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example</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/commutativity+of+limits+and+colimits">commutativity of limits and colimits</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/small+limit">small limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/filtered+colimit">filtered colimit</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/directed+colimit">directed colimit</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/sequential+colimit">sequential colimit</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sifted+colimit">sifted colimit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connected+limit">connected limit</a>, <a class="existingWikiWord" href="/nlab/show/wide+pullback">wide pullback</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/preserved+limit">preserved limit</a>, <a class="existingWikiWord" href="/nlab/show/reflected+limit">reflected limit</a>, <a class="existingWikiWord" href="/nlab/show/created+limit">created limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/product">product</a>, <a class="existingWikiWord" href="/nlab/show/fiber+product">fiber product</a>, <a class="existingWikiWord" href="/nlab/show/base+change">base change</a>, <a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a>, <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a>, <a class="existingWikiWord" href="/nlab/show/pushout">pushout</a>, <a class="existingWikiWord" href="/nlab/show/cobase+change">cobase change</a>, <a class="existingWikiWord" href="/nlab/show/equalizer">equalizer</a>, <a class="existingWikiWord" href="/nlab/show/coequalizer">coequalizer</a>, <a class="existingWikiWord" href="/nlab/show/join">join</a>, <a class="existingWikiWord" href="/nlab/show/meet">meet</a>, <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a>, <a class="existingWikiWord" href="/nlab/show/initial+object">initial object</a>, <a class="existingWikiWord" href="/nlab/show/direct+product">direct product</a>, <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/finite+limit">finite limit</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/exact+functor">exact functor</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kan+extension">Kan extension</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Yoneda+extension">Yoneda extension</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/weighted+limit">weighted limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/end">end and coend</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fibered+limit">fibered limit</a></p> </li> </ul> <h2 id="2categorical">2-Categorical</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-limit">2-limit</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/inserter">inserter</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/isoinserter">isoinserter</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equifier">equifier</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/inverter">inverter</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/PIE-limit">PIE-limit</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-pullback">2-pullback</a>, <a class="existingWikiWord" href="/nlab/show/comma+object">comma object</a></p> </li> </ul> <h2 id="1categorical_2">(∞,1)-Categorical</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-limit">(∞,1)-limit</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-pullback">(∞,1)-pullback</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fiber+sequence">fiber sequence</a></li> </ul> </li> </ul> </li> </ul> <h3 id="modelcategorical">Model-categorical</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+Kan+extension">homotopy Kan extension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+limit">homotopy limit</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+product">homotopy product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+equalizer">homotopy equalizer</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+fiber">homotopy fiber</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+totalization">homotopy totalization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+end">homotopy end</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+colimit">homotopy colimit</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coproduct">homotopy coproduct</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coequalizer">homotopy coequalizer</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+cofiber">homotopy cofiber</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cocone">mapping cocone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+pushout">homotopy pushout</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+realization">homotopy realization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coend">homotopy coend</a></p> </li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/infinity-limits+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#RelationToKernelPairs'>Relation to kernel pairs</a></li> <li><a href='#RelationToPushout'>Relation to pushouts</a></li> </ul> <li><a href='#examples'>Examples</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>The concept of <em>coequalizer</em> in a general <a class="existingWikiWord" href="/nlab/show/category">category</a> is the generalization of the construction where for two <a class="existingWikiWord" href="/nlab/show/functions">functions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>,</mo><mi>g</mi></mrow><annotation encoding="application/x-tex">f,g</annotation></semantics></math> between <a class="existingWikiWord" href="/nlab/show/sets">sets</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mover><munder><mo>⟶</mo><mi>g</mi></munder><mover><mo>⟶</mo><mi>f</mi></mover></mover><mi>Y</mi></mrow><annotation encoding="application/x-tex">X \stackrel{\overset{f}{\longrightarrow}}{\underset{g}{\longrightarrow}} Y</annotation></semantics></math></div> <p>one forms the set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><msub><mo stretchy="false">/</mo> <mo>∼</mo></msub></mrow><annotation encoding="application/x-tex">Y/_\sim</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/equivalence+classes">equivalence classes</a> induced by the <a class="existingWikiWord" href="/nlab/show/equivalence+relation">equivalence relation</a> generated by the relation</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>∼</mo><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)\sim g(x)</annotation></semantics></math></div> <p>for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x \in X</annotation></semantics></math>. This means that the <a class="existingWikiWord" href="/nlab/show/quotient">quotient</a> function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo lspace="verythinmathspace">:</mo><mi>Y</mi><mo>⟶</mo><mi>Y</mi><msub><mo stretchy="false">/</mo> <mo>∼</mo></msub></mrow><annotation encoding="application/x-tex">p \colon Y \longrightarrow Y/_\sim</annotation></semantics></math> satisfies</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>∘</mo><mi>f</mi><mo>=</mo><mi>p</mi><mo>∘</mo><mi>g</mi></mrow><annotation encoding="application/x-tex">p \circ f = p \circ g</annotation></semantics></math></div> <p>(a map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> satisfying this equation is said to “co-equalize” <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math>) and moreover <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/universal+property">universal</a> with this property.</p> <p>In this form this may be phrased generally in any <a class="existingWikiWord" href="/nlab/show/category">category</a>.</p> <h2 id="definition">Definition</h2> <div class="num_defn"> <h6 id="definition_2">Definition</h6> <p>In some <a class="existingWikiWord" href="/nlab/show/category">category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>, the <strong>coequalizer</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>coeq</mi><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">coeq(f,g)</annotation></semantics></math> of two <a class="existingWikiWord" href="/nlab/show/parallel+morphisms">parallel morphisms</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> between two <a class="existingWikiWord" href="/nlab/show/objects">objects</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> is (if it exists), the <em><a class="existingWikiWord" href="/nlab/show/colimit">colimit</a></em> under the <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a> formed by these two morphisms</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd></mtd> <mtd><mover><munder><mo>⟶</mo><mi>g</mi></munder><mover><mo>⟶</mo><mi>f</mi></mover></mover></mtd> <mtd></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mi>p</mi></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>coeq</mi><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mo>.</mo></mrow><annotation encoding="application/x-tex">\array{ X && \stackrel{\overset{f}{\longrightarrow}}{\underset{g}{\longrightarrow}} && Y \\ & \searrow && \swarrow_{\mathrlap{p}} \\ && coeq(f,g) }. </annotation></semantics></math></div></div> <p>Equivalently:</p> <div class="num_defn" id="CoequalizerDiagram"> <h6 id="definition_3">Definition</h6> <p>In a <a class="existingWikiWord" href="/nlab/show/category">category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><munderover><mo>⇉</mo><mi>g</mi><mi>f</mi></munderover><mi>Y</mi><mover><mo>→</mo><mi>p</mi></mover><mi>Z</mi></mrow><annotation encoding="application/x-tex">X \underoverset{g}{f}{\rightrightarrows} Y \overset{p}{\rightarrow} Z</annotation></semantics></math></div> <p>is called a <strong>coequalizer</strong> diagram if</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>∘</mo><mi>f</mi><mo>=</mo><mi>p</mi><mo>∘</mo><mi>g</mi></mrow><annotation encoding="application/x-tex">p \circ f = p \circ g</annotation></semantics></math>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/universal+property">universal</a> for this property: i.e. if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi><mo lspace="verythinmathspace">:</mo><mi>Y</mi><mo>→</mo><mi>W</mi></mrow><annotation encoding="application/x-tex">q \colon Y \to W</annotation></semantics></math> is a morphism of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi><mo>∘</mo><mi>f</mi><mo>=</mo><mi>q</mi><mo>∘</mo><mi>g</mi></mrow><annotation encoding="application/x-tex">q \circ f = q \circ g</annotation></semantics></math>, then there is a unique morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi><mo>′</mo><mo lspace="verythinmathspace">:</mo><mi>Z</mi><mo>→</mo><mi>W</mi></mrow><annotation encoding="application/x-tex">q' \colon Z \to W</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi><mo>′</mo><mo>∘</mo><mi>p</mi><mo>=</mo><mi>q</mi></mrow><annotation encoding="application/x-tex">q' \circ p = q</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd><mover><munder><mo>⟶</mo><mi>g</mi></munder><mover><mo>⟶</mo><mi>f</mi></mover></mover></mtd> <mtd><mi>Y</mi></mtd> <mtd><mover><mo>⟶</mo><mi>p</mi></mover></mtd> <mtd><mi>Z</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>q</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mrow><mo>∃</mo><mo>!</mo><mspace width="thinmathspace"></mspace><mi>q</mi><mo>′</mo></mrow></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>W</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ X &\stackrel{\overset{f}{\longrightarrow}}{\underset{g}{\longrightarrow}}& Y &\overset{p}{\longrightarrow}& Z \\ && {}^{\mathllap{q}}\downarrow & \swarrow_{\mathrlap{\exists ! \, q'}} \\ && W } </annotation></semantics></math></div></li> </ol> </div> <p> <div class='num_remark'> <h6>Remark</h6> <p>By <a class="existingWikiWord" href="/nlab/show/formal+duality">formal duality</a>, a coequalizer in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> is equivalently an <a class="existingWikiWord" href="/nlab/show/equalizer">equalizer</a> in the <a class="existingWikiWord" href="/nlab/show/opposite+category">opposite category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝒞</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">\mathcal{C}^{op}</annotation></semantics></math>.</p> </div> </p> <p> <div class='num_remark'> <h6>Remark</h6> <p>Morphisms that coequalize some pair of <a class="existingWikiWord" href="/nlab/show/parallel+morphisms">parallel morphisms</a> are called <em><a class="existingWikiWord" href="/nlab/show/regular+epimorphisms">regular epimorphisms</a></em>.</p> </div> </p> <h2 id="properties">Properties</h2> <h3 id="RelationToKernelPairs">Relation to kernel pairs</h3> <p> <div class='num_prop'> <h6>Proposition</h6> <p>In any category:</p> <ul> <li> <p>If a <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a></p> <ol> <li> <p>is the <a class="existingWikiWord" href="/nlab/show/coequalizer">coequalizer</a> of some pair of <a class="existingWikiWord" href="/nlab/show/parallel+morphisms">parallel morphisms</a> (hence: is a <a class="existingWikiWord" href="/nlab/show/regular+epimorphism">regular epimorphism</a>)</p> </li> <li> <p>has a <a class="existingWikiWord" href="/nlab/show/kernel+pair">kernel pair</a>,</p> </li> </ol> <p>then it is also the coequalizer of its kernel pair.</p> </li> <li> <p>If a <a class="existingWikiWord" href="/nlab/show/kernel+pair">kernel pair</a> has a coequalizer, then it is the kernel pair of its coequalizer.</p> </li> </ul> <p></p> </div> (e.g. <a href="#Borceux94">Borceux 1994, Prop. 2.5.7, 2.5.8</a>, <a href="#Taylor99">Taylor 1999, Lemma 5.6.6</a></p> <h3 id="RelationToPushout">Relation to pushouts</h3> <p>Coequalizers are closely related to <a class="existingWikiWord" href="/nlab/show/pushouts">pushouts</a>:</p> <p> <div class='num_prop' id='CoequalizedAsAPushout'> <h6>Proposition</h6> <p>A diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><munderover><mo>⇉</mo><mi>f</mi><mi>g</mi></munderover><mi>Y</mi><mover><mo>⟶</mo><mi>p</mi></mover><mi>Z</mi></mrow><annotation encoding="application/x-tex"> X \underoverset {f} {g} {\rightrightarrows} Y \overset{p}{\longrightarrow} Z </annotation></semantics></math></div> <p>is a coequalizer diagram, def. <a class="maruku-ref" href="#CoequalizerDiagram"></a>, precisely if</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi><mo>⊔</mo><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo></mtd> <mtd></mtd> <mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo><mrow><msup><mo></mo><mpadded width="0"><mi>p</mi></mpadded></msup></mrow></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><munder><mo>⟶</mo><mrow></mrow></munder></mtd> <mtd><mi>Z</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{ X \sqcup X &\overset{(f,g)}{\longrightarrow}& Y \\ \big\downarrow && \big\downarrow{^\mathrlap{p}} \\ X &\underset{}{\longrightarrow}& Z } </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/pushout">pushout</a> diagram.</p> </div> </p> <p>Conversely:</p> <p> <div class='num_prop' id='PushoutAsACoequalizer'> <h6>Proposition</h6> <p></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>f</mi> <mn>1</mn></msub></mrow></mover></mtd> <mtd><mi>B</mi></mtd></mtr> <mtr><mtd><mpadded width="0" lspace="-100%width"><mrow><msup><mrow></mrow> <mrow><msub><mi>f</mi> <mn>2</mn></msub></mrow></msup></mrow></mpadded><mo maxsize="1.8em" minsize="1.8em">↓</mo></mtd> <mtd></mtd> <mtd><mo maxsize="1.8em" minsize="1.8em">↓</mo><mrow><msup><mrow></mrow> <mpadded width="0"><mrow><msub><mi>p</mi> <mn>1</mn></msub></mrow></mpadded></msup></mrow></mtd></mtr> <mtr><mtd><mi>C</mi></mtd> <mtd><munder><mo>⟶</mo><mrow><msub><mi>p</mi> <mn>2</mn></msub></mrow></munder></mtd> <mtd><mi>D</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ A &\overset{f_1}{\longrightarrow}& B \\ {\mathllap{{}^{f_2}}}\Big\downarrow && \Big\downarrow{{}^\mathrlap{p_1}} \\ C &\underset{p_2}{\longrightarrow}& D } </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/pushout">pushout</a> square, precisely if</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>A</mi><munderover><mo>⇉</mo><mrow><msub><mi>q</mi> <mn>2</mn></msub><mo>∘</mo><msub><mi>f</mi> <mn>2</mn></msub></mrow><mrow><msub><mi>q</mi> <mn>1</mn></msub><mo>∘</mo><msub><mi>f</mi> <mn>1</mn></msub></mrow></munderover><mi>B</mi><mo>⊔</mo><mi>C</mi><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><msub><mi>p</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>p</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow></mover><mi>D</mi></mrow><annotation encoding="application/x-tex"> A \underoverset { q_2 \circ f_2 } { q_1 \circ f_1 } {\rightrightarrows} B \sqcup C \overset{(p_1,p_2)}{\longrightarrow} D </annotation></semantics></math></div> <p>is a coequalizer diagram.</p> </div> (Here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mover><mo>→</mo><mrow><msub><mi>q</mi> <mn>1</mn></msub></mrow></mover><mi>B</mi><mo>⊔</mo><mi>C</mi><mover><mo>←</mo><mrow><msub><mi>q</mi> <mn>2</mn></msub></mrow></mover><mi>C</mi></mrow><annotation encoding="application/x-tex">B \overset{q_1}{\to} B \sqcup C \overset{q_2}{\leftarrow} C</annotation></semantics></math> denotes the two <a class="existingWikiWord" href="/nlab/show/coprojections">coprojections</a> into the <a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a>.)</p> <h2 id="examples">Examples</h2> <div class="num_example" id="QuotientSet"> <h6 id="example">Example</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">\mathcal{C} =</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Set">Set</a>, the coequalizer of two <a class="existingWikiWord" href="/nlab/show/functions">functions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/quotient+set">quotient set</a> by the <a class="existingWikiWord" href="/nlab/show/equivalence+relation">equivalence relation</a> generated by the <a class="existingWikiWord" href="/nlab/show/relation">relation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>∼</mo><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x) \sim g(x)</annotation></semantics></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x \in X</annotation></semantics></math>.</p> </div> <div class="num_example"> <h6 id="example_2">Example</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">\mathcal{C} =</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Top">Top</a>, the coequalizer of two <a class="existingWikiWord" href="/nlab/show/continuous+functions">continuous functions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> whose underlying set is the <a class="existingWikiWord" href="/nlab/show/quotient+set">quotient set</a> from example <a class="maruku-ref" href="#QuotientSet"></a>, and whose topology is the corresponding <a class="existingWikiWord" href="/nlab/show/quotient+topology">quotient topology</a>.</p> </div> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/equalizer">equalizer</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/regular+epimorphism">regular epimorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/propositional+truncation+object">propositional truncation object</a></p> </li> </ul> <h2 id="references">References</h2> <p>Coequalizers were defined in the paper</p> <ul> <li id="EH"><a class="existingWikiWord" href="/nlab/show/Beno+Eckmann">Beno Eckmann</a>, <a class="existingWikiWord" href="/nlab/show/Peter+J.+Hilton">Peter J. Hilton</a>, <em>Group-like structures in general categories II. Equalizers, limits, lengths</em>. Mathematische Annalen 151:2 (1963), 150–186. <a href="https://doi.org/10.1007/bf01344176">doi:10.1007/bf01344176</a>.</li> </ul> <p>for any finite collection of parallel morphisms. The paper refers to them as <em>right equalizers</em>, whereas <a class="existingWikiWord" href="/nlab/show/equalizers">equalizers</a> are referred to as <em>left equalizers</em>.</p> <p>Textbook accounts:</p> <ul> <li id="Borceux94"> <p><a class="existingWikiWord" href="/nlab/show/Francis+Borceux">Francis Borceux</a>, Section 2.4 in Vol. 1: <em>Basic Category Theory</em> of: <em><a class="existingWikiWord" href="/nlab/show/Handbook+of+Categorical+Algebra">Handbook of Categorical Algebra</a></em>, Encyclopedia of Mathematics and its Applications <strong>50</strong> Cambridge University Press (1994) (<a href="https://doi.org/10.1017/CBO9780511525858">doi:10.1017/CBO9780511525858</a>)</p> </li> <li id="Taylor99"> <p><a class="existingWikiWord" href="/nlab/show/Paul+Taylor">Paul Taylor</a>, <em><a class="existingWikiWord" href="/nlab/show/Practical+Foundations+of+Mathematics">Practical Foundations of Mathematics</a></em>, Cambridge Studies in Advanced Mathematics 59, Cambridge University Press 1999 (<a href="http://www.paultaylor.eu/~pt/prafm/index.html">webpage</a>)</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on May 1, 2023 at 08:39:10. 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