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href="/nlab/show/natural+transformation">natural transformation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cat">Cat</a></p> </li> </ul> <h2 id="sidebar_universal_constructions">Universal constructions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+construction">universal construction</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/representable+functor">representable functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+functor">adjoint functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/limit">limit</a>/<a class="existingWikiWord" href="/nlab/show/colimit">colimit</a></p> </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</a>/<a class="existingWikiWord" href="/nlab/show/coend">coend</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kan+extension">Kan extension</a></p> </li> </ul> </li> </ul> <h2 id="sidebar_theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+construction">Grothendieck construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+functor+theorem">adjoint functor theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monadicity+theorem">monadicity theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+lifting+theorem">adjoint lifting theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gabriel-Ulmer+duality">Gabriel-Ulmer duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/small+object+argument">small object argument</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freyd-Mitchell+embedding+theorem">Freyd-Mitchell embedding theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relation+between+type+theory+and+category+theory">relation between type theory and category theory</a></p> </li> </ul> <h2 id="sidebar_extensions">Extensions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/sheaf+and+topos+theory">sheaf and topos theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a></p> </li> </ul> <h2 id="sidebar_applications">Applications</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/applications+of+%28higher%29+category+theory">applications of (higher) category theory</a></li> </ul> <div> <p> <a href="/nlab/edit/category+theory+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="limits_and_colimits">Limits and colimits</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/limit">limits and colimits</a></strong></p> <h2 id="1categorical">1-Categorical</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/limit">limit and colimit</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/limits+and+colimits+by+example">limits and colimits by 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="coproducts">Coproducts</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <li><a href='#examples'>Examples</a></li> <li><a href='#properties'>Properties</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 notion of <em>coproduct</em> is a generalization to arbitrary <a class="existingWikiWord" href="/nlab/show/categories">categories</a> of the notion of <a class="existingWikiWord" href="/nlab/show/disjoint+union">disjoint union</a> in the category <a class="existingWikiWord" href="/nlab/show/Set">Set</a>.</p> <h2 id="definition">Definition</h2> <p>For <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">C</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/category">category</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>Obj</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x, y \in Obj(C)</annotation></semantics></math> two <a class="existingWikiWord" 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fill-opacity="1"> <use xlink:href="#ixHDqqwmVROSGb09lEmKy91d7r0=-glyph-0-3" x="83.806" y="78.1"></use> </g> <path fill="none" stroke-width="0.47818" stroke-linecap="butt" stroke-linejoin="miter" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M -67.845406 25.990906 L -12.521187 -20.372375 " transform="matrix(1, 0, 0, -1, 89.279, 43.237)"></path> <path fill="none" stroke-width="0.47818" stroke-linecap="round" stroke-linejoin="round" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M -2.484257 2.869351 C -2.03136 1.149059 -1.018793 0.3361 -0.0000484778 0.00230956 C -1.022441 -0.334622 -2.034016 -1.146675 -2.485395 -2.870467 " transform="matrix(0.76643, 0.64229, 0.64229, -0.76643, 76.93996, 63.76352)"></path> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#ixHDqqwmVROSGb09lEmKy91d7r0=-glyph-1-1" x="39.578" y="51.017"></use> </g> <path fill="none" stroke-width="0.47818" stroke-linecap="butt" stroke-linejoin="miter" stroke="rgb(0%, 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rspace="thinmathspace">∐</mo><mi>y</mi><mo>→</mo><mi>Q</mi></mrow><annotation encoding="application/x-tex">(f,g) : x \coprod y \to Q</annotation></semantics></math> such that we have the following <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commuting diagram</a>:</p> <div style="text-align: center"> <svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="200.804" height="94.325" viewBox="0 0 200.804 94.325"> <defs> <g> <g id="Lj7HOnF5lycppnNY6j4y38yK3LU=-glyph-0-0"> </g> <g id="Lj7HOnF5lycppnNY6j4y38yK3LU=-glyph-0-1"> <path d="M 7.078125 -6.09375 C 6.609375 -6 6.421875 -5.640625 6.421875 -5.359375 C 6.421875 -5 6.703125 -4.890625 6.921875 -4.890625 C 7.359375 -4.890625 7.671875 -5.265625 7.671875 -5.671875 C 7.671875 -6.296875 6.953125 -6.59375 6.328125 -6.59375 C 5.421875 -6.59375 4.921875 -5.6875 4.78125 -5.40625 C 4.4375 -6.53125 3.515625 -6.59375 3.234375 -6.59375 C 1.71875 -6.59375 0.90625 -4.625 0.90625 -4.296875 C 0.90625 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1.147113 -1.021327 0.334613 -0.00179625 -0.001325 C -1.021327 -0.333356 -2.033046 -1.145856 -2.486171 -2.868512 " transform="matrix(-1, 0, 0, 1, 124.56461, 16.01695)"></path> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#Lj7HOnF5lycppnNY6j4y38yK3LU=-glyph-2-2" x="147.558" y="9.803"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#Lj7HOnF5lycppnNY6j4y38yK3LU=-glyph-3-2" x="151.16175" y="11.048"></use> </g> </svg> </div> <p>This morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(f,g)</annotation></semantics></math> is called the <strong><a class="existingWikiWord" href="/nlab/show/copairing">copairing</a></strong> of <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>. The morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>→</mo><mi>x</mi><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">x\to x\coprod y</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo>→</mo><mi>x</mi><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">y\to x\coprod y</annotation></semantics></math> are called <a class="existingWikiWord" href="/nlab/show/coprojections">coprojections</a> or sometimes “injections” or “inclusions”, although in general they may not be <a class="existingWikiWord" href="/nlab/show/monomorphisms">monomorphisms</a>.</p> <p><strong>Notation.</strong> The coproduct is also denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">a+b</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>⨿</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">a\amalg b</annotation></semantics></math>, especially when it is <a class="existingWikiWord" href="/nlab/show/disjoint+coproduct">disjoint</a> (or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>⊔</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">a \sqcup b</annotation></semantics></math> if your fonts don't include ‘<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⨿</mo></mrow><annotation encoding="application/x-tex">\amalg</annotation></semantics></math>’). The copairing is also denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[f,g]</annotation></semantics></math> or (when possible) given vertically: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo>{</mo><mfrac linethickness="0"><mrow><mi>f</mi></mrow><mrow><mi>g</mi></mrow></mfrac><mo>}</mo></mrow></mrow><annotation encoding="application/x-tex">\left\{{f \atop g}\right\}</annotation></semantics></math>.</p> <p>A coproduct is thus the <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a> over the <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a> that consists of just two objects.</p> <p>More generally, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> any <a class="existingWikiWord" href="/nlab/show/set">set</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><mi>S</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">F : S \to C</annotation></semantics></math> a collection of objects 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">C</annotation></semantics></math> indexed by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>, their <strong>coproduct</strong> is an object</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mrow><mi>s</mi><mo>∈</mo><mi>S</mi></mrow></munder><mi>F</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \coprod_{s \in S} F(s) </annotation></semantics></math></div> <p>equipped with maps</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>s</mi><mo stretchy="false">)</mo><mo>→</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mrow><mi>s</mi><mo>∈</mo><mi>S</mi></mrow></munder><mi>F</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> F(s) \to \coprod_{s \in S} F(s) </annotation></semantics></math></div> <p>such that this is <a class="existingWikiWord" href="/nlab/show/universal+property">universal</a> among all objects with maps from the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(s)</annotation></semantics></math>.</p> <h2 id="examples">Examples</h2> <ul> <li> <p>In <a class="existingWikiWord" href="/nlab/show/Set">Set</a>, the coproduct of a <a class="existingWikiWord" href="/nlab/show/family+of+sets">family of sets</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>C</mi> <mi>i</mi></msub><msub><mo stretchy="false">)</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding="application/x-tex">(C_i)_{i\in I}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/disjoint+union">disjoint union</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><msub><mi>C</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\coprod_{i\in I} C_i</annotation></semantics></math> of sets.</p> <p>This makes the coproduct a <a class="existingWikiWord" href="/nlab/show/categorification">categorification</a> of the operation of addition of <a class="existingWikiWord" href="/nlab/show/natural+number">natural number</a>s and more generally of <a class="existingWikiWord" href="/nlab/show/cardinal">cardinal</a> numbers: for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>,</mo><mi>T</mi><mo>∈</mo><mi>FinSet</mi></mrow><annotation encoding="application/x-tex">S,T \in FinSet</annotation></semantics></math> two finite sets and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">|</mo><mo>−</mo><mo stretchy="false">|</mo><mo>:</mo><mi>FinSet</mi><mo>→</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">|-| : FinSet \to \mathbb{N}</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/cardinality">cardinality</a> operation, we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">|</mo><mi>S</mi><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mi>T</mi><mo stretchy="false">|</mo><mo>=</mo><mo stretchy="false">|</mo><mi>S</mi><mo stretchy="false">|</mo><mo>+</mo><mo stretchy="false">|</mo><mi>T</mi><mo stretchy="false">|</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> |S \coprod T| = |S| + |T| \,. </annotation></semantics></math></div></li> <li> <p>In <a class="existingWikiWord" href="/nlab/show/Top">Top</a>, the coproduct of a family of spaces <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>C</mi> <mi>i</mi></msub><msub><mo stretchy="false">)</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding="application/x-tex">(C_i)_{i\in I}</annotation></semantics></math> is the space whose set of points is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><msub><mi>C</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\coprod_{i\in I} C_i</annotation></semantics></math> and whose open subspaces are of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><msub><mi>U</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\coprod_{i\in I} U_i</annotation></semantics></math> (the internal <a class="existingWikiWord" href="/nlab/show/disjoint+union">disjoint union</a>) where each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">U_i</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/open+subspace">open subspace</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">C_i</annotation></semantics></math>. This is typical of <a class="existingWikiWord" href="/nlab/show/topological+concrete+categories">topological concrete categories</a>.</p> </li> <li> <p>In <a class="existingWikiWord" href="/nlab/show/Grp">Grp</a>, the coproduct is the “<a class="existingWikiWord" href="/nlab/show/free+product+of+groups">free product of groups</a>”, whose <a class="existingWikiWord" href="/nlab/show/underlying+set">underlying set</a> is <em>not</em> a <a class="existingWikiWord" href="/nlab/show/disjoint+union">disjoint union</a>. This is typical of <a class="existingWikiWord" href="/nlab/show/algebraic+category">algebraic categories</a>.</p> </li> <li> <p>In <a class="existingWikiWord" href="/nlab/show/Ab">Ab</a>, in <a class="existingWikiWord" href="/nlab/show/Vect">Vect</a>, the coproduct is the <a class="existingWikiWord" href="/nlab/show/subobject">subobject</a> of the <a class="existingWikiWord" href="/nlab/show/product">product</a> consisting of those tuples of elements for which only finitely many are not 0.</p> </li> <li> <p>In <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a>, the coproduct of a family of categories <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>C</mi> <mi>i</mi></msub><msub><mo stretchy="false">)</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding="application/x-tex">(C_i)_{i\in I}</annotation></semantics></math> is the category with</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Obj</mi><mo stretchy="false">(</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>C</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><mi>Obj</mi><mo stretchy="false">(</mo><msub><mi>C</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Obj(\coprod_{i\in I} C_i) = \coprod_{i\in I} Obj(C_i)</annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mrow><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>C</mi> <mi>i</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mi>Hom</mi> <mrow><msub><mi>C</mi> <mi>i</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mtd> <mtd><mi>if</mi><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><msub><mi>C</mi> <mi>i</mi></msub></mtd></mtr> <mtr><mtd><mi>∅</mi></mtd> <mtd><mi>otherwise</mi></mtd></mtr></mtable></mrow></mrow></mrow><annotation encoding="application/x-tex"> Hom_{\coprod_{i\in I} C_i}(x,y) = \left\{ \begin{aligned} Hom_{C_i}(x,y) & if x,y \in C_i \\ \emptyset & otherwise \end{aligned} \right. </annotation></semantics></math></div></li> <li> <p>In <a class="existingWikiWord" href="/nlab/show/Grpd">Grpd</a>, the coproduct follows <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a> rather than <a class="existingWikiWord" href="/nlab/show/Grp">Grp</a>. This is typical of <a class="existingWikiWord" href="/nlab/show/oidifications">oidifications</a>: the coproduct becomes a disjoint union again.</p> </li> </ul> <h2 id="properties">Properties</h2> <ul> <li> <p>A coproduct 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">C</annotation></semantics></math> is the same as a <a class="existingWikiWord" href="/nlab/show/product">product</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">C^{op}</annotation></semantics></math>.</p> </li> <li> <p>When they exist, coproducts are unique up to unique canonical isomorphism, so we often say “<a class="existingWikiWord" href="/nlab/show/generalized+the">the</a> coproduct.”</p> </li> <li> <p>A coproduct indexed by the <a class="existingWikiWord" href="/nlab/show/empty+set">empty set</a> is an <a class="existingWikiWord" href="/nlab/show/initial+object">initial object</a> 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">C</annotation></semantics></math>.</p> </li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quasi-coproduct">quasi-coproduct</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sum+type">sum type</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/free+coproduct+completion">free coproduct completion</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/base+change">base change</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/dependent+sum">dependent sum</a>, <a class="existingWikiWord" href="/nlab/show/dependent+product">dependent product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dependent+sum+type">dependent sum type</a>, <a class="existingWikiWord" href="/nlab/show/dependent+product+type">dependent product type</a></p> </li> </ul> </li> </ul> <h2 id="references">References</h2> <p>Textbook account:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Francis+Borceux">Francis Borceux</a>, Section 2.2 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>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on September 23, 2023 at 23:52:26. 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