In mathematics, if $A$ is a subset of $B,$ then the inclusion map is the function$\iota$ that sends each element $x$ of $A$ to $x,$ treated as an element of $B:$

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

An inclusion map may also referred to as an inclusion function, an insertion,^{[1]} or a canonical injection.

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:

$\iota :A\hookrightarrow B.$

(However, some authors use this hooked arrow for any embedding.)

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 $\iota :A\to X$ 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 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 formsrestrict to submanifolds, giving a mapping in the other direction. Another example, more sophisticated, is that of affine schemes, for which the inclusions

Cofibration – continuous mapping between topological spacesPages displaying wikidata descriptions as a fallback

Identity function – In mathematics, a function that always returns the same value that was used as its argument

References

^MacLane, S.; Birkhoff, G. (1967). Algebra. Providence, RI: AMS Chelsea Publishing. p. 5. ISBN0-8218-1646-2. Note that "insertion" is a function S → U and "inclusion" a relation S ⊂ U; every inclusion relation gives rise to an insertion function.