family

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.

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.

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).

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.)

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 $I$.

A formal structural set theory with a formal definition/notion of family is known as a first-order set theory, while a set theory without a formal definition or notion of family is a zeroth-order set theory; this parallels the distinction between predicate and propositional logic.

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 $k$.

Last revised on March 3, 2021 at 00:36:40. See the history of this page for a list of all contributions to it.