CINXE.COM

<!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> abelian category in nLab </title> <meta http-equiv="Content-Type" content="text/html; charset=UTF-8" /> <meta name="robots" content="index,follow" /> <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[/*></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[//><!]]> </script> <script type="text/javascript"> <![CDATA[//><!]]> </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> abelian category </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/4094/#Item_11" 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"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="enriched_category_theory">Enriched category theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a></strong></p> <h2 id="background">Background</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a>, <a class="existingWikiWord" href="/nlab/show/closed+monoidal+category">closed monoidal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cosmos">cosmos</a>, <a class="existingWikiWord" href="/nlab/show/multicategory">multicategory</a>, <a class="existingWikiWord" href="/nlab/show/bicategory">bicategory</a>, <a class="existingWikiWord" href="/nlab/show/double+category">double category</a>, <a class="existingWikiWord" href="/nlab/show/virtual+double+category">virtual double category</a></p> </li> </ul> <h2 id="basic_concepts">Basic concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+category">enriched category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+functor">enriched functor</a>, <a class="existingWikiWord" href="/nlab/show/profunctor">profunctor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+natural+transformation">enriched natural transformation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+adjoint+functor">enriched adjoint functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+product+category">enriched product category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+functor+category">enriched functor category</a></p> </li> </ul> <h2 id="universal_constructions">Universal constructions</h2> <ul> <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> </ul> <h2 id="extra_stuff_structure_property">Extra stuff, structure, property</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/copowering">copowering</a> (<a class="existingWikiWord" href="/nlab/show/tensoring">tensoring</a>)</p> <p><a class="existingWikiWord" href="/nlab/show/powering">powering</a> (<a class="existingWikiWord" href="/nlab/show/cotensoring">cotensoring</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+enriched+category">monoidal enriched category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cartesian+closed+enriched+category">cartesian closed enriched category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+cartesian+closed+enriched+category">locally cartesian closed enriched category</a></p> </li> </ul> </li> </ul> <h3 id="homotopical_enrichment">Homotopical enrichment</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+homotopical+category">enriched homotopical category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+model+category">enriched model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+homotopical+presheaves">model structure on homotopical presheaves</a></p> </li> </ul> </div></div> <h4 id="additive_and_abelian_categories">Additive and abelian categories</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/additive+and+abelian+categories">additive and abelian categories</a></strong></p> <h2 id="context_and_background">Context and background</h2> <ul> <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/homological+algebra">homological algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable homotopy theory</a></p> </li> </ul> <h2 id="categories">Categories</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Ab-enriched+category">Ab-enriched category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pre-additive+category">pre-additive category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/pseudo-abelian+category">pseudo-abelian category</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/additive+category">additive category</a>,</p> </li> <li> <p>(AB1) <a class="existingWikiWord" href="/nlab/show/pre-abelian+category">pre-abelian category</a></p> </li> <li> <p>(AB2) <a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a></p> </li> <li> <p>(AB5) <a class="existingWikiWord" href="/nlab/show/Grothendieck+category">Grothendieck category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Quillen+exact+category">Quillen exact category</a></p> </li> </ul> <h2 id="functors">Functors</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/additive+functor">additive functor</a></p> </li> <li> <p>left/right <a class="existingWikiWord" href="/nlab/show/exact+functor">exact functor</a></p> </li> </ul> <h2 id="derived_categories">Derived categories</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+category">derived category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+functor">derived functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/triangulated+category">triangulated category</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/additive+and+abelian+categories+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="homological_algebra">Homological algebra</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/homological+algebra">homological algebra</a></strong></p> <p>(also <a class="existingWikiWord" href="/nlab/show/nonabelian+homological+algebra">nonabelian homological algebra</a>)</p> <p><em><a class="existingWikiWord" href="/schreiber/show/Introduction+to+Homological+Algebra">Introduction</a></em></p> <p><strong>Context</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/additive+and+abelian+categories">additive and abelian categories</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Ab-enriched+category">Ab-enriched category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pre-additive+category">pre-additive category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/additive+category">additive category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pre-abelian+category">pre-abelian category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+category">Grothendieck category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaves">abelian sheaves</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/semi-abelian+category">semi-abelian category</a></p> </li> </ul> <p><strong>Basic definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/kernel">kernel</a>, <a class="existingWikiWord" href="/nlab/show/cokernel">cokernel</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complex">complex</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential">differential</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homology">homology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+chain+complexes">category of chain complexes</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+map">chain map</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+homotopy">chain homotopy</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+homology+and+cohomology">chain homology and cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quasi-isomorphism">quasi-isomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homological+resolution">homological resolution</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cochain+on+a+simplicial+set">simplicial homology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+homology">generalized homology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/exact+sequence">exact sequence</a>,</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/short+exact+sequence">short exact sequence</a>, <a class="existingWikiWord" href="/nlab/show/long+exact+sequence">long exact sequence</a>, <a class="existingWikiWord" href="/nlab/show/split+exact+sequence">split exact sequence</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/injective+object">injective object</a>, <a class="existingWikiWord" href="/nlab/show/projective+object">projective object</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/injective+resolution">injective resolution</a>, <a class="existingWikiWord" href="/nlab/show/projective+resolution">projective resolution</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/flat+resolution">flat resolution</a></p> </li> </ul> </li> </ul> <p><strong>Stable homotopy theory notions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+category">derived category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/triangulated+category">triangulated category</a>, <a class="existingWikiWord" href="/nlab/show/enhanced+triangulated+category">enhanced triangulated category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+model+category">stable model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pretriangulated+dg-category">pretriangulated dg-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E-category">A-∞-category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+chain+complexes">(∞,1)-category of chain complexes</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+functor">derived functor</a>, <a class="existingWikiWord" href="/nlab/show/derived+functor+in+homological+algebra">derived functor in homological algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Tor">Tor</a>, <a class="existingWikiWord" href="/nlab/show/Ext">Ext</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+limit">homotopy limit</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+colimit">homotopy colimit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/lim%5E1+and+Milnor+sequences">lim^1 and Milnor sequences</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fr-code">fr-code</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a></p> </li> </ul> <p><strong>Constructions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/double+complex">double complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Koszul-Tate+resolution">Koszul-Tate resolution</a>, <a class="existingWikiWord" href="/nlab/show/BRST-BV+complex">BRST-BV complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+sequence">spectral sequence</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+sequence+of+a+filtered+complex">spectral sequence of a filtered complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+sequence+of+a+double+complex">spectral sequence of a double complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+spectral+sequence">Grothendieck spectral sequence</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Leray+spectral+sequence">Leray spectral sequence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Serre+spectral+sequence">Serre spectral sequence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hochschild-Serre+spectral+sequence">Hochschild-Serre spectral sequence</a></p> </li> </ul> </li> </ul> </li> </ul> <p><strong>Lemmas</strong></p> <p><a class="existingWikiWord" href="/nlab/show/diagram+chasing">diagram chasing</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/3x3+lemma">3x3 lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/four+lemma">four lemma</a>, <a class="existingWikiWord" href="/nlab/show/five+lemma">five lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/snake+lemma">snake lemma</a>, <a class="existingWikiWord" href="/nlab/show/connecting+homomorphism">connecting homomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/horseshoe+lemma">horseshoe lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Baer%27s+criterion">Baer's criterion</a></p> </li> </ul> <p><a class="existingWikiWord" href="/nlab/show/Schanuel%27s+lemma">Schanuel's lemma</a></p> <p><strong>Homology theories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/singular+homology">singular homology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cyclic+homology">cyclic homology</a></p> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Dold-Kan+correspondence">Dold-Kan correspondence</a> / <a class="existingWikiWord" href="/nlab/show/monoidal+Dold-Kan+correspondence">monoidal</a>, <a class="existingWikiWord" href="/nlab/show/operadic+Dold-Kan+correspondence">operadic</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Moore+complex">Moore complex</a>, <a class="existingWikiWord" href="/nlab/show/Alexander-Whitney+map">Alexander-Whitney map</a>, <a class="existingWikiWord" href="/nlab/show/Eilenberg-Zilber+map">Eilenberg-Zilber map</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Eilenberg-Zilber+theorem">Eilenberg-Zilber theorem</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/cochain+on+a+simplicial+set">cochain on a simplicial set</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+coefficient+theorem">universal coefficient theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K%C3%BCnneth+theorem">Künneth theorem</a></p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#PropertiesGeneral'>General</a></li> <li><a href='#FactorizationOfMorphisms'>Factorization of morphisms</a></li> <li><a href='#CanonicalAbEnrichment'>Canonical <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ab</mi></mrow><annotation encoding="application/x-tex">Ab</annotation></semantics></math>-enrichment</a></li> <li><a href='#RelationToToposes'>Relation to exactness properties of toposes</a></li> <li><a href='#EmbeddingTheorems'>Embedding theorems</a></li> <ul> <li><a href='#counterexamples'>Counterexamples</a></li> <li><a href='#EmbeddingIntoAb'>Embedding into <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ab</mi></mrow><annotation encoding="application/x-tex">Ab</annotation></semantics></math></a></li> <li><a href='#FreydMitchellEmbedding'>Freyd-Mitchell embedding into <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mi>Mod</mi></mrow><annotation encoding="application/x-tex">R Mod</annotation></semantics></math></a></li> <ul> <li><a href='#mitchells_embedding_theorem'>Mitchell’s Embedding Theorem</a></li> </ul> </ul> </ul> <li><a href='#examples'>Examples</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>The notion of <em>abelian category</em> is an abstraction of basic properties of the category <a class="existingWikiWord" href="/nlab/show/Ab">Ab</a> of <a class="existingWikiWord" href="/nlab/show/abelian+groups">abelian groups</a>, more generally of the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Mod">Mod</a> of <a class="existingWikiWord" href="/nlab/show/modules">modules</a> over some <a class="existingWikiWord" href="/nlab/show/ring">ring</a>, and still more generally of categories of <a class="existingWikiWord" href="/nlab/show/sheaves">sheaves</a> of abelian groups and of modules. It is such that much of the <a class="existingWikiWord" href="/nlab/show/homological+algebra">homological algebra</a> of <a class="existingWikiWord" href="/nlab/show/chain+complexes">chain complexes</a> can be developed inside every abelian category.</p> <p>The concept of abelian categories is one in a sequence of notions of <a class="existingWikiWord" href="/nlab/show/additive+and+abelian+categories">additive and abelian categories</a>.</p> <p>While additive categories differ significantly from <a class="existingWikiWord" href="/nlab/show/toposes">toposes</a>, there is an intimate relation between abelian categories and toposes. See <em><a class="existingWikiWord" href="/nlab/show/AT+category">AT category</a></em> for more on that.</p> <h2 id="definition">Definition</h2> <p>Recall the following fact about <a class="existingWikiWord" href="/nlab/show/pre-abelian+categories">pre-abelian categories</a> from <a href="pre-abelian+category#DecompositionOfMorphisms">this proposition</a>, discussed there:</p> <div class="num_prop" id="DecompositionOfMorphisms"> <h6 id="proposition">Proposition</h6> <p>Every <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a> <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>B</mi></mrow><annotation encoding="application/x-tex">f \colon A\to B</annotation></semantics></math> in a <a class="existingWikiWord" href="/nlab/show/pre-abelian+category">pre-abelian category</a> has a canonical de-<a class="existingWikiWord" href="/nlab/show/composition">composition</a></p> <div class="maruku-equation" id="eq:PreImageFactorization"><span class="maruku-eq-number">(1)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>A</mi><mover><mo>↠</mo><mrow><mspace width="thickmathspace"></mspace><mi>p</mi><mspace width="thickmathspace"></mspace></mrow></mover><mo lspace="0em" rspace="thinmathspace">coker</mo><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>ker</mi><mi>f</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mover><mo>→</mo><mrow><mspace width="thickmathspace"></mspace><mover><mi>f</mi><mo>¯</mo></mover><mspace width="thickmathspace"></mspace></mrow></mover><mi>ker</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mo lspace="0em" rspace="thinmathspace">coker</mo><mi>f</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mover><mo>↪</mo><mrow><mspace width="thickmathspace"></mspace><mi>i</mi><mspace width="thickmathspace"></mspace></mrow></mover><mi>B</mi></mrow><annotation encoding="application/x-tex"> A \overset{\; p \;}{\twoheadrightarrow} \coker\big(\ker f\big) \xrightarrow{\; \overline{f} \;} \ker\big(\coker f\big) \xhookrightarrow{\; i \;} B </annotation></semantics></math></div> <p>where:</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/cokernel">cokernel</a>, hence an <a class="existingWikiWord" href="/nlab/show/epimorphism">epimorphism</a>,</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/kernel">kernel</a>, hence a <a class="existingWikiWord" href="/nlab/show/monomorphism">monomorphism</a>.</p> </li> </ul> </div> <div class="num_defn" id="AbelianCategory"> <h6 id="definition_2">Definition</h6> <p>An <strong>abelian category</strong> is a <a class="existingWikiWord" href="/nlab/show/pre-abelian+category">pre-abelian category</a> satisfying the following equivalent conditions.</p> <ol> <li> <p>For every <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math>, the canonical morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>f</mi><mo stretchy="false">¯</mo></mover><mo lspace="verythinmathspace">:</mo><mi>coker</mi><mo stretchy="false">(</mo><mi>ker</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><mi>ker</mi><mo stretchy="false">(</mo><mi>coker</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\bar{f} \colon coker(ker(f)) \to ker(coker(f))</annotation></semantics></math> <a class="maruku-eqref" href="#eq:PreImageFactorization">(1)</a> from prop. <a class="maruku-ref" href="#DecompositionOfMorphisms"></a> is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> (hence providing an <a class="existingWikiWord" href="/nlab/show/image">image</a> factorization <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>↠</mo><mi>im</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>↪</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A \twoheadrightarrow im(f) \hookrightarrow B</annotation></semantics></math>).</p> </li> <li> <p>Every <a class="existingWikiWord" href="/nlab/show/monomorphism">monomorphism</a> is a <a class="existingWikiWord" href="/nlab/show/kernel">kernel</a> and every <a class="existingWikiWord" href="/nlab/show/epimorphism">epimorphism</a> is a <a class="existingWikiWord" href="/nlab/show/cokernel">cokernel</a>.</p> </li> </ol> </div> <div class="num_prop"> <h6 id="proposition_2">Proposition</h6> <p>These two conditions are indeed equivalent.</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>The first condition implies that if <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> is a <a class="existingWikiWord" href="/nlab/show/monomorphism">monomorphism</a> then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>≅</mo><mi>ker</mi><mo stretchy="false">(</mo><mo lspace="0em" rspace="thinmathspace">coker</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f \cong \ker(\coker(f))</annotation></semantics></math> (in the category of objects over <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>) so <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> is a kernel. Dually if <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> is an <a class="existingWikiWord" href="/nlab/show/epimorphism">epimorphism</a> it follows that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>≅</mo><mi>coker</mi><mo stretchy="false">(</mo><mi>ker</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f \cong coker(ker(f))</annotation></semantics></math>. So (1) implies (2).</p> <p>The converse can be found in, among other places, Chapter VIII of (<a href="#MacLane">MacLane</a>).</p> </div> <h2 id="properties">Properties</h2> <h3 id="PropertiesGeneral">General</h3> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>The notion of abelian category is self-dual: <a class="existingWikiWord" href="/nlab/show/opposite+category">opposite</a> of any abelian category is abelian.</p> </div> <div class="num_remark" id="RegularEpisAndMonos"> <h6 id="remark_2">Remark</h6> <p>By the second formulation of the definition <a class="maruku-ref" href="#AbelianCategory"></a>, in an abelian category</p> <ul> <li> <p>every <a class="existingWikiWord" href="/nlab/show/monomorphism">monomorphism</a> is a <a class="existingWikiWord" href="/nlab/show/regular+monomorphism">regular monomorphism</a>;</p> </li> <li> <p>every <a class="existingWikiWord" href="/nlab/show/epimorphism">epimorphism</a> is a <a class="existingWikiWord" href="/nlab/show/regular+epimorphism">regular epimorphism</a>.</p> </li> </ul> <p>It follows that every abelian category is a <em><a class="existingWikiWord" href="/nlab/show/balanced+category">balanced category</a></em>.</p> </div> <p> <div class='num_prop' id='PullbackPreservesEpimorphisms'> <h6>Proposition</h6> <p>In an <a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a>, <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> preserves <a class="existingWikiWord" href="/nlab/show/epimorphisms">epimorphisms</a> and <a class="existingWikiWord" href="/nlab/show/pushout">pushout</a> preserves <a class="existingWikiWord" href="/nlab/show/monomorphisms">monomorphisms</a>.</p> </div> Because every abelian category is a <a class="existingWikiWord" href="/nlab/show/regular+category">regular category</a>. For an explicit proof see, e.g., <a href="#Selick">Selick, Prop. 1.3.13</a>.</p> <h3 id="FactorizationOfMorphisms">Factorization of morphisms</h3> <div class="num_prop"> <h6 id="proposition_3">Proposition</h6> <p>In an abelian category every morphism decomposes <a class="existingWikiWord" href="/nlab/show/generalized+the">uniquely up to a unique isomorphism</a> into the composition of an <a class="existingWikiWord" href="/nlab/show/epimorphism">epimorphism</a> and a <a class="existingWikiWord" href="/nlab/show/monomorphism">monomorphism</a>, via prop <a class="maruku-ref" href="#DecompositionOfMorphisms"></a> combined with def. <a class="maruku-ref" href="#AbelianCategory"></a>.</p> <p>Since by remark <a class="maruku-ref" href="#RegularEpisAndMonos"></a> every monic is <a class="existingWikiWord" href="/nlab/show/regular+monomorphism">regular</a>, hence <a class="existingWikiWord" href="/nlab/show/strong+monomorphism">strong</a>, it follows that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>epi</mi><mo>,</mo><mi>mono</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(epi, mono)</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/orthogonal+factorization+system">orthogonal factorization system</a> in an abelian category; see at <em><a class="existingWikiWord" href="/nlab/show/%28epi%2C+mono%29+factorization+system">(epi, mono) factorization system</a></em>.</p> </div> <div class="num_remark"> <h6 id="remark_3">Remark</h6> <p>Some references claim that this property characterizes abelian categories among pre-abelian ones, but it is not clear to the authors of this page why this should be so, although we do not currently have a counterexample; see <a href="http://nforum.mathforge.org/discussion/4094/?Focus=33415#Comment_33415">this discussion</a>.</p> </div> <h3 id="CanonicalAbEnrichment">Canonical <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ab</mi></mrow><annotation encoding="application/x-tex">Ab</annotation></semantics></math>-enrichment</h3> <p>The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ab</mi></mrow><annotation encoding="application/x-tex">Ab</annotation></semantics></math>-enrichment of an abelian category need not be specified a priori. If an arbitrary (not necessarily pre-additive) <a class="existingWikiWord" href="/nlab/show/locally+small+category">locally small</a> 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> has a <a class="existingWikiWord" href="/nlab/show/zero+object">zero object</a>, binary products and coproducts, kernels, cokernels and the property that every monic is a kernel arrow and every epi is a cokernel arrow (so that all monos and epis are <a class="existingWikiWord" href="/nlab/show/normal+monomorphism">normal</a>), then it can be equipped with a unique addition on the morphism sets such that composition is bilinear and <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 abelian with respect to this structure. However, in most examples, the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ab</mi></mrow><annotation encoding="application/x-tex">Ab</annotation></semantics></math>-enrichment is evident from the start and does not need to be constructed in this way. (A similar statement is true for <a class="existingWikiWord" href="/nlab/show/additive+categories">additive categories</a>, although the most natural result in that case gives only enrichment over abelian <a class="existingWikiWord" href="/nlab/show/monoids">monoids</a>; see <a class="existingWikiWord" href="/nlab/show/semiadditive+category">semiadditive category</a>.)</p> <p>The last point is of relevance in particular for <a class="existingWikiWord" href="/nlab/show/infinity-category">higher categorical</a> generalizations of additive categories. See for instance <a href="http://www.math.harvard.edu/~lurie/papers/DAG-I.pdf#page=5">remark 2.14, p. 5</a> of <a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>‘s <a class="existingWikiWord" href="/nlab/show/Stable+Infinity-Categories">Stable Infinity-Categories</a>.</p> <h3 id="RelationToToposes">Relation to exactness properties of toposes</h3> <p>The <a class="existingWikiWord" href="/nlab/show/exactness+properties">exactness properties</a> of abelian categories have many features in common with exactness properties of <a class="existingWikiWord" href="/nlab/show/toposes">toposes</a> or of <a class="existingWikiWord" href="/nlab/show/pretoposes">pretoposes</a>.</p> <p><a href="AT+category#Freyd99">Freyd (1999)</a> gave a sharp description of the properties shared by these categories, introducing a new concept called <em><a class="existingWikiWord" href="/nlab/show/AT+categories">AT categories</a></em> (for “abelian-topos”), and showing convincingly that the difference between the A and the T can be concentrated precisely in the difference of the behavior of the initial object.</p> <h3 id="EmbeddingTheorems">Embedding theorems</h3> <p>Not every <a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a> is a <a class="existingWikiWord" href="/nlab/show/concrete+category">concrete category</a> such as <a class="existingWikiWord" href="/nlab/show/Ab">Ab</a> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Mod">Mod</a>. But for many proofs in <a class="existingWikiWord" href="/nlab/show/homological+algebra">homological algebra</a> it is very convenient to have a concrete abelian category, for that allows one to check the behaviour of morphisms on actual <em>elements</em> of the sets underlying the <a class="existingWikiWord" href="/nlab/show/objects">objects</a>.</p> <p>The following <em>embedding theorems</em>, however, show that under good conditions an abelian category can be <em>embedded</em> into <a class="existingWikiWord" href="/nlab/show/Ab">Ab</a> as a <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> by an <a class="existingWikiWord" href="/nlab/show/exact+functor">exact functor</a>, and generally can be embedded this way into <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mi>Mod</mi></mrow><annotation encoding="application/x-tex">R Mod</annotation></semantics></math>, for some ring <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>. This is the celebrated <em><a class="existingWikiWord" href="/nlab/show/Freyd-Mitchell+embedding+theorem">Freyd-Mitchell embedding theorem</a></em> discussed <a href="#FreydMitchellEmbedding">below</a>.</p> <p>This implies for instance that proofs about <a class="existingWikiWord" href="/nlab/show/exact+sequence">exactness of sequences</a> in an abelian category can always be obtained by a naive argument on elements – called a “<a class="existingWikiWord" href="/nlab/show/diagram+chasing">diagram chase</a>” – because that does hold true after such an embedding, and the exactness of the embedding means that the notion of exact sequences is preserved by it.</p> <p>Alternatively, one can reason with <a class="existingWikiWord" href="/nlab/show/generalized+elements">generalized elements</a> in an abelian category, without explicitly embedding it into a larger concrete category, see at <em><a class="existingWikiWord" href="/nlab/show/element+in+an+abelian+category">element in an abelian category</a></em>. But under suitable conditions this comes down to working subject to an embedding into <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ab</mi></mrow><annotation encoding="application/x-tex">Ab</annotation></semantics></math>, see the discussion at <em><a href="#EmbeddingIntoAb">Embedding into Ab</a></em> below.</p> <h4 id="counterexamples">Counterexamples</h4> <p>First of all, it’s easy to see that not every abelian category is equivalent to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Mod">Mod</a> for some ring <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>. The reason is that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mi>Mod</mi></mrow><annotation encoding="application/x-tex">R Mod</annotation></semantics></math> has all <a class="existingWikiWord" href="/nlab/show/small+category">small category</a> <a class="existingWikiWord" href="/nlab/show/limits">limits</a> and <a class="existingWikiWord" href="/nlab/show/colimits">colimits</a>. For a <a class="existingWikiWord" href="/nlab/show/Noetherian+ring">Noetherian ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> the category of <a class="existingWikiWord" href="/nlab/show/finitely+generated+module">finitely generated</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-modules is an abelian category that lacks these properties.</p> <h4 id="EmbeddingIntoAb">Embedding into <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ab</mi></mrow><annotation encoding="application/x-tex">Ab</annotation></semantics></math></h4> <p>(…)</p> <p>(<a href="#Bergman">Bergman 1974</a>)</p> <p>(…)</p> <h4 id="FreydMitchellEmbedding">Freyd-Mitchell embedding into <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mi>Mod</mi></mrow><annotation encoding="application/x-tex">R Mod</annotation></semantics></math></h4> <div class="num_thm"> <h6 id="mitchells_embedding_theorem">Mitchell’s Embedding Theorem</h6> <p>Every small abelian category admits a <a class="existingWikiWord" href="/nlab/show/full+functor">full</a>, <a class="existingWikiWord" href="/nlab/show/faithful+functor">faithful</a> and <a class="existingWikiWord" href="/nlab/show/exact+functor">exact</a> functor to the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mi>Mod</mi></mrow><annotation encoding="application/x-tex">R Mod</annotation></semantics></math> for some ring <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>.</p> </div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>This result can be found as Theorem 7.34 on page 150 of Peter Freyd’s book <a href="http://www.emis.de/journals/TAC/reprints/articles/3/tr3.pdf#page=176">Abelian Categories</a>. His terminology is a bit outdated, in that he calls an abelian category “fully abelian” if admits a full and faithful exact functor to a category of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-modules. See also the <a href="http://en.wikipedia.org/wiki/Mitchell%27s_embedding_theorem">Wikipedia article</a> for the idea of the proof.</p> </div> <p>For more see at <em><a class="existingWikiWord" href="/nlab/show/Freyd-Mitchell+embedding+theorem">Freyd-Mitchell embedding theorem</a></em>.</p> <p>We can also characterize which abelian categories <em>are</em> equivalent to a category of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-modules:</p> <div class="num_theorem"> <h6 id="theorem">Theorem</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">C</annotation></semantics></math> be an abelian category. 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">C</annotation></semantics></math> has all <a class="existingWikiWord" href="/nlab/show/small+category">small</a> <a class="existingWikiWord" href="/nlab/show/coproducts">coproducts</a> and has a <a class="existingWikiWord" href="/nlab/show/compact+object">compact</a> <a class="existingWikiWord" href="/nlab/show/projective+object">projective</a> <a class="existingWikiWord" href="/nlab/show/generator">generator</a>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>≃</mo><mi>R</mi><mi>Mod</mi></mrow><annotation encoding="application/x-tex">C \simeq R Mod</annotation></semantics></math> for some ring <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>. In fact, in this situation we can take <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>=</mo><mi>C</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>x</mi><msup><mo stretchy="false">)</mo> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">R = C(x,x)^{op}</annotation></semantics></math> where <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> is any compact projective generator. Conversely, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>≃</mo><mi>R</mi><mi>Mod</mi></mrow><annotation encoding="application/x-tex">C \simeq R Mod</annotation></semantics></math>, then <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> has all small coproducts and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>=</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">x = R</annotation></semantics></math> is a compact projective generator.</p> </div> <div class="proof"> <h6 id="proof_3">Proof</h6> <p>This theorem, minus the explicit description of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>, can be found as Exercise F on page 103 of Peter Freyd’s book <a href="http://www.emis.de/journals/TAC/reprints/articles/3/tr3.pdf#page=132">Abelian Categories</a>. The first part of this theorem can also be found as Prop. 2.1.7. of Victor Ginzburg’s <a href="http://arxiv.org/PS_cache/math/pdf/0506/0506603v1.pdf#page=4">Lectures on noncommutative geometry</a>. Conversely, it is easy to see that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> is a compact projective generator of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mi>Mod</mi></mrow><annotation encoding="application/x-tex">R Mod</annotation></semantics></math>.</p> </div> <p>One can characterize functors between categories of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-modules that are either (isomorphic) to functors of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><msub><mo>⊗</mo> <mi>R</mi></msub><mo>−</mo></mrow><annotation encoding="application/x-tex">B \otimes_R -</annotation></semantics></math> where <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> is a bimodule or those which look as Hom-modules. For the characterization of the tensoring functors see <a class="existingWikiWord" href="/nlab/show/Eilenberg-Watts+theorem">Eilenberg-Watts theorem</a>.</p> <p>Going still further one should be able to obtain a nice theorem describing the image of the embedding of the weak 2-category of</p> <ul> <li>rings</li> <li>bimodules</li> <li>bimodule homomorphisms</li> </ul> <p>into the strict 2-category of</p> <ul> <li>abelian categories</li> <li>right exact functors</li> <li>natural transformations.</li> </ul> <p>For more discussion see the <a href="http://golem.ph.utexas.edu/category/2007/08/questions_about_modules.html"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-Cafe</a>.</p> <h2 id="examples">Examples</h2> <ul> <li> <p>Of course, <a class="existingWikiWord" href="/nlab/show/Ab">Ab</a> is abelian,</p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/category+of+modules">category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>R</mi> </mrow> <annotation encoding="application/x-tex">R</annotation> </semantics> </math>Mod of (left) modules</a> over any <a class="existingWikiWord" href="/nlab/show/ring">ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> is abelian</p> </li> <li> <p>Therefore in particular the category <a class="existingWikiWord" href="/nlab/show/Vect">Vect</a> of vector spaces over any field is an abelian category</p> </li> <li> <p>The full subcategory of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>Mod whose objects are the Noetherian left <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-modules is abelian, since it contains any submodule or quotient module of any of its objects (see Theorem 2.3.8 p.103 of Berrick and Keating in the textbook references below).</p> </li> <li> <p>Similarly, the full subcategory of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>Mod whose objects are the Artinian left <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-modules is abelian, since it contains any submodule or quotient module of any of its objects (loc. cit.).</p> </li> <li> <p>Also similarly, the full subcategory of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>Mod whose objects are the Artinian semisimple modules is abelian, since it contains any submodule or quotient module of any of its objects (loc. cit.) .</p> </li> <li> <p>as is the <a class="existingWikiWord" href="/nlab/show/category+of+representations">category of representations</a> of a <a class="existingWikiWord" href="/nlab/show/group">group</a> (e.g. <a href="https://unapologetic.wordpress.com/2008/12/15/the-category-of-representations-is-abelian/">here</a>)</p> </li> <li> <p>The <a class="existingWikiWord" href="/nlab/show/category+of+sheaves">category of</a> <a class="existingWikiWord" href="/nlab/show/sheaves+of+abelian+groups">sheaves of abelian groups</a> on any <a class="existingWikiWord" href="/nlab/show/site">site</a> is abelian.</p> </li> </ul> <p>Counter-examples:</p> <ul> <li>The category of <a class="existingWikiWord" href="/nlab/show/torsion+subgroup">torsion-free</a> abelian groups is pre-abelian, but not abelian: the monomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><mo>:</mo><mi>ℤ</mi><mo>→</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">2:\mathbb{Z}\to\mathbb{Z}</annotation></semantics></math> is not a kernel.</li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/additive+and+abelian+categories">additive and abelian categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+subcategory">abelian subcategory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Deligne+tensor+product+of+abelian+categories">Deligne tensor product of abelian categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pseudo-abelian+category">pseudo-abelian category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quasi-abelian+category">quasi-abelian category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/semi-abelian+category">semi-abelian category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/length+of+an+object">length of an object</a></p> </li> </ul> <h2 id="references">References</h2> <p>Maybe the first reference on abelian categories, then still called <em>exact categories</em> is</p> <ul> <li>D. A. Buchsbaum, <em>Exact categories and duality</em>, Transactions of the American Mathematical Society Vol. 80, No. 1 (1955), pp. 1-34 (<a href="http://www.jstor.org/stable/1993003">JSTOR</a>)</li> </ul> <p>Further foundations of the theory were then laid in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Alexander+Grothendieck">Alexander Grothendieck</a>, <em><a class="existingWikiWord" href="/nlab/show/Tohoku">Sur quelques points d'algèbre homologique</a>, Tôhoku Math. J. vol 9, n.2, 3, 1957.</em></li> </ul> <p>Other classic references:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Pierre+Gabriel">Pierre Gabriel</a>, <em><a class="existingWikiWord" href="/nlab/show/Des+Cat%C3%A9gories+Ab%C3%A9liennes">Des Catégories Abéliennes</a></em>, Bulletin de la Société Mathématique de France <strong>90</strong> (1962) 323-448 [<a href="http://www.numdam.org/item?id=BSMF_1962__90__323_0">numdam:BSMF_1962__90__323_0</a>]</p> </li> <li id="Freyd64"> <p><a class="existingWikiWord" href="/nlab/show/Peter+Freyd">Peter Freyd</a>, <em>Abelian Categories – An Introduction to the theory of functors</em>, originally published by Harper and Row, New York(1964), Reprints in Theory and Applications of Categories <strong>3</strong> (2003) [<a href="http://www.emis.de/journals/TAC/reprints/articles/3/tr3abs.html">tac:tr3</a>, <a href="https://www.emis.de/journals/TAC/reprints/articles/3/tr3.pdf">pdf</a>]</p> </li> </ul> <p>Textbook accounts:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Nicolae+Popescu">Nicolae Popescu</a>, <em><a class="existingWikiWord" href="/nlab/show/Abelian+categories+with+applications+to+rings+and+modules">Abelian categories with applications to rings and modules</a></em>, London Math. Soc. Monographs <strong>3</strong>, Academic Press (1973) [<a href="http://www.ams.org/mathscinet-getitem?mr=0340375">MR0340375</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Saunders+MacLane">Saunders MacLane</a>, Chapter VIII of: <em><a class="existingWikiWord" href="/nlab/show/Categories+for+the+Working+Mathematician">Categories for the Working Mathematician</a></em> (1978)</p> </li> <li> <p>A. J. Berrick and M. E. Keating, <em>Categories and Modules, with K-theory in View</em>, Cambridge Studies in Advanced Mathematics <strong>67</strong>, Cambridge University Press (2000)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Masaki+Kashiwara">Masaki Kashiwara</a>, <a class="existingWikiWord" href="/nlab/show/Pierre+Schapira">Pierre Schapira</a>, Section 8 of: <em><a class="existingWikiWord" href="/nlab/show/Categories+and+Sheaves">Categories and Sheaves</a></em>, Grundlehren der Mathematischen Wissenschaften <strong>332</strong>, Springer (2006) [<a href="https://link.springer.com/book/10.1007/3-540-27950-4">doi:10.1007/3-540-27950-4</a>, <a href="https://www.maths.ed.ac.uk/~v1ranick/papers/kashiwara2.pdf">pdf</a>]</p> </li> <li id="EGNO15"> <p><a class="existingWikiWord" href="/nlab/show/Pavel+Etingof">Pavel Etingof</a>, <a class="existingWikiWord" href="/nlab/show/Shlomo+Gelaki">Shlomo Gelaki</a>, <a class="existingWikiWord" href="/nlab/show/Dmitri+Nikshych">Dmitri Nikshych</a>, <a class="existingWikiWord" href="/nlab/show/Victor+Ostrik">Victor Ostrik</a>, Chapter 1 of: <em>Tensor Categories</em>, AMS Mathematical Surveys and Monographs <strong>205</strong> (2015) [<a href="https://bookstore.ams.org/surv-205">ISBN:978-1-4704-3441-0</a>, <a href="http://www-math.mit.edu/~etingof/egnobookfinal.pdf">pdf</a>]</p> </li> </ul> <p>Further review:</p> <ul> <li> <p>Rankey Datta, <em>An introduction to abelian categories</em> (2010) (<a href="http://www-bcf.usc.edu/~lauda/teaching/rankeya.pdf">pdf</a>)</p> </li> <li id="Selick"> <p>Paul Selick, <em>Homological Algebra Notes</em> (<a href="www.math.toronto.edu/selick/mat1352/1350notes.pdf">pdf</a>,<a class="existingWikiWord" href="/nlab/files/Selick_HomologicalAlgebra.pdf" title="pdf">pdf</a>)</p> </li> </ul> <p>On <a class="existingWikiWord" href="/nlab/show/diagram+chasing">diagram chasing</a> and <a class="existingWikiWord" href="/nlab/show/elements+in+an+abelian+category">elements in an abelian category</a>:</p> <ul> <li id="Bergman"><a class="existingWikiWord" href="/nlab/show/George+Bergman">George Bergman</a>, <em>A note on abelian categories – translating element-chasing proofs, and exact embedding in abelian groups</em> (1974) [<a href="http://math.berkeley.edu/~gbergman/papers/unpub/elem-chase.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Bergman-ElementChasing.pdf" title="pdf">pdf</a>]</li> </ul> <p>For more discussion of the <em><a class="existingWikiWord" href="/nlab/show/Freyd-Mitchell+embedding+theorem">Freyd-Mitchell embedding theorem</a></em> see there.</p> <p>The proof that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mi>Mod</mi></mrow><annotation encoding="application/x-tex">R Mod</annotation></semantics></math> is an abelian category is spelled out for instance in</p> <ul> <li>Rankeya Datta, <em>The category of modules over a commutative ring and abelian categories</em> (<a href="http://www.math.columbia.edu/~ums/pdf/Rankeya_R-mod_and_Abelian_Categories.pdf">pdf</a>)</li> </ul> <p>A discussion about to which extent abelian categories are a general context for <a class="existingWikiWord" href="/nlab/show/homological+algebra">homological algebra</a> is archived at nForum <a href="http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=2052&Focus=17680#Comment_17680">here</a>.</p> <p>See also the <a href="http://www.mta.ca/~cat-dist/catlist/1999/atcat">catlist 1999 discussion</a> on comparison between abelian categories and topoi (<a class="existingWikiWord" href="/nlab/show/AT+categories">AT categories</a>).</p> <p>Formalization of abelian categories as <a class="existingWikiWord" href="/nlab/show/univalent+categories">univalent categories</a> in <a class="existingWikiWord" href="/nlab/show/univalent+foundations+of+mathematics">univalent foundations of mathematics</a> (<a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a>):</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/unimath">unimath</a>, <em>Abelian Categories</em> [<a href="https://unimath.github.io/doc/UniMath/4dd5c17/UniMath.CategoryTheory.Abelian.html">UniMath.CategoryTheory.Abelian</a>]</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on June 6, 2024 at 08:00:37. See the <a href="/nlab/history/abelian+category" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/abelian+category" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/4094/#Item_11">Discuss</a><span class="backintime"><a href="/nlab/revision/abelian+category/81" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/abelian+category" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/abelian+category" accesskey="S" class="navlink" id="history" rel="nofollow">History (81 revisions)</a> <a href="/nlab/show/abelian+category/cite" style="color: black">Cite</a> <a href="/nlab/print/abelian+category" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/abelian+category" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>