CINXE.COM
coproduct (Rev #10) in nLab
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0 plus SVG 1.1//EN" "http://www.w3.org/2002/04/xhtml-math-svg/xhtml-math-svg-flat.dtd" > <html xmlns="http://www.w3.org/1999/xhtml"> <head> <title> coproduct (Rev #10) in nLab </title> <meta http-equiv="Content-Type" content="text/html; charset=UTF-8" /> <meta name="robots" content="noindex,nofollow" /> <meta name="viewport" content="width=device-width, initial-scale=1" /> <link href="/stylesheets/instiki.css?1676280126" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/mathematics.css?1660229990" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/syntax.css?1660229990" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/nlab.css?1676280126" media="all" rel="stylesheet" type="text/css" /> <link rel="stylesheet" type="text/css" href="https://cdn.jsdelivr.net/gh/dreampulse/computer-modern-web-font@master/fonts.css"/> <style type="text/css"> h1#pageName, div.info, .newWikiWord a, a.existingWikiWord, .newWikiWord a:hover, [actiontype="toggle"]:hover, #TextileHelp h3 { color: #226622; } a:visited.existingWikiWord { color: #164416; } </style> <style type="text/css"><!--/*--><![CDATA[/*><!--*/ .toc ul {margin: 0; padding: 0;} .toc ul ul {margin: 0; padding: 0 0 0 10px;} .toc li > p {margin: 0} .toc ul li {list-style-type: none; position: relative;} .toc div {border-top:1px dotted #ccc;} .rightHandSide h2 {font-size: 1.5em;color:#008B26} table.plaintable { border-collapse:collapse; margin-left:30px; border:0; } .plaintable td {border:1px solid #000; padding: 3px;} .plaintable th {padding: 3px;} .plaintable caption { font-weight: bold; font-size:1.1em; text-align:center; margin-left:30px; } /* Query boxes for questioning and answering mechanism */ div.query{ background: #f6fff3; border: solid #ce9; border-width: 2px 1px; padding: 0 1em; margin: 0 1em; max-height: 20em; overflow: auto; } /* Standout boxes for putting important text */ div.standout{ background: #fff1f1; border: solid black; border-width: 2px 1px; padding: 0 1em; margin: 0 1em; overflow: auto; } /* Icon for links to n-category arXiv documents (commented out for now i.e. disabled) a[href*="http://arxiv.org/"] { background-image: url(../files/arXiv_icon.gif); background-repeat: no-repeat; background-position: right bottom; padding-right: 22px; } */ /* Icon for links to n-category cafe posts (disabled) a[href*="http://golem.ph.utexas.edu/category"] { background-image: url(../files/n-cafe_5.gif); background-repeat: no-repeat; background-position: right bottom; padding-right: 25px; } */ /* Icon for links to pdf files (disabled) a[href$=".pdf"] { background-image: url(../files/pdficon_small.gif); background-repeat: no-repeat; background-position: right bottom; padding-right: 25px; } */ /* Icon for links to pages, etc. -inside- pdf files (disabled) a[href*=".pdf#"] { background-image: url(../files/pdf_entry.gif); background-repeat: no-repeat; background-position: right bottom; padding-right: 25px; } */ a.existingWikiWord { color: #226622; } a.existingWikiWord:visited { color: #226622; } a.existingWikiWord[title] { border: 0px; color: #aa0505; text-decoration: none; } a.existingWikiWord[title]:visited { border: 0px; color: #551111; text-decoration: none; } a[href^="http://"] { border: 0px; color: #003399; } a[href^="http://"]:visited { border: 0px; color: #330066; } a[href^="https://"] { border: 0px; color: #003399; } a[href^="https://"]:visited { border: 0px; color: #330066; } div.dropDown .hide { display: none; } div.dropDown:hover .hide { display:block; } div.clickDown .hide { display: none; } div.clickDown:focus { outline:none; } div.clickDown:focus .hide, div.clickDown:hover .hide { display: block; } div.clickDown .clickToReveal, div.clickDown:focus .clickToHide { display:block; } div.clickDown:focus .clickToReveal, div.clickDown .clickToHide { display:none; } div.clickDown .clickToReveal:after { content: "A(Hover to reveal, click to "hold")"; font-size: 60%; } div.clickDown .clickToHide:after { content: "A(Click to hide)"; font-size: 60%; } div.clickDown .clickToHide, div.clickDown .clickToReveal { white-space: pre-wrap; } .un_theorem, .num_theorem, .un_lemma, .num_lemma, .un_prop, .num_prop, .un_cor, .num_cor, .un_defn, .num_defn, .un_example, .num_example, .un_note, .num_note, .un_remark, .num_remark { margin-left: 1em; } span.theorem_label { margin-left: -1em; } .proof span.theorem_label { margin-left: 0em; } :target { background-color: #BBBBBB; border-radius: 5pt; } /*]]>*/--></style> <script src="/javascripts/prototype.js?1660229990" type="text/javascript"></script> <script src="/javascripts/effects.js?1660229990" type="text/javascript"></script> <script src="/javascripts/dragdrop.js?1660229990" type="text/javascript"></script> <script src="/javascripts/controls.js?1660229990" type="text/javascript"></script> <script src="/javascripts/application.js?1660229990" type="text/javascript"></script> <script src="/javascripts/page_helper.js?1660229990" type="text/javascript"></script> <script src="/javascripts/thm_numbering.js?1660229990" type="text/javascript"></script> <script type="text/x-mathjax-config"> <!--//--><![CDATA[//><!-- MathJax.Ajax.config.path["Contrib"] = "/MathJax"; MathJax.Hub.Config({ MathML: { useMathMLspacing: true }, "HTML-CSS": { scale: 90, extensions: ["handle-floats.js"] } }); MathJax.Hub.Queue( function () { var fos = document.getElementsByTagName('foreignObject'); for (var i = 0; i < fos.length; i++) { MathJax.Hub.Typeset(fos[i]); } }); //--><!]]> </script> <script type="text/javascript"> <!--//--><![CDATA[//><!-- window.addEventListener("DOMContentLoaded", function () { var div = document.createElement('div'); var math = document.createElementNS('http://www.w3.org/1998/Math/MathML', 'math'); document.body.appendChild(div); div.appendChild(math); // Test for MathML support comparable to WebKit version https://trac.webkit.org/changeset/203640 or higher. div.setAttribute('style', 'font-style: italic'); var mathml_unsupported = !(window.getComputedStyle(div.firstChild).getPropertyValue('font-style') === 'normal'); div.parentNode.removeChild(div); if (mathml_unsupported) { // MathML does not seem to be supported... var s = document.createElement('script'); s.src = "https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.7/MathJax.js?config=MML_HTMLorMML-full"; document.querySelector('head').appendChild(s); } else { document.head.insertAdjacentHTML("beforeend", '<style>svg[viewBox] {max-width: 100%}</style>'); } }); //--><!]]> </script> <link href="https://ncatlab.org/nlab/atom_with_headlines" rel="alternate" title="Atom with headlines" type="application/atom+xml" /> <link href="https://ncatlab.org/nlab/atom_with_content" rel="alternate" title="Atom with full content" type="application/atom+xml" /> <script type="text/javascript"> document.observe("dom:loaded", function() { generateThmNumbers(); }); </script> </head> <body> <div id="Container"> <div id="Content"> <h1 id="pageName"> <span style="float: left; margin: 0.5em 0.25em -0.25em 0"> <svg xmlns="http://www.w3.org/2000/svg" width="1.872em" height="1.8em" viewBox="0 0 190 181"> <path fill="#226622" d="M72.8 145c-1.6 17.3-15.7 10-23.6 20.2-5.6 7.3 4.8 15 11.4 15 11.5-.2 19-13.4 26.4-20.3 3.3-3 8.2-4 11.2-7.2a14 14 0 0 0 2.9-11.1c-1.4-9.6-12.4-18.6-16.9-27.2-5-9.6-10.7-27.4-24.1-27.7-17.4-.3-.4 26 4.7 30.7 2.4 2.3 5.4 4.1 7.3 6.9 1.6 2.3 2.1 5.8-1 7.2-5.9 2.6-12.4-6.3-15.5-10-8.8-10.6-15.5-23-26.2-31.8-5.2-4.3-11.8-8-18-3.7-7.3 4.9-4.2 12.9.2 18.5a81 81 0 0 0 30.7 23c3.3 1.5 12.8 5.6 10 10.7-2.5 5.2-11.7 3-15.6 1.1-8.4-3.8-24.3-21.3-34.4-13.7-3.5 2.6-2.3 7.6-1.2 11.1 2.8 9 12.2 17.2 20.9 20.5 17.3 6.7 34.3-8 50.8-12.1z"/> <path fill="#a41e32" d="M145.9 121.3c-.2-7.5 0-19.6-4.5-26-5.4-7.5-12.9-1-14.1 5.8-1.4 7.8 2.7 14.1 4.8 21.3 3.4 12 5.8 29-.8 40.1-3.6-6.7-5.2-13-7-20.4-2.1-8.2-12.8-13.2-15.1-1.9-2 9.7 9 21.2 12 30.1 1.2 4 2 8.8 6.4 10.3 6.9 2.3 13.3-4.7 17.7-8.8 12.2-11.5 36.6-20.7 43.4-36.4 6.7-15.7-13.7-14-21.3-7.2-9.1 8-11.9 20.5-23.6 25.1 7.5-23.7 31.8-37.6 38.4-61.4 2-7.3-.8-29.6-13-19.8-14.5 11.6-6.6 37.6-23.3 49.2z"/> <path fill="#193c78" d="M86.3 47.5c0-13-10.2-27.6-5.8-40.4 2.8-8.4 14.1-10.1 17-1 3.8 11.6-.3 26.3-1.8 38 11.7-.7 10.5-16 14.8-24.3 2.1-4.2 5.7-9.1 11-6.7 6 2.7 7.4 9.2 6.6 15.1-2.2 14-12.2 18.8-22.4 27-3.4 2.7-8 6.6-5.9 11.6 2 4.4 7 4.5 10.7 2.8 7.4-3.3 13.4-16.5 21.7-16 14.6.7 12 21.9.9 26.2-5 1.9-10.2 2.3-15.2 3.9-5.8 1.8-9.4 8.7-15.7 8.9-6.1.1-9-6.9-14.3-9-14.4-6-33.3-2-44.7-14.7-3.7-4.2-9.6-12-4.9-17.4 9.3-10.7 28 7.2 35.7 12 2 1.1 11 6.9 11.4 1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> </span> <span class="webName">nLab</span> coproduct (Rev #10) </h1> <div class="navigation"> <span class="skipNav"><a href='#navEnd'>Skip the Navigation Links</a> | </span> <span style="display:inline-block; width: 0.3em;"></span> <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/1939/#Item_3" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <div class='rightHandSide toc'> **<a class='existingWikiWord' href='/nlab/show/limit'>limits and colimits</a>** ## 1-Categorical * <a class='existingWikiWord' href='/nlab/show/limit'>limit and colimit</a> * <a class='existingWikiWord' href='/nlab/show/limits+and+colimits+by+example'>limits and colimits by example</a> * <a class='existingWikiWord' href='/nlab/show/commutativity+of+limits+and+colimits'>commutativity of limits and colimits</a> * <a class='existingWikiWord' href='/nlab/show/small+limit'>small limit</a> * <a class='existingWikiWord' href='/nlab/show/filtered+limit'>filtered colimit</a> * <a class='existingWikiWord' href='/nlab/show/directed+colimit'>directed colimit</a> * <a class='existingWikiWord' href='/nlab/show/sequential+limit'>sequential colimit</a> * <a class='existingWikiWord' href='/nlab/show/sifted+colimit'>sifted colimit</a> * <a class='existingWikiWord' href='/nlab/show/connected+limit'>connected limit</a>, <a class='existingWikiWord' href='/nlab/show/wide+pullback'>wide pullback</a> * <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> * <a class='existingWikiWord' href='/nlab/show/cartesian+product'>product</a>, <a class='existingWikiWord' href='/nlab/show/pullback'>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> * <a class='existingWikiWord' href='/nlab/show/finite+limit'>finite limit</a> * <a class='existingWikiWord' href='/nlab/show/exact+functor'>exact functor</a> * <a class='existingWikiWord' href='/nlab/show/Kan+extension'>Kan extension</a> * <a class='existingWikiWord' href='/nlab/show/Yoneda+extension'>Yoneda extension</a> * <a class='existingWikiWord' href='/nlab/show/weighted+limit'>weighted limit</a> * <a class='existingWikiWord' href='/nlab/show/end'>end and coend</a> * <a class='existingWikiWord' href='/nlab/show/fibered+limit'>fibered limit</a> ## 2-Categorical * <a class='existingWikiWord' href='/nlab/show/2-limit'>2-limit</a> * <a class='existingWikiWord' href='/nlab/show/inserter'>inserter</a> * <a class='existingWikiWord' href='/nlab/show/isoinserter'>isoinserter</a> * <a class='existingWikiWord' href='/nlab/show/equifier'>equifier</a> * <a class='existingWikiWord' href='/nlab/show/inverter'>inverter</a> * <a class='existingWikiWord' href='/nlab/show/PIE-limit'>PIE-limit</a> * <a class='existingWikiWord' href='/nlab/show/2-pullback'>2-pullback</a>, <a class='existingWikiWord' href='/nlab/show/comma+object'>comma object</a> ## (∞,1)-Categorical * <a class='existingWikiWord' href='/nlab/show/%28%E2%88%9E%2C1%29-limit'>(∞,1)-limit</a> * <a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29-pullback'>(∞,1)-pullback</a> * <a class='existingWikiWord' href='/nlab/show/fiber+sequence'>fiber sequence</a> ### Model-categorical * <a class='existingWikiWord' href='/nlab/show/homotopy+Kan+extension'>homotopy Kan extension</a> * <a class='existingWikiWord' href='/nlab/show/homotopy+limit'>homotopy limit</a> * <a class='existingWikiWord' href='/nlab/show/homotopy+product'>homotopy product</a> * <a class='existingWikiWord' href='/nlab/show/homotopy+equalizer'>homotopy equalizer</a> * <a class='existingWikiWord' href='/nlab/show/fiber+sequence'>homotopy fiber</a> * <a class='existingWikiWord' href='/nlab/show/mapping+cone'>mapping cone</a> * <a class='existingWikiWord' href='/nlab/show/homotopy+pullback'>homotopy pullback</a> * <a class='existingWikiWord' href='/nlab/show/homotopy+totalization'>homotopy totalization</a> * <a class='existingWikiWord' href='/nlab/show/homotopy+coend'>homotopy end</a> * <a class='existingWikiWord' href='/nlab/show/homotopy+limit'>homotopy colimit</a> * <a class='existingWikiWord' href='/nlab/show/homotopy+coproduct'>homotopy coproduct</a> * <a class='existingWikiWord' href='/nlab/show/homotopy+coequalizer'>homotopy coequalizer</a> * <a class='existingWikiWord' href='/nlab/show/cofiber+sequence'>homotopy cofiber</a> * <a class='existingWikiWord' href='/nlab/show/mapping+cocone'>mapping cocone</a> * <a class='existingWikiWord' href='/nlab/show/homotopy+pushout'>homotopy pushout</a> * <a class='existingWikiWord' href='/nlab/show/homotopy+realization'>homotopy realization</a> * <a class='existingWikiWord' href='/nlab/show/homotopy+coend'>homotopy coend</a> <div> <p> <a href='/nlab/edit/infinity-limits+-+contents'>Edit this sidebar</a> </p> </div> </div> <h1 id='coproducts'>Coproducts</h1> <div class='maruku_toc'><ul><li><a href='#definition'>Definition</a></li><li><a href='#examples'>Examples</a></li><li><a href='#remarks'>Remarks</a></li></ul></div> <h2 id='definition'>Definition</h2> <p>A <strong>coproduct</strong> of two <a class='existingWikiWord' href='/nlab/show/object'>objects</a> <math class='maruku-mathml' display='inline' id='mathml_65e1d69b26c35cf62113e4f924cf8ffb47a2dab6_1' xmlns='http://www.w3.org/1998/Math/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> <math class='maruku-mathml' display='inline' id='mathml_65e1d69b26c35cf62113e4f924cf8ffb47a2dab6_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> is an object <math class='maruku-mathml' display='inline' id='mathml_65e1d69b26c35cf62113e4f924cf8ffb47a2dab6_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow><annotation encoding='application/x-tex'>a+b</annotation></semantics></math> (also written <math class='maruku-mathml' display='inline' id='mathml_65e1d69b26c35cf62113e4f924cf8ffb47a2dab6_4' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_65e1d69b26c35cf62113e4f924cf8ffb47a2dab6_5' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_65e1d69b26c35cf62113e4f924cf8ffb47a2dab6_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>⨿</mo></mrow><annotation encoding='application/x-tex'>\amalg</annotation></semantics></math>’) equipped with <a class='existingWikiWord' href='/nlab/show/morphism'>morphisms</a> <math class='maruku-mathml' display='inline' id='mathml_65e1d69b26c35cf62113e4f924cf8ffb47a2dab6_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mo>:</mo><mi>a</mi><mo>→</mo><mi>a</mi><mo>+</mo><mi>b</mi></mrow><annotation encoding='application/x-tex'>i:a\to a+b</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_65e1d69b26c35cf62113e4f924cf8ffb47a2dab6_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>j</mi><mo>:</mo><mi>b</mi><mo>→</mo><mi>a</mi><mo>+</mo><mi>b</mi></mrow><annotation encoding='application/x-tex'>j:b\to a+b</annotation></semantics></math> such that for any object <math class='maruku-mathml' display='inline' id='mathml_65e1d69b26c35cf62113e4f924cf8ffb47a2dab6_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>c</mi></mrow><annotation encoding='application/x-tex'>c</annotation></semantics></math> with morphisms <math class='maruku-mathml' display='inline' id='mathml_65e1d69b26c35cf62113e4f924cf8ffb47a2dab6_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>:</mo><mi>a</mi><mo>→</mo><mi>c</mi></mrow><annotation encoding='application/x-tex'>f:a\to c</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_65e1d69b26c35cf62113e4f924cf8ffb47a2dab6_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>g</mi><mo>:</mo><mi>b</mi><mo>→</mo><mi>c</mi></mrow><annotation encoding='application/x-tex'>g:b\to c</annotation></semantics></math>, there is a unique morphism <math class='maruku-mathml' display='inline' id='mathml_65e1d69b26c35cf62113e4f924cf8ffb47a2dab6_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi><mo>:</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo>→</mo><mi>c</mi></mrow><annotation encoding='application/x-tex'>p:a+b\to c</annotation></semantics></math> such that <math class='maruku-mathml' display='inline' id='mathml_65e1d69b26c35cf62113e4f924cf8ffb47a2dab6_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi><mo>∘</mo><mi>i</mi><mo>=</mo><mi>f</mi></mrow><annotation encoding='application/x-tex'>p \circ i=f</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_65e1d69b26c35cf62113e4f924cf8ffb47a2dab6_14' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi><mo>∘</mo><mi>j</mi><mo>=</mo><mi>g</mi></mrow><annotation encoding='application/x-tex'>p \circ j=g</annotation></semantics></math>.</p> <p>This map <math class='maruku-mathml' display='inline' id='mathml_65e1d69b26c35cf62113e4f924cf8ffb47a2dab6_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi></mrow><annotation encoding='application/x-tex'>p</annotation></semantics></math> is called the <strong><a class='existingWikiWord' href='/nlab/show/copairing'>copairing</a></strong> of <math class='maruku-mathml' display='inline' id='mathml_65e1d69b26c35cf62113e4f924cf8ffb47a2dab6_16' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi></mrow><annotation encoding='application/x-tex'>f</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_65e1d69b26c35cf62113e4f924cf8ffb47a2dab6_17' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>g</mi></mrow><annotation encoding='application/x-tex'>g</annotation></semantics></math> and is often denoted <math class='maruku-mathml' display='inline' id='mathml_65e1d69b26c35cf62113e4f924cf8ffb47a2dab6_18' xmlns='http://www.w3.org/1998/Math/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 <math class='maruku-mathml' display='inline' id='mathml_65e1d69b26c35cf62113e4f924cf8ffb47a2dab6_19' xmlns='http://www.w3.org/1998/Math/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:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_65e1d69b26c35cf62113e4f924cf8ffb47a2dab6_20' xmlns='http://www.w3.org/1998/Math/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></div> <h2 id='examples'>Examples</h2> <ul> <li> <p>In <a class='existingWikiWord' href='/nlab/show/Set'>Set</a>, the coproduct of a family of sets <math class='maruku-mathml' display='inline' id='mathml_65e1d69b26c35cf62113e4f924cf8ffb47a2dab6_21' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_65e1d69b26c35cf62113e4f924cf8ffb47a2dab6_22' xmlns='http://www.w3.org/1998/Math/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> </li> <li> <p>In <a class='existingWikiWord' href='/nlab/show/Top'>Top</a>, the coproduct of a family of spaces <math class='maruku-mathml' display='inline' id='mathml_65e1d69b26c35cf62113e4f924cf8ffb47a2dab6_23' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_65e1d69b26c35cf62113e4f924cf8ffb47a2dab6_24' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_65e1d69b26c35cf62113e4f924cf8ffb47a2dab6_25' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_65e1d69b26c35cf62113e4f924cf8ffb47a2dab6_26' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_65e1d69b26c35cf62113e4f924cf8ffb47a2dab6_27' xmlns='http://www.w3.org/1998/Math/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+category'>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</a>, whose underlying set is <em>not</em> a disjoint union. 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/Cat'>Cat</a>, the coproduct of a family of categories <math class='maruku-mathml' display='inline' id='mathml_65e1d69b26c35cf62113e4f924cf8ffb47a2dab6_28' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='block' id='mathml_65e1d69b26c35cf62113e4f924cf8ffb47a2dab6_29' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='block' id='mathml_65e1d69b26c35cf62113e4f924cf8ffb47a2dab6_30' xmlns='http://www.w3.org/1998/Math/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 columnalign='right left right left right left right left right left' columnspacing='0em' displaystyle='true'><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/horizontal+categorification'>oidifications</a>: the coproduct becomes a disjoint union again.</p> </li> </ul> <h2 id='remarks'>Remarks</h2> <ul> <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>One can define in a similar way a coproduct of any family of objects. A coproduct of the <a class='existingWikiWord' href='/nlab/show/empty+set'>empty</a> family is an <a class='existingWikiWord' href='/nlab/show/initial+object'>initial object</a>.</p> </li> <li> <p>A coproduct is a special case of a <a class='existingWikiWord' href='/nlab/show/colimit'>colimit</a> in which the diagram category is <a class='existingWikiWord' href='/nlab/show/discrete+category'>discrete</a>.</p> </li> <li> <p>A coproduct in <math class='maruku-mathml' display='inline' id='mathml_65e1d69b26c35cf62113e4f924cf8ffb47a2dab6_31' xmlns='http://www.w3.org/1998/Math/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/cartesian+product'>product</a> in the <a class='existingWikiWord' href='/nlab/show/opposite+category'>opposite category</a> <math class='maruku-mathml' display='inline' id='mathml_65e1d69b26c35cf62113e4f924cf8ffb47a2dab6_32' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>C</mi> <mi>op</mi></msup></mrow><annotation encoding='application/x-tex'>C^{op}</annotation></semantics></math>.</p> </li> </ul> <p> </p> </div> <!-- Revision --> <div class="revisedby"> <p> Revision on May 19, 2010 at 08:54:54 by <a href="/nlab/author/Urs+Schreiber" style="color: #005c19">Urs Schreiber</a> See the <a href="/nlab/history/coproduct" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="https://nforum.ncatlab.org/discussion/1939/#Item_3">Discuss</a><span class="backintime"><a href="/nlab/revision/coproduct/11" accesskey="F" class="navlinkbackintime" id="to_next_revision" rel="nofollow">Next revision</a> (13 more)</span><span class="backintime"><a href="/nlab/revision/coproduct/9" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a> (9 more)</span><a href="/nlab/show/coproduct" class="navlink" id="to_current_revision">Current version of page</a><a href="/nlab/revision/diff/coproduct/10" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/coproduct" accesskey="S" class="navlink" id="history" rel="nofollow">History (22 revisions)</a><a href="/nlab/rollback/coproduct?rev=10" class="navlink" id="rollback" rel="nofollow">Rollback</a> <a href="/nlab/revision/coproduct/10/cite" style="color: black">Cite</a> <a href="/nlab/source/coproduct/10" id="view_source" rel="nofollow">Source</a> </div> </div> <!-- Content --> </div> <!-- Container --> </body> </html>