Inclusion map

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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 be referred to as an inclusion function, an insertion,{{cite book| first1 = S. | last1 = MacLane | first2 = G. | last2 = Birkhoff | title = Algebra | publisher = AMS Chelsea Publishing |location=Providence, RI | year = 1967| isbn = 0-8218-1646-2 | page = 5 | quote = Note that “insertion” is a function {{math|S → U}} and "inclusion" a relation {{math|S ⊂ U}}; every inclusion relation gives rise to an insertion function.}} or a canonical injection.

A "hooked arrow" ({{unichar|21AA|RIGHTWARDS ARROW WITH HOOK|ulink=Unicode}}){{cite web| title = Arrows – Unicode| url = https://www.unicode.org/charts/PDF/U2190.pdf| access-date = 2017-02-07|publisher=Unicode Consortium}} 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{{cite book| first = C. | last = Chevalley | title = Fundamental Concepts of Algebra | url = https://archive.org/details/fundamentalconce00chev_0 | url-access = registration | publisher = Academic Press|location= New York, NY | year = 1956| isbn = 0-12-172050-0 |page= [https://archive.org/details/fundamentalconce00chev_0/page/1 1]}} 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 $R.$

References

Category:Basic concepts in set theory

Category:Functions and mappings

Inclusion map

Applications of inclusion maps

See also

References