Inclusion map

In mathematics, if $$A$$ is a subset of $$B,$$ then the inclusion map is the function ι|$\iota$ that sends each element $$x$$ of $$A$$ to $$x,$$ treated as an element of $$B:$$ $$\iota : A\rightarrow B, \qquad \iota(x)=x.$$

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

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 $$\iota : A \to X$$ into the domain $$X$$, then one can form the restriction $$f\circ \iota$$ 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 $$I$$ is an ideal of $$R.$$