nLab
family

Indexed families

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 II (whose elements are the indices of the family) and, for each index kk, some element x kx_k. One can also speak of an II-indexed family. An ordered pair is a family indexed by 2={0,1}2 = \{0,1\}; an infinite sequence is an \mathbb{N}-indexed family.

Notation

As a whole, this family may be denoted (x k|k:I)(x_k \;|\; k\colon I), (x k) k:I(x_k)_{k\colon I}, (x k) k(x_k)_k, or simply xx. 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 λk:I.x k\lambda\, k\colon I.\; x_k or (kx k)(k \mapsto x_k). Finally, instead of k:Ik\colon I, one can see the type of kk indicated using any other method, especially kIk \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 SS; then an II-indexed family of elements of SS, also called an indexed subset of SS, is simply a function to SS from II. If the elements are from a category CC, then an II-indexed family of objects of CC is a functor (or anafunctor) to CC from the discrete category on II, and an II-indexed family of morphisms of CC, also called an indexed subcategory, is similarly a functor to the arrow category of CC. In general, the elements ought to be from (at the very least) some sort of \infty-groupoid GG, in which case the family is a functor to GG from the discrete \infty-groupoid on II.

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 SS is simply a binary relation between SS and some index set II, writing ax ka \in x_k to denote that the SS-element aa is related to the index kk.

Revised on June 20, 2013 11:10:50 by Toby Bartels (98.19.42.235)