In mathematics, if ${\displaystyle A}$ is a subset of ${\displaystyle B,}$ then the inclusion map is the function ${\displaystyle \iota }$ that sends each element ${\displaystyle x}$ of ${\displaystyle A}$ to ${\displaystyle x,}$ treated as an element of ${\displaystyle B:}$

$${\displaystyle \iota :A\rightarrow B,\qquad \iota (x)=x.}$$

An inclusion map may also referred to as an inclusion function, an insertion,^[1] or a canonical injection.

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:

$${\displaystyle \iota :A\hookrightarrow B.}$$

(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 ${\displaystyle \iota :A\to X}$ into the domain ${\displaystyle X}$, then one can form the restriction ${\displaystyle f\circ \iota }$ 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

$${\displaystyle \iota (x\star y)=\iota (x)\star \iota (y)}$$

${\displaystyle \star }$

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

$${\displaystyle \operatorname {Spec} \left(R/I\right)\to \operatorname {Spec} (R)}$$

$${\displaystyle \operatorname {Spec} \left(R/I^{2}\right)\to \operatorname {Spec} (R)}$$

${\displaystyle R}$

${\displaystyle I}$

${\displaystyle R.}$