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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> <ul> <li><a href='#in_a_category'>In a category</a></li> <li><a href='#in_a_fibration'>In a fibration</a></li> </ul> <li><a href='#examples'>Examples</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#characterization_of_sheaf_toposes'>Characterization of sheaf toposes</a></li> <li><a href='#in_coherent_categories'>In coherent categories</a></li> </ul> <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>disjoint coproduct</em> is a generalization to arbitrary <a class="existingWikiWord" href="/nlab/show/categories">categories</a> of that of <em><a class="existingWikiWord" href="/nlab/show/disjoint+union">disjoint union</a></em> of sets.</p> <p>Informally, one says that a <a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>+</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X + Y</annotation></semantics></math> of 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><mo>,</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X, Y</annotation></semantics></math> 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> is <em>disjoint</em> if both <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> inject into it and the <a class="existingWikiWord" href="/nlab/show/intersection">intersection</a> of <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> with <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> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>+</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X + Y</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/initial+object">empty</a>. In this case one often writes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mi>Y</mi><mo>≔</mo><mi>X</mi><mo>+</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X \coprod Y \coloneqq X + Y</annotation></semantics></math> for the coproduct, particularly if the coproduct is <a class="existingWikiWord" href="/nlab/show/pullback-stable+colimit">stable under pullbacks</a>, and there one speaks of the <em><a class="existingWikiWord" href="/nlab/show/disjoint+union">disjoint union</a></em> of <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> with <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> <h2 id="definition">Definition</h2> <h3 id="in_a_category">In a category</h3> <p>A binary <a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a> <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> in a <a class="existingWikiWord" href="/nlab/show/category">category</a> is <strong>disjoint</strong> if</p> <ol> <li> <p>the coprojections <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>→</mo><mi>a</mi><mo>+</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">a\to a+b</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>→</mo><mi>a</mi><mo>+</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">b\to a+b</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/monomorphism">monic</a>, and</p> </li> <li> <p>their <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> (exists and) is an <a class="existingWikiWord" href="/nlab/show/initial+object">initial object</a> (which is therefore also their <a class="existingWikiWord" href="/nlab/show/intersection">intersection</a> in the poset of subobjects of <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>).</p> </li> </ol> <p>Equivalently, this means we have <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> squares</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><mo>→</mo></mtd> <mtd><mi>a</mi></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>b</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>b</mi></mtd> <mtd></mtd> <mtd></mtd> <mtd><mn>0</mn></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>b</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>a</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>a</mi><mo>+</mo><mi>b</mi></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>b</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>a</mi><mo>+</mo><mi>b</mi></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>a</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>a</mi><mo>+</mo><mi>b</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ a & \to & a &&& b & \to & b &&& 0 & \to & b\\ \downarrow && \downarrow &&& \downarrow && \downarrow &&& \downarrow && \downarrow \\ a & \to & a+b &&& b & \to & a+b &&& a & \to & a+b} </annotation></semantics></math></div> <p>An arbitrary coproduct <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mi>i</mi></msub><msub><mi>a</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\sum_i a_i</annotation></semantics></math> is disjoint if each coprojection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>a</mi> <mi>i</mi></msub><mo>→</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mi>k</mi></msub><msub><mi>a</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">a_i\to \sum_k a_k</annotation></semantics></math> is monic and the intersection of any two distinct ones is initial.</p> <p>Note that every 0-ary coproduct (that, is initial object) is disjoint. However, disjointness of coproducts is most commonly considered in categories with a <a class="existingWikiWord" href="/nlab/show/strict+initial+object">strict initial object</a>, in which case disjointness can be expressed as “a map into a coproduct from a noninitial object factors through a coprojection in at most one way”.</p> <p>A more <a class="existingWikiWord" href="/nlab/show/constructive+mathematics">constructive</a> way to phrase disjointness of an arbitrary coproduct is that the pullback of any two <a class="existingWikiWord" href="/nlab/show/coprojections">coprojections</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>a</mi> <mi>i</mi></msub><mo>→</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mi>k</mi></msub><msub><mi>a</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">a_i\to \sum_k a_k</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>a</mi> <mi>j</mi></msub><mo>→</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mi>k</mi></msub><msub><mi>a</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">a_j\to \sum_k a_k</annotation></semantics></math> is the coproduct <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>j</mi></mrow></msub><msub><mi>a</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\sum_{i=j} a_i</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>=</mo><mi>j</mi></mrow><annotation encoding="application/x-tex">i=j</annotation></semantics></math> denotes the <a class="existingWikiWord" href="/nlab/show/subsingleton">subsingleton</a> corresponding to the <a class="existingWikiWord" href="/nlab/show/proposition">proposition</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>=</mo><mi>j</mi></mrow><annotation encoding="application/x-tex">i=j</annotation></semantics></math>, a.k.a. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mo>*</mo><mo lspace="mediummathspace" rspace="mediummathspace">∣</mo><mi>i</mi><mo>=</mo><mi>j</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{ \ast \mid i=j \}</annotation></semantics></math>. (Since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>a</mi> <mi>i</mi></msub><mo>=</mo><msub><mi>a</mi> <mi>j</mi></msub></mrow><annotation encoding="application/x-tex">a_i=a_j</annotation></semantics></math> as soon as this indexing set is <a class="existingWikiWord" href="/nlab/show/inhabited+set">inhabited</a>, this coproduct could equally be written <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>j</mi></mrow></msub><msub><mi>a</mi> <mi>j</mi></msub></mrow><annotation encoding="application/x-tex">\sum_{i=j} a_j</annotation></semantics></math>.)</p> <h3 id="in_a_fibration">In a fibration</h3> <p>Suppose <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>π</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">\pi:C\to S</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/Grothendieck+fibration">Grothendieck fibration</a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">u:A\to B</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/indexed+coproduct">indexed coproduct</a>, which is to say an <a class="existingWikiWord" href="/nlab/show/opcartesian+arrow">opcartesian arrow</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> lying over some <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f:X\to Y</annotation></semantics></math> in <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>. Then this coproduct is <strong>disjoint</strong> if the <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><msub><mo>×</mo> <mi>B</mi></msub><mi>A</mi></mrow><annotation encoding="application/x-tex">A\times_B A</annotation></semantics></math> exists 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 <a class="existingWikiWord" href="/nlab/show/preserved+limit">preserved</a> by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math>, and the induced <a class="existingWikiWord" href="/nlab/show/diagonal+morphism">diagonal morphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>→</mo><mi>A</mi><msub><mo>×</mo> <mi>B</mi></msub><mi>A</mi></mrow><annotation encoding="application/x-tex">A\to A\times_B A</annotation></semantics></math> is also opcartesian (over the diagonal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mi>X</mi><msub><mo>×</mo> <mi>Y</mi></msub><mi>X</mi></mrow><annotation encoding="application/x-tex">X\to X\times_Y X</annotation></semantics></math>).</p> <p>This specializes to the above definition for arbitrary coproducts (in its constructive phrasing) when an ordinary category <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 represented by the corresponding <a class="existingWikiWord" href="/nlab/show/family+fibration">family fibration</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Set</mi></mrow><annotation encoding="application/x-tex">Set</annotation></semantics></math>, whose fiber over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">X\in Set</annotation></semantics></math> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mi>X</mi></msup></mrow><annotation encoding="application/x-tex">C^X</annotation></semantics></math>.</p> <h2 id="examples">Examples</h2> <ul> <li> <p>In the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Set</mi></mrow><annotation encoding="application/x-tex">Set</annotation></semantics></math> of sets and functions, the coproduct is given by <a class="existingWikiWord" href="/nlab/show/disjoint+union">disjoint union</a> and is, unsurprisingly, disjoint. In the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Pfn</mi></mrow><annotation encoding="application/x-tex">Pfn</annotation></semantics></math> of sets and <a class="existingWikiWord" href="/nlab/show/partial+function">partial functions</a> the coproduct is equally given by disjoint union and total injections and is disjoint as well.</p> </li> <li> <p>Since having all finitary disjoint coproducts is half of the condition for a category to be <a class="existingWikiWord" href="/nlab/show/extensive+category">extensive</a>, extensive categories provide <a class="existingWikiWord" href="/nlab/show/extensive+category#examples">examples for categories with disjoint finite coproducts</a>. In the preceding discussion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Set</mi></mrow><annotation encoding="application/x-tex">Set</annotation></semantics></math> instantiates this case whereas <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Pfn</mi></mrow><annotation encoding="application/x-tex">Pfn</annotation></semantics></math> does not: since the initial object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>∅</mi></mrow><annotation encoding="application/x-tex">\emptyset</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Pfn</mi></mrow><annotation encoding="application/x-tex">Pfn</annotation></semantics></math> is not <a class="existingWikiWord" href="/nlab/show/strict+initial+object">strict</a>, the latter category is not extensive.</p> </li> <li> <p>In the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Vect</mi></mrow><annotation encoding="application/x-tex">Vect</annotation></semantics></math> of (real) <a class="existingWikiWord" href="/nlab/show/vector+space">vector spaces</a> coproducts are given by <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a> and are disjoint but not <a class="existingWikiWord" href="/nlab/show/pullback-stable+colimit">stable under pullback</a>: pulling back the colimit diagram <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi><mo>→</mo><mi>ℝ</mi><mo>⊕</mo><mi>ℝ</mi><mo>←</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}\to\mathbb{R}\oplus\mathbb{R}\leftarrow\mathbb{R}</annotation></semantics></math> along the diagonal morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo>:</mo><mi>ℝ</mi><mo>→</mo><mi>ℝ</mi><mo>⊕</mo><mi>ℝ</mi><mspace width="thickmathspace"></mspace><mo>,</mo><mspace width="thickmathspace"></mspace><mi>x</mi><mo>↦</mo><mi>x</mi><mo>⊕</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">\Delta:\mathbb{R}\to\mathbb{R}\oplus\mathbb{R}\; ,\; x\mapsto x\oplus x</annotation></semantics></math> yields <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><mi>ℝ</mi><mo>←</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">0\to\mathbb{R}\leftarrow 0</annotation></semantics></math> which is not a colimit diagram. Whence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Vect</mi></mrow><annotation encoding="application/x-tex">Vect</annotation></semantics></math> is not <a class="existingWikiWord" href="/nlab/show/extensive+category">extensive</a>.</p> </li> <li> <p>Non-example: the <a class="existingWikiWord" href="/nlab/show/interval+category">interval category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo>{</mo><mn>0</mn><mo>→</mo><mn>1</mn><mo>}</mo></mrow></mrow><annotation encoding="application/x-tex">\left\{ 0 \to 1 \right\}</annotation></semantics></math> has coproducts but they are not all disjoint: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>+</mo><mn>1</mn><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">1+1=1</annotation></semantics></math>. There are plenty more examples of <a class="existingWikiWord" href="/nlab/show/posets">posets</a> that have non-disjoint coproducts besides this one. In a <a class="existingWikiWord" href="/nlab/show/Boolean+algebra">Boolean algebra</a>, two elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math> are disjoint in the sense that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∧</mo><mi>b</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">a\wedge b=0</annotation></semantics></math> if and only if <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\vee b</annotation></semantics></math> is their disjoint coproduct.</p> </li> </ul> <h2 id="properties">Properties</h2> <h3 id="characterization_of_sheaf_toposes">Characterization of sheaf toposes</h3> <p>Having all small disjoint coproducts is one of the conditions in <a class="existingWikiWord" href="/nlab/show/Giraud%27s+theorem">Giraud's theorem</a> characterizing <a class="existingWikiWord" href="/nlab/show/sheaf+toposes">sheaf toposes</a>.</p> <h3 id="in_coherent_categories">In coherent categories</h3> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>Let <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> be a <a class="existingWikiWord" href="/nlab/show/coherent+category">coherent category</a>. If <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>Z</mi></mrow><annotation encoding="application/x-tex">X, Y \hookrightarrow Z</annotation></semantics></math> are two <a class="existingWikiWord" href="/nlab/show/subobjects">subobjects</a> of some <a class="existingWikiWord" href="/nlab/show/object">object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Z</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">Z \in \mathcal{C}</annotation></semantics></math> and are disjoint, in that their <a class="existingWikiWord" href="/nlab/show/intersection">intersection</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Z</mi></mrow><annotation encoding="application/x-tex">Z</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/initial+object">empty</a>, <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>∅</mi></mrow><annotation encoding="application/x-tex">X \cap Y \simeq\emptyset</annotation></semantics></math>, then their <a class="existingWikiWord" href="/nlab/show/union">union</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∪</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X \cup Y</annotation></semantics></math> is their (disjoint) <a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a>.</p> </div> <p>This apears as (<a href="#Johnstone">Johnstone, cor. A1.4.4</a>).</p> <div class="num_defn" id="PositiveCategory"> <h6 id="definition_2">Definition</h6> <p>A coherent category in which all coproducts are disjoint is also called a <strong><a class="existingWikiWord" href="/nlab/show/positive+category">positive coherent category</a></strong>.</p> </div> <p>(<a href="#Johnstone">Johnstone, p. 34</a>)</p> <div class="num_example"> <h6 id="example">Example</h6> <p>Every <a class="existingWikiWord" href="/nlab/show/extensive+category">extensive category</a> is in particular positive, by definition.</p> </div> <p>In a positive coherent category, every morphism into a coproduct factors through the coproduct coprojections:</p> <div class="num_prop"> <h6 id="proposition_2">Proposition</h6> <p>Let <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> be a postive coherent category, def. <a class="maruku-ref" href="#PositiveCategory"></a>, and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo>→</mo><mi>X</mi><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f \colon A \to X \coprod Y</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a>. Then the two <a class="existingWikiWord" href="/nlab/show/subobjects">subobjects</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>↪</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">f^*(X) \hookrightarrow A</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>↪</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f^*(Y) \hookrightarrow Y</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>, being the <a class="existingWikiWord" href="/nlab/show/pullbacks">pullbacks</a> in</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><msup><mi>f</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>i</mi> <mi>X</mi></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>A</mi></mtd> <mtd><mover><mo>→</mo><mi>f</mi></mover></mtd> <mtd><mi>X</mi><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mtable><mtr><mtd><msup><mi>f</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>i</mi> <mi>Y</mi></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>A</mi></mtd> <mtd><mover><mo>→</mo><mi>f</mi></mover></mtd> <mtd><mi>X</mi><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mi>Y</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ f^* (X) &\to& X \\ \downarrow && \downarrow^{\mathrlap{i_X}} \\ A &\stackrel{f}{\to}& X \coprod Y } \;\;\;\; \array{ f^* (Y) &\to& Y \\ \downarrow && \downarrow^{\mathrlap{i_Y}} \\ A &\stackrel{f}{\to}& X \coprod Y } </annotation></semantics></math></div> <p>are disjoint in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is their disjoint coproduct</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>≃</mo><msup><mi>f</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><msup><mi>f</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> A \simeq f^*(X) \coprod f^*(Y) \,. </annotation></semantics></math></div></div> <p>This appears in (<a href="#Johnstone">Johnstone, p. 34</a>).</p> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>This means that if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">A \in \mathcal{C}</annotation></semantics></math> itself is indecomposable in that it is not a coproduct of two objects in a non-trivial way, for instance if <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 an <a class="existingWikiWord" href="/nlab/show/extensive+category">extensive category</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">A \in \mathcal{C}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/connected+object">connected object</a>, then every morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>→</mo><mi>X</mi><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">A \to X \coprod Y</annotation></semantics></math> into a disjoint coproduct factors through one of the two canonical inclusions.</p> </div> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/disjoint+subset">disjoint subset</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/disjunctive+logic">disjunctive logic</a></p> </li> </ul> <h2 id="references">References</h2> <ul> <li id="CarboniLackWalters93"> <p><a class="existingWikiWord" href="/nlab/show/Aurelio+Carboni">Aurelio Carboni</a>, <a class="existingWikiWord" href="/nlab/show/Stephen+Lack">Stephen Lack</a>, <a class="existingWikiWord" href="/nlab/show/Bob+Walters">R. F. C. Walters</a>, around Def. 2.5 in: <em>Introduction to extensive and distributive categories</em>, JPAA <strong>84</strong> (1993) pp. 145-158 (<a href="https://doi.org/10.1016/0022-4049(93)90035-R">doi:10.1016/0022-4049(93)90035-R</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Peter+Johnstone">Peter Johnstone</a>, Sec. A1.4.4, p. 34 in: <em><a class="existingWikiWord" href="/nlab/show/Sketches+of+an+Elephant">Sketches of an Elephant</a></em></p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on January 27, 2025 at 18:08:06. 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