Indexed family - Definition

In mathematics, an index set is another name for a function domain. A collection indexed by I, often written A_i for i in I (can be said 'for i running over I ') is in effect a function A(i) into some codomain.

Usage for index sets

Index sets are often used in sums (sigma notation) and other such operations; and are common when the A_i are themselves sets rather than numbers, in indexed intersections and unions.

Families

A family is another description of an indexed collection, often used of a family of sets. In contrast to a set of elements, a family can contain an element more than once (that is, the underlying function need not be injective).

Examples

• An n-tuple can be considered as a family over the finite index set {1, 2, ..., n}
• A sequence is a family over the natural numbers.

Usage in category theory

More generally, a functor can be considered as giving rise to an indexed family of objects in a category D, indexed by another category C, and related by morphisms depending on two indices.

See also

• disjoint union
• tagged union
• index notation
• array