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 (whose elements are the indices of the family) and, for each index , a thing . One can also speak of an -indexed family of things. An ordered pair is a -indexed family; an infinite sequence is an -indexed family.
As a whole, this family may be denoted , , , or simply . 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 or . Finally, instead of , one can see the type of indicated using any other method, especially (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 ; then an -indexed family of elements of is simply a function to from . If the things form a category , then an -indexed family of objects of is a functor (or anafunctor) to from the discrete category on , and an -indexed family of morphisms of is similarly a functor to the arrow category of . In general, things ought to form (at the very least) some sort of -groupoid , in which case a family of things is a functor to from the discrete groupoid on .
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 is simply a binary relation between and some index set , writing to denote that the -element is related to the index .