# Inclusion map

In mathematics, if $A$ is a subset of $B,$ then the inclusion map (also inclusion function, insertion, or canonical injection) is the function $\iota$ that sends each element $x$ of $A$ to $x,$ treated as an element of $B:$  $A$ is a subset of $B,$ and $B$ is a superset of $A.$ $\iota :A\rightarrow B,\qquad \iota (x)=x.$ A "hooked arrow" (U+21AA RIGHTWARDS ARROW WITH HOOK) is sometimes used in place of the function arrow above to denote an inclusion map; thus:

$\iota :A\hookrightarrow B.$ (However, some authors use this hooked arrow for any embedding.)

This and other analogous injective functions 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 $\iota :A\to X,$ then one can form the restriction $f\,\iota$ of $f.$ In many instances, one can also construct a canonical inclusion into the codomain $R\to Y$ known as the range of $f.$ ## 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 $\star ,$  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 (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

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

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

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