# Inclusion map $\displaystyle{ A }$ is a subset of $\displaystyle{ B, }$ and $\displaystyle{ B }$ is a superset of $\displaystyle{ A. }$

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

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

(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 $\displaystyle{ f }$ 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 $\displaystyle{ \iota(x\star y) = \iota(x) \star \iota(y) }$ 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 $\displaystyle{ \operatorname{Spec}\left(R/I\right) \to \operatorname{Spec}(R) }$ and $\displaystyle{ \operatorname{Spec}\left(R/I^2\right) \to \operatorname{Spec}(R) }$ may be different morphisms, where $\displaystyle{ R }$ is a commutative ring and $\displaystyle{ R }$ is an ideal of $\displaystyle{ R. }$