Inclusion map

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A is a subset of B, and B is a superset of A.

In mathematics, if A is a subset of B, then the inclusion map (also inclusion function, insertion, or canonical injection) [1] is the function \iota that sends each element, x of A to x, treated as an element of B:

\iota: A\rightarrow B, \qquad \iota(x)=x.

A "hooked arrow" \hookrightarrow is sometimes used in place of the function arrow above to denote an inclusion map.

This and other analogous injective functions [2] from substructures are sometimes called natural injections.

Given any morphism f between objects X and Y, if there is an inclusion map into the domain \iota : A\rightarrow X, then one can form the restriction fi of f. In many instances, one can also construct a canonical inclusion into the codomain RY known as the range of f.

Applications of inclusion maps[edit]

Inclusion maps tend to be homomorphisms of algebraic structures; thus, such inclusion maps are embeddings. More precisely, given a sub-structure closed under some operations, the inclusion map will be an embedding for tautological reasons. For example, for a binary operation \star, to require that

\iota(x\star y)=\iota(x)\star \iota(y)

is simply to say that \star is consistently computed in the sub-structure and the large structure. The case of a unary operation is similar; but one should also look at nullary operations, which pick out a constant element. Here the point is that closure means such constants must already be given in the substructure.

Inclusion maps are seen in algebraic topology where if A is a strong deformation retract of X, the inclusion map yields an isomorphism between all homotopy groups (i.e. is a homotopy equivalence)

Inclusion maps in geometry come in different kinds: for example embeddings of submanifolds. Contravariant objects such as differential forms restrict to submanifolds, giving a mapping in the other direction. Another example, more sophisticated, is that of affine schemes, for which the inclusions




may be different morphisms, where R is a commutative ring and I an ideal.

See also[edit]


  1. ^ Mac Lane, S.; Birkhoff, G. (1967), Algebra , page 5
  2. ^ Chevalley, C. (1956), Fundamental Concepts of Algebra , page 1


  • Chevalley, C. (1956), Fundamental Concepts of Algebra, Academic Press, New York, ISBN 0-12-172050-0 .
  • Mac Lane, S.; Birkhoff, G. (1967), Algebra, AMS Chelsea Publishing, Providence, Rhode Island, ISBN 0-8218-1646-2 .