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

A "hooked arrow" ( U+21AA ↪ RIGHTWARDS ARROW WITH HOOK) ^[2] is sometimes used in place of the function arrow above to denote an inclusion map; thus:

(However, some authors use this hooked arrow for any embedding.)

This and other analogous injective functions ^[3] from substructures are sometimes called natural injections.

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

Applications of inclusion maps

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

is simply to say that ${\displaystyle \star }$ 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 ${\displaystyle A}$ is a strong deformation retract of ${\displaystyle X,}$ the inclusion map yields an isomorphism between all homotopy groups (that is, it is a homotopy equivalence).

Inclusion maps in geometry come in different kinds: for example embeddings of submanifolds. Contravariant objects (which is to say, objects that have pullbacks; these are called covariant in an older and unrelated terminology) 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

and may be different morphisms, where ${\displaystyle R}$ is a commutative ring and ${\displaystyle R}$ is an ideal of ${\displaystyle R.}$

See also

• Cofibration
• Identity function – In mathematics, a function that always returns the same value that was used as its argument

References