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Inclusion map

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In <a href="Mathematics" title="Mathematics">mathematics</a>, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

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</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> is a <a href="Subset" title="Subset">subset</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B,}">

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<annotation encoding="application/x-tex">{\displaystyle B,}</annotation>

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</math><img src="./075d661417b8ca5a991a2a7bd4991cc1ab856d9d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.411ex; height:2.509ex;" alt="{\displaystyle B,}" loading="lazy"> then the inclusion map is the <a href="Function_(mathematics)" title="Function (mathematics)">function</a> <a href="%CE%99" class="mw-redirect" title="Ι"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \iota }">

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<annotation encoding="application/x-tex">{\displaystyle \iota }</annotation>

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</math><img src="./bce48dd56254d0a7c33e987c7c8eeb44c963ac04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.823ex; height:1.676ex;" alt="{\displaystyle \iota }" loading="lazy"></a> that sends each element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

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</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,}">

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<annotation encoding="application/x-tex">{\displaystyle x,}</annotation>

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</math><img src="./feff4d40084c7351bf57b11ba2427f6331f5bdbe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.977ex; height:2.009ex;" alt="{\displaystyle x,}" loading="lazy"> treated as an element of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B:}">

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<annotation encoding="application/x-tex">{\displaystyle B:}</annotation>

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</math><img src="./7b032bec438ad620f69c91105798a7e17b372996.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.056ex; height:2.176ex;" alt="{\displaystyle B:}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle \iota :A\rightarrow B,\qquad \iota (x)=x.}</annotation>

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An inclusion map may also be referred to as an inclusion function, an insertion,<a href="#cite_note-1">[1]</a> or a canonical injection.

A "hooked arrow" (<style data-mw-deduplicate="TemplateStyles:r886049734">

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</style><a href="Unicode" title="Unicode">U+</a>21AA ↪ RIGHTWARDS ARROW WITH HOOK)<a href="#cite_note-Unicode_Arrows-2">[2]</a> is sometimes used in place of the function arrow above to denote an inclusion map; thus:

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<annotation encoding="application/x-tex">{\displaystyle \iota :A\hookrightarrow B.}</annotation>

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(However, some authors use this hooked arrow for any <a href="Embedding" title="Embedding">embedding</a>.)

This and other analogous <a href="Injective" class="mw-redirect" title="Injective">injective</a> functions<a href="#cite_note-3">[3]</a> from <a href="Substructure_(mathematics)" title="Substructure (mathematics)">substructures</a> are sometimes called natural injections.

Given any <a href="Morphism" title="Morphism">morphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> between <a href="Object_(category_theory)" class="mw-redirect" title="Object (category theory)">objects</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

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<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

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</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy">, if there is an inclusion map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \iota :A\to X}">

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<annotation encoding="application/x-tex">{\displaystyle \iota :A\to X}</annotation>

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</math><img src="./000ce517d6b34a70ae983bee062d358cc6700a4d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.098ex; height:2.176ex;" alt="{\displaystyle \iota :A\to X}" loading="lazy"> into the <a href="Domain_of_a_function" title="Domain of a function">domain</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">, then one can form the <a href="Restriction_(mathematics)" title="Restriction (mathematics)">restriction</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\circ \iota }">

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<annotation encoding="application/x-tex">{\displaystyle f\circ \iota }</annotation>

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</math><img src="./00cab020ce236b477285f18e75d42282fa4a1661.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.297ex; height:2.509ex;" alt="{\displaystyle f\circ \iota }" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f.}">

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<annotation encoding="application/x-tex">{\displaystyle f.}</annotation>

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</math><img src="./ecb3ed2e17fa8f336dcc0fd4b3eddbfb02a50ef3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.925ex; height:2.509ex;" alt="{\displaystyle f.}" loading="lazy"> In many instances, one can also construct a canonical inclusion into the <a href="Codomain" title="Codomain">codomain</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R\to Y}">

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<annotation encoding="application/x-tex">{\displaystyle R\to Y}</annotation>

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</math><img src="./72a42d8855865f38ef53a82a185cff1c6de4a436.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.151ex; height:2.176ex;" alt="{\displaystyle R\to Y}" loading="lazy"> known as the <a href="Range_of_a_function" title="Range of a function">range</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f.}">

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<annotation encoding="application/x-tex">{\displaystyle f.}</annotation>

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</math><img src="./ecb3ed2e17fa8f336dcc0fd4b3eddbfb02a50ef3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.925ex; height:2.509ex;" alt="{\displaystyle f.}" loading="lazy">

<div class="mw-heading mw-heading2"><h2 id="Applications_of_inclusion_maps">Applications of inclusion maps</h2></div>

Inclusion maps tend to be <a href="Homomorphism" title="Homomorphism">homomorphisms</a> of <a href="Algebraic_structure" title="Algebraic structure">algebraic structures</a>; thus, such inclusion maps are <a href="Embedding" title="Embedding">embeddings</a>. 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 xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \star ,}">

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<annotation encoding="application/x-tex">{\displaystyle \star ,}</annotation>

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</math><img src="./15d7330e74e4bd2cc1de8d198ad280f3695b8353.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.809ex; height:2.009ex;" alt="{\displaystyle \star ,}" loading="lazy"> to require that

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<annotation encoding="application/x-tex">{\displaystyle \iota (x\star y)=\iota (x)\star \iota (y)}</annotation>

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is simply to say that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \star }">

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<annotation encoding="application/x-tex">{\displaystyle \star }</annotation>

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</math><img src="./bd316a21eeb5079a850f223b1d096a06bfa788c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.035ex; margin-bottom: -0.206ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle \star }" loading="lazy"> is consistently computed in the sub-structure and the large structure. The case of a <a href="Unary_operation" title="Unary operation">unary operation</a> is similar; but one should also look at <a href="Nullary" class="mw-redirect" title="Nullary">nullary</a> operations, which pick out a constant element. Here the point is that <a href="Closure_(mathematics)" title="Closure (mathematics)">closure</a> means such constants must already be given in the substructure.

Inclusion maps are seen in <a href="Algebraic_topology" title="Algebraic topology">algebraic topology</a> where if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

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</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> is a <a href="Strong_deformation_retract" class="mw-redirect" title="Strong deformation retract">strong deformation retract</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,}">

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<annotation encoding="application/x-tex">{\displaystyle X,}</annotation>

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</math><img src="./09ba32eeb405f7f5f2bac1eb12987c47d2fd42df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.627ex; height:2.509ex;" alt="{\displaystyle X,}" loading="lazy"> the inclusion map yields an <a href="Group_isomorphism" title="Group isomorphism">isomorphism</a> between all <a href="Homotopy_groups" class="mw-redirect" title="Homotopy groups">homotopy groups</a> (that is, it is a <a href="Homotopy" title="Homotopy">homotopy equivalence</a>).

Inclusion maps in <a href="Geometry" title="Geometry">geometry</a> come in different kinds: for example <a href="Embedding" title="Embedding">embeddings</a> of <a href="Submanifold" title="Submanifold">submanifolds</a>. <a href="Covariance_and_contravariance_of_functors" class="mw-redirect" title="Covariance and contravariance of functors">Contravariant</a> objects (which is to say, objects that have <a href="Pullback" title="Pullback">pullbacks</a>; these are called <a href="Covariance_and_contravariance_of_vectors" title="Covariance and contravariance of vectors">covariant</a> in an older and unrelated terminology) such as <a href="Differential_form" title="Differential form">differential forms</a> restrict to submanifolds, giving a mapping in the other direction. Another example, more sophisticated, is that of <a href="Affine_scheme" class="mw-redirect" title="Affine scheme">affine schemes</a>, for which the inclusions

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Spec} \left(R/I\right)\to \operatorname {Spec} (R)}</annotation>

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and

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Spec} \left(R/I^{2}\right)\to \operatorname {Spec} (R)}</annotation>

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may be different <a href="Morphism" title="Morphism">morphisms</a>, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

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<annotation encoding="application/x-tex">{\displaystyle R}</annotation>

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</math><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"> is a <a href="Commutative_ring" title="Commutative ring">commutative ring</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}">

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<annotation encoding="application/x-tex">{\displaystyle I}</annotation>

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</math><img src="./535ea7fc4134a31cbe2251d9d3511374bc41be9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}" loading="lazy"> is an <a href="Ideal_(ring_theory)" title="Ideal (ring theory)">ideal</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R.}">

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<annotation encoding="application/x-tex">{\displaystyle R.}</annotation>

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</math><img src="./fdcae6b33a27f86c7961318cd7ee3d789d3bcdd2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.411ex; height:2.176ex;" alt="{\displaystyle R.}" loading="lazy">

<ul><li><a href="Cofibration" title="Cofibration">Cofibration</a></li>

<li><a href="Identity_function" title="Identity function">Identity function</a> – Function that returns its argument unchanged</li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFMacLaneBirkhoff1967" class="citation book cs1">MacLane, S.; Birkhoff, G. (1967). Algebra. Providence, RI: AMS Chelsea Publishing. p. 5. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-8218-1646-2</bdi>. <q>Note that "insertion" is a function S → U and "inclusion" a relation S ⊂ U; every inclusion relation gives rise to an insertion function.</q></cite>

</li>

<li id="cite_note-Unicode_Arrows-2"><a href="#cite_ref-Unicode_Arrows_2-0">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.unicode.org/charts/PDF/U2190.pdf">"Arrows – Unicode"</a> (PDF). <a href="Unicode_Consortium" title="Unicode Consortium">Unicode Consortium</a>. Retrieved 2017-02-07.</cite>

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> <cite id="CITEREFChevalley1956" class="citation book cs1">Chevalley, C. (1956). <a rel="nofollow" class="external text" href="https://archive.org/details/fundamentalconce00chev_0">Fundamental Concepts of Algebra</a>. New York, NY: Academic Press. p. <a rel="nofollow" class="external text" href="https://archive.org/details/fundamentalconce00chev_0/page/1">1</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-12-172050-0</bdi>.</cite> <code class="cs1-code">{{cite book}}</code>: ISBN / Date incompatibility (help)

</li>

</ol></div></div></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2024-09-27" href="https://en.wikipedia.org/wiki/?title=Inclusion_map&oldid=1247977499">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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