@prefix psr: <http://data.loterre.fr/ark:/67375/PSR> .
@prefix skos: <http://www.w3.org/2004/02/skos/core#> .
@prefix dc: <http://purl.org/dc/terms/> .
@prefix xsd: <http://www.w3.org/2001/XMLSchema#> .

psr: a skos:ConceptScheme .
psr:-WHZKH3TW-6
  skos:prefLabel "morphism"@en, "morphisme"@fr ;
  a skos:Concept ;
  skos:narrower psr:-SWDW5W2B-R .

psr:-SWDW5W2B-R
  dc:modified "2024-10-18"^^xsd:date ;
  skos:inScheme psr: ;
  a skos:Concept ;
  skos:prefLabel "monomorphisme"@fr, "monomorphism"@en ;
  skos:definition """In the context of abstract algebra or universal algebra, a <b>monomorphism</b> is an injective homomorphism. A monomorphism from <span class="texhtml mvar" style="font-style:italic;">X</span> to <span class="texhtml mvar" style="font-style:italic;">Y</span> is often denoted with the notation  <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle X\\\\hookrightarrow Y}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>X</mi>         <mo stretchy="false">↪<!-- ↪ --></mo>         <mi>Y</mi>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle X\\\\hookrightarrow Y}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5d77a675aa10c404233db9c24ab7b9dfbcde726" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.66ex; height:2.176ex;" alt="{\\\\displaystyle X\\\\hookrightarrow Y}"></span>. In the more general setting of category theory, a <b>monomorphism</b> (also called a <b>monic morphism</b> or a <b>mono</b>) is a left-cancellative morphism. That is, an arrow <span class="texhtml"><i>f</i> : <i>X</i> → <i>Y</i></span> such that for all objects <span class="texhtml"><i>Z</i></span> and all morphisms <span class="texhtml"><i>g</i><sub>1</sub>, <i>g</i><sub>2</sub>: <i>Z</i> → <i>X</i></span>,  <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle f\\\\circ g_{1}=f\\\\circ g_{2}\\\\implies g_{1}=g_{2}.}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>f</mi>         <mo>∘<!-- ∘ --></mo>         <msub>           <mi>g</mi>           <mrow class="MJX-TeXAtom-ORD">             <mn>1</mn>           </mrow>         </msub>         <mo>=</mo>         <mi>f</mi>         <mo>∘<!-- ∘ --></mo>         <msub>           <mi>g</mi>           <mrow class="MJX-TeXAtom-ORD">             <mn>2</mn>           </mrow>         </msub>         <mspace width="thickmathspace"></mspace>         <mo stretchy="false">⟹<!-- ⟹ --></mo>         <mspace width="thickmathspace"></mspace>         <msub>           <mi>g</mi>           <mrow class="MJX-TeXAtom-ORD">             <mn>1</mn>           </mrow>         </msub>         <mo>=</mo>         <msub>           <mi>g</mi>           <mrow class="MJX-TeXAtom-ORD">             <mn>2</mn>           </mrow>         </msub>         <mo>.</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle f\\\\circ g_{1}=f\\\\circ g_{2}\\\\implies g_{1}=g_{2}.}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce07e9ad6f488f906905b64a7f6845c65653448c" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:28.83ex; height:2.509ex;" alt="{\\\\displaystyle f\\\\circ g_{1}=f\\\\circ g_{2}\\\\implies g_{1}=g_{2}.}"> 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Monomorphism">https://en.wikipedia.org/wiki/Monomorphism</a>)"""@en, """Dans le cadre de l'algèbre générale ou de l'algèbre universelle, un <b>monomorphisme</b> est simplement un morphisme injectif. Dans le cadre plus général de la théorie des catégories, un monomorphisme est un morphisme simplifiable à gauche, c'est-à-dire un morphisme <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle f\\\\colon X\\	o Y}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>f</mi>         <mo>:<!-- : --></mo>         <mi>X</mi>         <mo stretchy="false">→<!-- → --></mo>         <mi>Y</mi>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle f\\\\colon X\\	o Y}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07b9ff205beb51e7899846aeae788ae5e5546a3e" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.68ex; height:2.509ex;" alt="{\\\\displaystyle f\\\\colon X\\	o Y}"></span> tel que pour tout <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle Z}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>Z</mi>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle Z}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\\\\displaystyle Z}"></span>,  <center><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\orall g_{1},g_{2}\\\\colon Z\\	o X\\\\quad \\\\left(f\\\\circ g_{1}=f\\\\circ g_{2}\\\\quad \\\\Rightarrow \\\\quad g_{1}=g_{2}\\
ight),}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi mathvariant="normal">∀<!-- ∀ --></mi>         <msub>           <mi>g</mi>           <mrow class="MJX-TeXAtom-ORD">             <mn>1</mn>           </mrow>         </msub>         <mo>,</mo>         <msub>           <mi>g</mi>           <mrow class="MJX-TeXAtom-ORD">             <mn>2</mn>           </mrow>         </msub>         <mo>:<!-- : --></mo>         <mi>Z</mi>         <mo stretchy="false">→<!-- → --></mo>         <mi>X</mi>         <mspace width="1em"></mspace>         <mrow>           <mo>(</mo>           <mrow>             <mi>f</mi>             <mo>∘<!-- ∘ --></mo>             <msub>               <mi>g</mi>               <mrow class="MJX-TeXAtom-ORD">                 <mn>1</mn>               </mrow>             </msub>             <mo>=</mo>             <mi>f</mi>             <mo>∘<!-- ∘ --></mo>             <msub>               <mi>g</mi>               <mrow class="MJX-TeXAtom-ORD">                 <mn>2</mn>               </mrow>             </msub>             <mspace width="1em"></mspace>             <mo stretchy="false">⇒<!-- ⇒ --></mo>             <mspace width="1em"></mspace>             <msub>               <mi>g</mi>               <mrow class="MJX-TeXAtom-ORD">                 <mn>1</mn>               </mrow>             </msub>             <mo>=</mo>             <msub>               <mi>g</mi>               <mrow class="MJX-TeXAtom-ORD">                 <mn>2</mn>               </mrow>             </msub>           </mrow>           <mo>)</mo>         </mrow>         <mo>,</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\orall g_{1},g_{2}\\\\colon Z\\	o X\\\\quad \\\\left(f\\\\circ g_{1}=f\\\\circ g_{2}\\\\quad \\\\Rightarrow \\\\quad g_{1}=g_{2}\\
ight),}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/747d8180eae03b9c531f46d4eea781b969f3efa1" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:50.57ex; height:2.843ex;" alt="{\\\\displaystyle \\orall g_{1},g_{2}\\\\colon Z\\	o X\\\\quad \\\\left(f\\\\circ g_{1}=f\\\\circ g_{2}\\\\quad \\\\Rightarrow \\\\quad g_{1}=g_{2}\\
ight),}"></span></center> ou encore : l'application   <center><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle f_{*}\\\\colon \\\\operatorname {Hom} (Z,X)\\	o \\\\operatorname {Hom} (Z,Y),~g\\\\mapsto f\\\\circ g{\\	ext{ est injective.}}}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <msub>           <mi>f</mi>           <mrow class="MJX-TeXAtom-ORD">             <mo>∗<!-- ∗ --></mo>           </mrow>         </msub>         <mo>:<!-- : --></mo>         <mi>Hom</mi>         <mo>⁡<!-- ⁡ --></mo>         <mo stretchy="false">(</mo>         <mi>Z</mi>         <mo>,</mo>         <mi>X</mi>         <mo stretchy="false">)</mo>         <mo stretchy="false">→<!-- → --></mo>         <mi>Hom</mi>         <mo>⁡<!-- ⁡ --></mo>         <mo stretchy="false">(</mo>         <mi>Z</mi>         <mo>,</mo>         <mi>Y</mi>         <mo stretchy="false">)</mo>         <mo>,</mo>         <mtext> </mtext>         <mi>g</mi>         <mo stretchy="false">↦<!-- ↦ --></mo>         <mi>f</mi>         <mo>∘<!-- ∘ --></mo>         <mi>g</mi>         <mrow class="MJX-TeXAtom-ORD">           <mtext> est injective.</mtext>         </mrow>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle f_{*}\\\\colon \\\\operatorname {Hom} (Z,X)\\	o \\\\operatorname {Hom} (Z,Y),~g\\\\mapsto f\\\\circ g{\\	ext{ est injective.}}}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5b1303d16711893ed42bbb1634d0242fb89acd7" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:53.448ex; height:2.843ex;" alt="{\\\\displaystyle f_{*}\\\\colon \\\\operatorname {Hom} (Z,X)\\	o \\\\operatorname {Hom} (Z,Y),~g\\\\mapsto f\\\\circ g{\\	ext{ est injective.}}}"></span></center> Les monomorphismes sont la généralisation aux catégories des fonctions injectives ; dans certaines catégories, les deux notions coïncident d'ailleurs. Mais les monomorphismes restent des objets plus généraux (voir l'exemple ci-dessous). Le dual d'un monomorphisme est un épimorphisme (c'est-à-dire qu'un monomorphisme dans la catégorie <i>C</i> est un épimorphisme dans la catégorie duale <i>C</i><sup>op</sup>).  
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Monomorphisme">https://fr.wikipedia.org/wiki/Monomorphisme</a>)"""@fr ;
  skos:exactMatch <https://fr.wikipedia.org/wiki/Monomorphisme>, <https://en.wikipedia.org/wiki/Monomorphism> ;
  skos:broader psr:-WHZKH3TW-6 .