nLab family

Idea

The term ‘family’ is often used as a synonym for ‘collection’ (especially in the sense of subset). However, we can use it more precisely, for the concept of an indexed family.

Definitions

A family, or indexed family (or sometimes indexed set, etc), consists of an index set $I$ (whose elements are the indices of the family) and, for each index $k$, some element $x_k$. One can also speak of an $I$-indexed family. An ordered pair is a family indexed by $2 = \{0,1\}$; an infinite sequence is an $\mathbb{N}$-indexed family.

In concrete categories

In a concrete category $C$, given an object $A:Ob(C)$ an $A$-indexed family of objects in $C$ is a function $B:El(A) \to Ob(C)$.

Notation

As a whole, this family may be denoted $(x_k \;|\; k\colon I)$, $(x_k)_{k\colon I}$, $(x_k)_k$, 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 $\lambda\, k\colon I.\; x_k$ or $(k \mapsto x_k)$. Finally, instead of $k\colon I$, one can see the type of $k$ indicated using any other method, especially $k \in I$ (which ultimately derives from material set theory).

Families vs collections

Formally, a family of things should be distinguished from a collection of things; properly, it is the range 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.)

Formalization

We have been vague so far about what a family is a family of (what the elements are). The easiest case is when the elements are from a set $S$; then an $I$-indexed family of elements of $S$, also called an indexed subset of $S$, is simply a function to $S$ from $I$. If the elements are from 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 $C$, also called an indexed subcategory, is similarly a functor to the arrow category of $C$. In general, the elements ought to be from (at the very least) some sort of $\infty$-groupoid $G$, in which case the family is a functor to $G$ from the discrete $\infty$-groupoid on $I$.

In foundations without proper classes or higher categories, 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 $k$.

Related concepts

• free coproduct completion

• dependent type

• empty family

• quantum set