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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,^[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 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

^ MacLane, S.; Birkhoff, G. (1967). Algebra. Providence, RI: AMS Chelsea Publishing. p. 5. ISBN 0-8218-1646-2. Note that "insertion" is a function $S \to U$ and "inclusion" a relation $S \subset U$ ; every inclusion relation gives rise to an insertion function.
^ "Arrows – Unicode" (PDF). Unicode Consortium. Retrieved 2017-02-07.
^ Chevalley, C. (1956). Fundamental Concepts of Algebra. New York, NY: Academic Press. p. 1. ISBN 0-12-172050-0.

[1] MacLane, S.; Birkhoff, G. (1967). Algebra. Providence, RI: AMS Chelsea Publishing. p. 5. ISBN 0-8218-1646-2. Note that "insertion" is a function $S \to U$ and "inclusion" a relation $S \subset U$ ; every inclusion relation gives rise to an insertion function.

[Unicode_Arrows-2] "Arrows – Unicode" (PDF). Unicode Consortium. Retrieved 2017-02-07.

[3] Chevalley, C. (1956). Fundamental Concepts of Algebra. New York, NY: Academic Press. p. 1. ISBN 0-12-172050-0.

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[2]

[3]

@@ Line 1: / Line 1: @@
+{{Short description|Set-theoretic function}}
-[[File:Venn A subset B.svg|150px|thumb|right|{{mvar|A}} is a subset of {{mvar|B}}, and {{mvar|B}} is a superset of {{mvar|A}}.]]
+[[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 {{mvar|A}} is a [[subset]] of {{mvar|B}}, then the '''inclusion map''' (also '''inclusion function''', '''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 '''canonical injection''') is the [[function (mathematics)|function]] {{mvar|[[ι]]}} that sends each element {{mvar|x}} of {{mvar|A}} to {{mvar|x}}, treated as an element of {{mvar|B}}:
+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'''.
-:<math>\iota: A\rightarrow B, \qquad \iota(x)=x.</math>
-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| accessdate = 2017-02-07|publisher=[[Unicode Consortium]]}}</ref> is sometimes used in place of the function arrow above to denote an inclusion map; thus:
+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]].)
-:<math>\iota: A\hookrightarrow B.</math>
-(On the other hand, this notation is sometimes reserved for [[embedding]]s.)
 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]] {{mvar|f}} between [[object (category theory)|objects]] {{mvar|X}} and {{mvar|Y}}, if there is an inclusion map into the [[domain of a function|domain]] {{math|''ι'' : ''A'' → ''X''}}, then one can form the [[Function (mathematics)#Restrictions and extensions|restriction]] {{mvar|f&thinsp;ι}} of {{mvar|f}}.  In many instances, one can also construct a canonical inclusion into the [[codomain]] {{math|''R'' → ''Y''}} known as the [[range of a function|range]] of {{mvar|f}}.
+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 ==
+==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|⋆}}, to require that
+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]]).
-:<math>\iota(x\star y)=\iota(x)\star \iota(y)</math>
-is simply to say that {{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 {{mvar|A}} is a [[strong deformation retract]] of {{mvar|X}}, 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>
-:<math>\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==
-:<math>\operatorname{Spec}\left(R/I^2\right) \to \operatorname{Spec}(R)</math>
+* {{annotated link|Cofibration}}
-may be different [[morphism]]s, where {{mvar|R}} is a [[commutative ring]] and {{mvar|I}} is an [[ideal (ring theory)|ideal]] of {{mvar|R}}.
+* {{annotated link|Identity function}}
-== See also ==
+==References==
-*[[Cofibration]]
-*[[Identity function]]
-== References ==
 {{reflist}}
 {{DEFAULTSORT:Inclusion Map}}
-[[Category:Functions and mappings]]
 [[Category:Basic concepts in set theory]]
+[[Category:Functions and mappings]]