# Inclusion map

In mathematics, if A is a subset of B, then the inclusion map (also inclusion function, insertion,[1] or canonical injection) is the function ι 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.
${\displaystyle \iota :A\rightarrow B,\qquad \iota (x)=x.}$

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:

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

(On the other hand, this notation is sometimes reserved for embeddings.)

This and other analogous injective functions[3] 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 ι : AX, then one can form the restriction f ι 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

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 , to require that

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

is simply to say that 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

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

and

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

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