nLab family of sets

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In set theory

A family of sets consists of an index set II and, for each element kk of II, a set S kS_k.

Given II, a set FF and a function p:FIp\colon F \to I, we get a family of sets by defining S kS_k to be the preimage p *(k)p^*(k).

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

kS k={(k,x)|kI,xS k} \biguplus_k S_k = \{ (k,x) | k \in I, x \in S_k \}

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

(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 II called the index set, and a functor F:ISetF:I \to Set, where Set is the large category of sets.

See also

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