Editing Inclusion map

{{Short description|Set-theoretic function}}
[[File:Venn A subset B.svg|150px|thumb|right|<math>A</math> is a [[subset]] of <math>B,</math> and <math>B</math> is a [[Subset|superset]] of <math>A.</math>]]
In [[mathematics]], if <math>A</math> is a [[subset]] of <math>B,</math> then the '''inclusion map''' is the [[function (mathematics)|function]] [[ι|<math>\iota</math>]] that sends each element <math>x</math> of <math>A</math> to <math>x,</math> treated as an element of <math>B:</math>
<math display=block>\iota : A\rightarrow B, \qquad \iota(x)=x.</math>

An inclusion map may also be referred to as an '''inclusion function''', an '''insertion''',<ref>{{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.}}</ref> or a '''canonical injection'''.

A "hooked arrow" ({{unichar|21AA|RIGHTWARDS ARROW WITH HOOK|ulink=Unicode}})<ref name="Unicode Arrows">{{cite web| title = Arrows – Unicode| url = https://www.unicode.org/charts/PDF/U2190.pdf| access-date = 2017-02-07|publisher=[[Unicode Consortium]]}}</ref> is sometimes used in place of the function arrow above to denote an inclusion map; thus:
<math display=block>\iota: A\hookrightarrow B.</math>

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

This and other analogous [[injective]] functions<ref>{{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]}}</ref> from [[substructure (mathematics)|substructures]] are sometimes called '''natural injections'''.

Given any [[morphism]] <math>f</math> between [[object (category theory)|objects]] <math>X</math> and <math>Y</math>, if there is an inclusion map <math>\iota : A \to X</math> into the [[Domain of a function|domain]] <math>X</math>, then one can form the [[Restriction (mathematics)|restriction]] <math>f\circ \iota</math> of <math>f.</math>  In many instances, one can also construct a canonical inclusion into the [[codomain]] <math>R \to Y</math> known as the [[range of a function|range]] of <math>f.</math>

==Applications of inclusion maps==
Inclusion maps tend to be [[homomorphism]]s of [[algebraic structure]]s; thus, such inclusion maps are [[embedding]]s.  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 <math>\star,</math> to require that
<math display=block>\iota(x\star y) = \iota(x) \star \iota(y)</math>
is simply to say that <math>\star</math> 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 (mathematics)|closure]] means such constants must already be given in the substructure.

Inclusion maps are seen in [[algebraic topology]] where if <math>A</math> is a [[strong deformation retract]] of <math>X,</math> the inclusion map yields an [[Group isomorphism|isomorphism]] between all [[homotopy groups]] (that is, it is a [[Homotopy|homotopy equivalence]]).

Inclusion maps in [[geometry]] come in different kinds: for example [[embedding]]s of [[submanifold]]s. [[Covariance and contravariance of functors|Contravariant]] objects (which is to say, objects that have [[pullback]]s; these are called [[covariance and contravariance of vectors|covariant]] in an older and unrelated terminology) such as [[differential form]]s ''restrict'' to submanifolds, giving a mapping in the ''other direction''. Another example, more sophisticated, is that of [[affine scheme]]s, for which the inclusions
<math display=block>\operatorname{Spec}\left(R/I\right) \to \operatorname{Spec}(R)</math>
and
<math display=block>\operatorname{Spec}\left(R/I^2\right) \to \operatorname{Spec}(R)</math>
may be different [[morphism]]s, where <math>R</math> is a [[commutative ring]] and <math>I</math> is an [[Ideal (ring theory)|ideal]] of <math>R.</math>

==See also==

* {{annotated link|Cofibration}}
* {{annotated link|Identity function}}

==References==
{{reflist}}

{{DEFAULTSORT:Inclusion Map}}

[[Category:Basic concepts in set theory]]
[[Category:Functions and mappings]]