Canonical Injection

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

A "hooked arrow" () 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
$\operatorname\left(R/I\right) \to \operatorname(R)$
and
$\operatorname\left(R/I^2\right) \to \operatorname(R)$
may be different morphisms, where $R$ is a commutative ring and $R$ is an ideal of $R.$

See also

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References

{{DEFAULTSORT:Inclusion Map

Basic concepts in set theory
Functions and mappings