nLab
family

The term ‘family’ is often used as a synonym for ‘collection’ (especially in the sense of subset). However, if we use it more precisely, then a family of things consists of an index set $I$ (whose elements are the indices of the family) and, for each index $k$, a thing $x_k$. One can also speak of an $I$-indexed family of things. An ordered pair is a $2$-indexed family; an infinite sequence is an $\mathbb{N}$-indexed family.

As a whole, this family may be denoted $(x_k\mid k:I)$, $(x_k{)}_{k:I}$, $(x_k{)}_k$, or simply $x$. Sometimes one sees braces used instead of parentheses, giving the same notation for a family as for a collection, although this is falling out of fashion; the parentheses ultimately come from notation for ordered pairs. One can also use notation for functions, such as $\lambda k:I.x_k$ or $(k\mapsto x_k)$. Finally, instead of $k:I$, one can see the type of $k$ indicated using any other method, especially $k\in I$ (which ultimately derives from material set theory).

Formally, a family of things should be distinguished from a collection of things; properly, it is the image of a family of things that is a collection, such as a subset of an appropriate ambient set of things. On the other hand, often the difference between a family and a collection is unimportant, and the two may be used interchangeably. (For example, one can take the union of either a family of subsets or a collection of subsets, with equivalent results; but one can take the sum of only a family of cardinal numbers.)

We have been vague about ‘thing’ so far. The easiest case is when the things form a set $S$; then an $I$-indexed family of elements of $S$ is simply a function to $S$ from $I$. If the things form a category $C$, then an $I$-indexed family of objects of $C$ is a functor (or anafunctor) to $C$ from the discrete category on $I$, and an $I$-indexed family of morphisms of $C$ is similarly a functor to the arrow category of $C$. In general, things ought to form (at the very least) some sort of $\mathrm{\infty }$-groupoid $G$, in which case a family of things is a functor to $G$ from the discrete groupoid on $I$.

In foundations without proper classes, it may be tricky to specify exactly what a family of sets is, if one cannot literally speak of a functor from a discrete category to the large category Set; see the article. On the other hand, there is no difficulty in speaking of a family of subsets of a given set; even in predicative mathematics (where one cannot speak of the power set), a family of subsets of $S$ is simply a binary relation between $S$ and some index set $I$, writing $a\in x_k$ to denote that the $S$-element $a$ is related to the index $k$.