \to Set$, where Set is the large category of sets.

In dependent type theory

A family of sets consists of a type $I$ called the index type, and a type family $S(x)$ indexed by elements $x:I$, such that every dependent type $S(x)$ is an h-set by the typal axiom K definition or the typal uniqueness of identity proofs definition.

Equivalently, it is a type family $(I, S)$ which satisfies the familial axiom K.

Inconsistency of the universal family of sets

A universal family of sets is a set $U$ with a set $I$ and a function $E:U \to I$ such that for all sets $A$, there is a unique element $i_A \in I$ and a bijection $\delta_A:A \cong E^*(i_A)$ from $A$ to the fiber of $E$ at $i_A$. This is inconsistent in any set theory, due to the set-theoretic Girard's paradox.

See also

• family
• family of subsets