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> crossed module 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> crossed module </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/3860/#Item_16" 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>Content</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="higher_category_theory">Higher category theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></li> <li><a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a></li> </ul> <h2 id="basic_concepts">Basic concepts</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/k-morphism">k-morphism</a>, <a class="existingWikiWord" href="/nlab/show/coherence">coherence</a></li> <li><a class="existingWikiWord" href="/nlab/show/looping+and+delooping">looping and delooping</a></li> <li><a class="existingWikiWord" href="/nlab/show/stabilization">looping and suspension</a></li> </ul> <h2 id="basic_theorems">Basic theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis</a>-theorem</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/delooping+hypothesis">delooping hypothesis</a>-theorem</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/periodic+table">periodic table</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stabilization+hypothesis">stabilization hypothesis</a>-theorem</p> </li> <li> <p><a class="existingWikiWord" href="/michaelshulman/show/exactness+hypothesis">exactness hypothesis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/holographic+principle+of+higher+category+theory">holographic principle</a></p> </li> </ul> <h2 id="applications">Applications</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/applications+of+%28higher%29+category+theory">applications of (higher) category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+category+theory+and+physics">higher category theory and physics</a></p> </li> </ul> <h2 id="models">Models</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/%28n%2Cr%29-category">(n,r)-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Theta-space">Theta-space</a></li> <li><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-category">∞-category</a>/<a class="existingWikiWord" href="/nlab/show/%E2%88%9E-category">∞-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category">(∞,n)-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/n-fold+complete+Segal+space">n-fold complete Segal space</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C2%29-category">(∞,2)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/quasi-category">quasi-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/algebraic+quasi-category">algebraic quasi-category</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/simplicially+enriched+category">simplicially enriched category</a></li> <li><a class="existingWikiWord" href="/nlab/show/complete+Segal+space">complete Segal space</a></li> <li><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C0%29-category">(∞,0)-category</a>/<a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/algebraic+Kan+complex">algebraic Kan complex</a></li> <li><a class="existingWikiWord" href="/nlab/show/simplicial+T-complex">simplicial T-complex</a></li> </ul> </li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2CZ%29-category">(∞,Z)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/n-category">n-category</a> = (n,n)-category <ul> <li><a class="existingWikiWord" href="/nlab/show/2-category">2-category</a>, <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-category">(2,1)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/1-category">1-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/0-category">0-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/%28-1%29-category">(-1)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/%28-2%29-category">(-2)-category</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/n-poset">n-poset</a> = <a class="existingWikiWord" href="/nlab/show/n-poset">(n-1,n)-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/poset">poset</a> = <a class="existingWikiWord" href="/nlab/show/%280%2C1%29-category">(0,1)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/2-poset">2-poset</a> = <a class="existingWikiWord" href="/nlab/show/%281%2C2%29-category">(1,2)-category</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/n-groupoid">n-groupoid</a> = (n,0)-category <ul> <li><a class="existingWikiWord" href="/nlab/show/2-groupoid">2-groupoid</a>, <a class="existingWikiWord" href="/nlab/show/3-groupoid">3-groupoid</a></li> </ul> </li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/categorification">categorification</a>/<a class="existingWikiWord" href="/nlab/show/decategorification">decategorification</a></li> <li><a class="existingWikiWord" href="/nlab/show/geometric+definition+of+higher+category">geometric definition of higher category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a></li> <li><a class="existingWikiWord" href="/nlab/show/quasi-category">quasi-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/simplicial+model+for+weak+%E2%88%9E-categories">simplicial model for weak ∞-categories</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/complicial+set">complicial set</a></li> <li><a class="existingWikiWord" href="/nlab/show/weak+complicial+set">weak complicial set</a></li> </ul> </li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/algebraic+definition+of+higher+category">algebraic definition of higher category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/bicategory">bicategory</a></li> <li><a class="existingWikiWord" href="/nlab/show/bigroupoid">bigroupoid</a></li> <li><a class="existingWikiWord" href="/nlab/show/tricategory">tricategory</a></li> <li><a class="existingWikiWord" href="/nlab/show/tetracategory">tetracategory</a></li> <li><a class="existingWikiWord" href="/nlab/show/strict+%E2%88%9E-category">strict ∞-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/Batanin+%E2%88%9E-category">Batanin ∞-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/Trimble+n-category">Trimble ∞-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/Grothendieck-Maltsiniotis+%E2%88%9E-categories">Grothendieck-Maltsiniotis ∞-categories</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable homotopy theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a></li> <li><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28%E2%88%9E%2C1%29-category">symmetric monoidal (∞,1)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/dg-category">dg-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+category">A-∞ category</a></li> <li><a class="existingWikiWord" href="/nlab/show/triangulated+category">triangulated category</a></li> </ul> </li> </ul> </li> </ul> <h2 id="morphisms">Morphisms</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/k-morphism">k-morphism</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/2-morphism">2-morphism</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/transfor">transfor</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a></li> <li><a class="existingWikiWord" href="/nlab/show/modification">modification</a></li> </ul> </li> </ul> <h2 id="functors">Functors</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/functor">functor</a></li> <li><a class="existingWikiWord" href="/nlab/show/2-functor">2-functor</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/pseudofunctor">pseudofunctor</a></li> <li><a class="existingWikiWord" href="/nlab/show/lax+functor">lax functor</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a></li> </ul> <h2 id="universal_constructions">Universal constructions</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/2-limit">2-limit</a></li> <li><a class="existingWikiWord" href="/nlab/show/adjoint+%28%E2%88%9E%2C1%29-functor">(∞,1)-adjunction</a></li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-Kan+extension">(∞,1)-Kan extension</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/limit+in+a+quasi-category">(∞,1)-limit</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-Grothendieck+construction">(∞,1)-Grothendieck construction</a></li> </ul> <h2 id="extra_properties_and_structure">Extra properties and structure</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/cosmic+cube">cosmic cube</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/k-tuply+monoidal+n-category">k-tuply monoidal n-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/strict+%E2%88%9E-category">strict ∞-category</a>, <a class="existingWikiWord" href="/nlab/show/strict+%E2%88%9E-groupoid">strict ∞-groupoid</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a></li> </ul> <h2 id="1categorical_presentations">1-categorical presentations</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/homotopical+category">homotopical category</a></li> <li><a class="existingWikiWord" href="/nlab/show/model+category">model category theory</a></li> <li><a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a></li> </ul> </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="content">Content</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <ul> <li><a href='#diagrammatic_definition'>Diagrammatic definition</a></li> <li><a href='#definition_in_terms_of_equations'>Definition in terms of equations</a></li> <li><a href='#Morphisms'>Morphisms</a></li> </ul> <li><a href='#examples'>Examples</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#References'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>The concept of <em>crossed modules</em> of groups (<a href="#Whitehead41">Whitehead 41</a>, <a href="#Whitehead49">Whitehead 49</a>) is a basic concept in <a class="existingWikiWord" href="/nlab/show/homotopical+algebra">homotopical algebra</a> and <a class="existingWikiWord" href="/nlab/show/homological+algebra">homological algebra</a>: It is (from the <a class="existingWikiWord" href="/nlab/show/nPOV">nPOV</a>) a convenient way of encoding a <a class="existingWikiWord" href="/nlab/show/strict+2-group">strict 2-group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> in terms of a <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∂</mo><mo>:</mo><msub><mi>G</mi> <mn>2</mn></msub><mo>→</mo><msub><mi>G</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\partial : G_2 \to G_1</annotation></semantics></math> of two ordinary <a class="existingWikiWord" href="/nlab/show/groups">groups</a>.</p> <p>From other points of view it is:</p> <ul> <li> <p>like the inclusion of a <a class="existingWikiWord" href="/nlab/show/normal+subgroup">normal subgroup</a>, but isn't an inclusion in general;</p> </li> <li> <p>like a <a class="existingWikiWord" href="/nlab/show/module">module</a> with a twisted ‘multiplication’;</p> </li> <li> <p>like an <a class="existingWikiWord" href="/nlab/show/action">action</a> by <a class="existingWikiWord" href="/nlab/show/automorphisms">automorphisms</a> on a group;</p> </li> <li> <p>a <a class="existingWikiWord" href="/nlab/show/crossed+complex">crossed complex</a> concentrated in degrees <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>;</p> </li> <li> <p>a nonabelian <a class="existingWikiWord" href="/nlab/show/chain+complex">chain-complex</a> of length 2;</p> </li> <li> <p>a <a class="existingWikiWord" href="/nlab/show/Moore+complex">Moore complex</a> of certain <a class="existingWikiWord" href="/nlab/show/simplicial+groups">simplicial groups</a>.</p> </li> </ul> <p>Historically, crossed modules were among the first examples of <a class="existingWikiWord" href="/nlab/show/higher+dimensional+algebra">higher dimensional algebra</a> to be studied.</p> <h2 id="definition">Definition</h2> <h3 id="diagrammatic_definition">Diagrammatic definition</h3> <p> <div class='num_defn' id='DiagrammaticDefinition'> <h6>Definition</h6> <p>A <strong>crossed module of groups</strong> is</p> <ul> <li> <p>a <a class="existingWikiWord" href="/nlab/show/pair">pair</a> of <a class="existingWikiWord" href="/nlab/show/groups">groups</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>G</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">G_2, G_1</annotation></semantics></math>,</p> </li> <li> <p>a <a class="existingWikiWord" href="/nlab/show/group+homomorphism">group homomorphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub><mover><mo>⟶</mo><mi>δ</mi></mover><msub><mi>G</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex"> G_2 \overset{ \delta }{ \longrightarrow } G_1 </annotation></semantics></math></div></li> <li> <p>a <a class="existingWikiWord" href="/nlab/show/group+homomorphism">group homomorphism</a> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">G_1</annotation></semantics></math> to the <a class="existingWikiWord" href="/nlab/show/automorphism+group">automorphism group</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>1</mn></msub><mover><mo>⟶</mo><mi>α</mi></mover><mi>Aut</mi><mo stretchy="false">(</mo><msub><mi>G</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> G_1 \overset{ \alpha }{ \longrightarrow } Aut(G_2) \,, </annotation></semantics></math></div> <p>which we may equivalently regard as a <a class="existingWikiWord" href="/nlab/show/function">function</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>α</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>G</mi> <mn>1</mn></msub><mo>×</mo><msub><mi>G</mi> <mn>2</mn></msub><mo>⟶</mo><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex"> \alpha \;\colon\; G_1 \times G_2 \longrightarrow G_2 </annotation></semantics></math></div> <p>(out of the <a class="existingWikiWord" href="/nlab/show/Cartesian+product">Cartesian product</a> of <a class="existingWikiWord" href="/nlab/show/underlying">underlying</a> <a class="existingWikiWord" href="/nlab/show/sets">sets</a>/<a class="existingWikiWord" href="/nlab/show/objects">objects</a>)</p> <p>that satisfies the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">G_1</annotation></semantics></math>-<a href="action#eq:ActionProperty">action property</a> and is such that for every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mn>1</mn></msub><mo>∈</mo><msub><mi>G</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">g_1 \in G_1</annotation></semantics></math> it is a group <a class="existingWikiWord" href="/nlab/show/automorphism">automorphism</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math>;</p> </li> </ul> <p>such that the following <a class="existingWikiWord" href="/nlab/show/commuting+diagram">diagrams commute</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>G</mi> <mn>2</mn></msub><mo>×</mo><msub><mi>G</mi> <mn>2</mn></msub></mtd> <mtd></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>δ</mi><mo>×</mo><mi>Id</mi></mrow></mover></mtd> <mtd></mtd> <mtd><msub><mi>G</mi> <mn>1</mn></msub><mo>×</mo><msub><mi>G</mi> <mn>2</mn></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>Ad</mi></mpadded></msub><mo>↘</mo></mtd> <mtd></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mi>α</mi></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msub><mi>G</mi> <mn>2</mn></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ G_2 \times G_2 && \overset{ \delta \times Id }{ \longrightarrow } && G_1 \times G_2 \\ & {}_{\mathllap{Ad}} \searrow && \swarrow_{\mathrlap{\alpha}} \\ && G_2 } </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>G</mi> <mn>1</mn></msub><mo>×</mo><msub><mi>G</mi> <mn>2</mn></msub></mtd> <mtd><mover><mo>⟶</mo><mi>α</mi></mover></mtd> <mtd><msub><mi>G</mi> <mn>2</mn></msub></mtd></mtr> <mtr><mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo><mpadded width="0"><mrow><msup><mrow></mrow> <mrow><mi>Id</mi><mo>×</mo><mi>δ</mi></mrow></msup></mrow></mpadded></mtd> <mtd></mtd> <mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo><mpadded width="0"><mrow><msup><mrow></mrow> <mi>δ</mi></msup></mrow></mpadded></mtd></mtr> <mtr><mtd><msub><mi>G</mi> <mn>1</mn></msub><mo>×</mo><msub><mi>G</mi> <mn>1</mn></msub></mtd> <mtd><mover><mo>⟶</mo><mi>Ad</mi></mover></mtd> <mtd><msub><mi>G</mi> <mn>1</mn></msub><mspace width="thinmathspace"></mspace><mo>,</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ G_1 \times G_2 & \stackrel{ \alpha }{ \longrightarrow } & G_2 \\ \big\downarrow\mathrlap{{}^{ {Id \times \delta} }} && \big\downarrow\mathrlap{{}^{ {\delta} }} \\ G_1 \times G_1 & \stackrel{ Ad }{ \longrightarrow } & G_1 \,, } </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ad</mi></mrow><annotation encoding="application/x-tex">Ad</annotation></semantics></math> denotes the <a class="existingWikiWord" href="/nlab/show/adjoint+action">adjoint action</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math> on itself.</p> </div> </p> <p> <div class='num_remark'> <h6>Remark</h6> <p>The diagrammatic Def. <a class="maruku-ref" href="#DiagrammaticDefinition"></a> makes sense <a class="existingWikiWord" href="/nlab/show/internalization">internal</a> to any <a class="existingWikiWord" href="/nlab/show/cartesian+monoidal+category">cartesian monoidal category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>:</p> <ul> <li> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">\mathcal{C}=</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Sets">Sets</a> we get bare crossed modules of <a class="existingWikiWord" href="/nlab/show/discrete+groups">discrete groups</a>;</p> </li> <li> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">\mathcal{C}=</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/SmoothManifolds">SmoothManifolds</a> we get crossed modules of <a class="existingWikiWord" href="/nlab/show/Lie+groups">Lie groups</a>;</p> <p>if one allows <a class="existingWikiWord" href="/nlab/show/infinite-dimensional+manifolds">infinite-dimensional manifolds</a> here then this subsumes crossed modules of <a class="existingWikiWord" href="/nlab/show/Kac-Moody+groups">Kac-Moody groups</a> such as appear in models for the <a class="existingWikiWord" href="/nlab/show/String+2-group">String 2-group</a>.</p> </li> </ul> <p></p> </div> </p> <p>Alternatively, one can take another tack, and define crossed module objects in categories that support enough structure without using internal groups, the most general case of which, in practice, are <a class="existingWikiWord" href="/nlab/show/semiabelian+categories">semiabelian categories</a>. There one considers the objects to behave ‘like groups’ in the sense that the category they form looks very much like the category of groups. Janelidze (<a href="#Janelidze_03">Janelidze 2003</a>) defined the notion of internal crossed module in a semiabelian category (so that in the prototypical example of the category of groups, they reduce to the above notion).</p> <p>A key result, also due to (<a href="#Janelidze_03">Janelidze 2003</a>) and generalising the Brown-Spencer theorem from the case of ordinary crossed modules, is the following:</p> <div class="num_theorem"> <h6 id="theorem">Theorem</h6> <p><strong>(Janelidze’s Brown-Spencer theorem).</strong> 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 a <a class="existingWikiWord" href="/nlab/show/semiabelian+category">semiabelian category</a>. Then the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>XMod</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">XMod(C)</annotation></semantics></math> of crossed modules in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is equivalent to the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Gpd</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Gpd(C)</annotation></semantics></math> of internal groupoids in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>.</p> </div> <p>Here the notion of <a class="existingWikiWord" href="/nlab/show/internal+groupoid">internal groupoid</a> is the usual diagrammatic notion.</p> <h3 id="definition_in_terms_of_equations">Definition in terms of equations</h3> <p>The two <a class="existingWikiWord" href="/nlab/show/diagrams">diagrams</a> can be translated into <a class="existingWikiWord" href="/nlab/show/equations">equations</a>, which may often be helpful.</p> <ul> <li> <p>If we write the effect of acting with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mn>1</mn></msub><mo>∈</mo><msub><mi>G</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">g_1\in G_1</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mn>2</mn></msub><mo>∈</mo><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">g_2\in G_2</annotation></semantics></math> as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mrow></mrow> <mrow><msub><mi>g</mi> <mn>1</mn></msub></mrow></msup><msub><mi>g</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">{}^{g_1}g_2</annotation></semantics></math>, then the second diagram translates as the equation:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>δ</mi><mo stretchy="false">(</mo><msup><mrow></mrow> <mrow><msub><mi>g</mi> <mn>1</mn></msub></mrow></msup><msub><mi>g</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>g</mi> <mn>1</mn></msub><mi>δ</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><msubsup><mi>g</mi> <mn>1</mn> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo>.</mo></mrow><annotation encoding="application/x-tex">\delta({}^{g_1}g_2) = g_1\delta(g_2)g_1^{-1}.</annotation></semantics></math></div> <p>In other words, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>δ</mi></mrow><annotation encoding="application/x-tex">\delta</annotation></semantics></math> is equivariant for the action of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">G_1</annotation></semantics></math>.</p> </li> <li> <p>The first diagram is slightly more subtle. The group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math> can act on itself in two different ways, (i) by the usual conjugation action, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mrow></mrow> <mrow><msub><mi>g</mi> <mn>2</mn></msub></mrow></msup><msubsup><mi>g</mi> <mn>2</mn> <mstyle scriptlevel="0"><mo>′</mo></mstyle></msubsup><mo>=</mo><msub><mi>g</mi> <mn>2</mn></msub><msubsup><mi>g</mi> <mn>2</mn> <mstyle scriptlevel="0"><mo>′</mo></mstyle></msubsup><msubsup><mi>g</mi> <mn>2</mn> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup></mrow><annotation encoding="application/x-tex">{}^{g_2}g^\prime_2=g_2g^\prime_2g_2^{-1} </annotation></semantics></math> and (ii) by first mapping <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">g_2</annotation></semantics></math> down to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">G_1</annotation></semantics></math> and then using the action of that group back on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math>. The first diagram says that the two actions coincide. Equationally this gives:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mrow></mrow> <mrow><mi>δ</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow></msup><msubsup><mi>g</mi> <mn>2</mn> <mstyle scriptlevel="0"><mo>′</mo></mstyle></msubsup><mo>=</mo><msub><mi>g</mi> <mn>2</mn></msub><msubsup><mi>g</mi> <mn>2</mn> <mstyle scriptlevel="0"><mo>′</mo></mstyle></msubsup><msubsup><mi>g</mi> <mn>2</mn> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo>.</mo></mrow><annotation encoding="application/x-tex">{}^{\delta(g_2)}g^\prime_2 = g_2g^\prime_2g_2^{-1}.</annotation></semantics></math></div> <p>This equation is known as the <strong>Peiffer rule</strong> in the literature. Another way to interpret it is to rewrite it slightly:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mrow></mrow> <mrow><mi>δ</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow></msup><msubsup><mi>g</mi> <mn>2</mn> <mstyle scriptlevel="0"><mo>′</mo></mstyle></msubsup><msub><mi>g</mi> <mn>2</mn></msub><mo>=</mo><msub><mi>g</mi> <mn>2</mn></msub><msubsup><mi>g</mi> <mn>2</mn> <mstyle scriptlevel="0"><mo>′</mo></mstyle></msubsup></mrow><annotation encoding="application/x-tex">{}^{\delta(g_2)}g^\prime_2 g_2 = g_2g^\prime_2</annotation></semantics></math></div> <p>The Peiffer rule can thus be seen as a ‘twisted commutativity law’ for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math>.</p> </li> </ul> <h3 id="Morphisms">Morphisms</h3> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math> two <a class="existingWikiWord" href="/nlab/show/strict+2-group"> strict 2-groups</a> coming from crossed modules <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>G</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[G]</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>H</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[H]</annotation></semantics></math>, a morphism of strict 2-groups <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>G</mi><mo>→</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">f : G \to H</annotation></semantics></math>, and hence a morphism of crossed modules <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>f</mi><mo stretchy="false">]</mo><mo>:</mo><mo stretchy="false">[</mo><mi>G</mi><mo stretchy="false">]</mo><mo>→</mo><mo stretchy="false">[</mo><mi>H</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[f] : [G] \to [H]</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/2-functor">2-functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>f</mi><mo>:</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>H</mi></mrow><annotation encoding="application/x-tex"> \mathbf{B}f : \mathbf{B}G \to \mathbf{B}H </annotation></semantics></math></div> <p>between the corresponding <a class="existingWikiWord" href="/nlab/show/delooping">delooped</a> <a class="existingWikiWord" href="/nlab/show/2-groupoid"> 2-groupoids</a>. Expressing this in terms of a diagram of the ordinary groups appearing in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>G</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[G]</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>H</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[H]</annotation></semantics></math> yields a diagram called a <a class="existingWikiWord" href="/nlab/show/butterfly">butterfly</a>. See there for more details.</p> <h2 id="examples">Examples</h2> <ul> <li> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math> any group, its <a class="existingWikiWord" href="/nlab/show/automorphism">automorphism</a> crossed module is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>AUT</mi><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo><mo>≔</mo><mo stretchy="false">(</mo><msub><mi>G</mi> <mn>2</mn></msub><mo>=</mo><mi>H</mi><mo>,</mo><msub><mi>G</mi> <mn>1</mn></msub><mo>=</mo><mi>Aut</mi><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo><mo>,</mo><mi>δ</mi><mo>=</mo><mi>Id</mi><mo>,</mo><mi>α</mi><mo>=</mo><mi>Ad</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> AUT(H) \coloneqq (G_2 = H, G_1 = Aut(H), \delta = Id, \alpha = Ad) \,. </annotation></semantics></math></div> <p>Under the equivalence of crossed modules with <a class="existingWikiWord" href="/nlab/show/strict+2-group"> strict 2-groups</a> this corresponds to the <a class="existingWikiWord" href="/nlab/show/automorphism+2-group">automorphism 2-group</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Aut</mi> <mi>Grpd</mi></msub><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>H</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Aut_{Grpd}(\mathbf{B}H) </annotation></semantics></math></div> <p>of <a class="existingWikiWord" href="/nlab/show/automorphism"> automorphisms</a> in the category <a class="existingWikiWord" href="/nlab/show/Grpd">Grpd</a> of <a class="existingWikiWord" href="/nlab/show/groupoid"> groupoids</a> on the one-object <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>H</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}H</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math>.</p> </li> <li> <p>Almost the canonical example of a crossed module is given by a group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> and a <a class="existingWikiWord" href="/nlab/show/normal+subgroup">normal subgroup</a> <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> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>. We take <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub><mo>=</mo><mi>N</mi></mrow><annotation encoding="application/x-tex">G_2 = N</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>1</mn></msub><mo>=</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">G_1 = G</annotation></semantics></math> with the action <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math> being the <a class="existingWikiWord" href="/nlab/show/conjugation+action">conjugation action</a>, whilst <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>δ</mi></mrow><annotation encoding="application/x-tex">\delta</annotation></semantics></math> is the given inclusion, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo>↪</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">N \hookrightarrow G</annotation></semantics></math>.</p> </li> </ul> <p>This is ‘almost canonical’, since if we replace the groups by simplicial groups <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mo>.</mo></msub></mrow><annotation encoding="application/x-tex">G_.</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>N</mi> <mo>.</mo></msub></mrow><annotation encoding="application/x-tex">N_.</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><msub><mi>G</mi> <mo>.</mo></msub><mo stretchy="false">)</mo><mo>,</mo><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><msub><mi>N</mi> <mo>.</mo></msub><mo stretchy="false">)</mo><mo>,</mo><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>inc</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\pi_0(G_.),\pi_0(N_.),\pi_0(inc))</annotation></semantics></math> is a crossed module, and given any crossed module, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mi>P</mi><mo>,</mo><mi>δ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(C,P,\delta)</annotation></semantics></math>, there is a simplicial group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mo>.</mo></msub></mrow><annotation encoding="application/x-tex">G_.</annotation></semantics></math> and a normal subgroup <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>N</mi> <mo>.</mo></msub></mrow><annotation encoding="application/x-tex">N_.</annotation></semantics></math>, such that the construction above gives the given crossed module up to isomorphism.</p> <ul> <li> <p>Another standard example of a crossed module is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><msup><mo>→</mo> <mn>0</mn></msup><mi>P</mi></mrow><annotation encoding="application/x-tex">M \to ^0 P</annotation></semantics></math> where <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 group and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> is a <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>-module. Thus the category of modules over groups embeds in the category of crossed modules.</p> </li> <li> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>μ</mi><mo>:</mo><mi>M</mi><mo>→</mo><mi>P</mi></mrow><annotation encoding="application/x-tex">\mu: M \to P</annotation></semantics></math> is a crossed module with cokernel <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> is abelian, then the operation of <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> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> factors through <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>. In fact such crossed modules in which both <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> and <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> are abelian should not be sneezed at! A good example is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>μ</mi><mo>:</mo><msub><mi>C</mi> <mn>2</mn></msub><mo>×</mo><msub><mi>C</mi> <mn>2</mn></msub><mo>→</mo><msub><mi>C</mi> <mn>4</mn></msub></mrow><annotation encoding="application/x-tex">\mu: C_2 \times C_2 \to C_4</annotation></semantics></math> where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">C_n</annotation></semantics></math> denotes the cyclic group of order <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>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>μ</mi></mrow><annotation encoding="application/x-tex">\mu</annotation></semantics></math> is injective on each factor, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mn>4</mn></msub></mrow><annotation encoding="application/x-tex">C_4</annotation></semantics></math> acts on the product by the twist. This crossed module has a <a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> with fundamental and second homotopy groups <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">C_2</annotation></semantics></math> and non trivial <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-invariant in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mn>3</mn></msup><mo stretchy="false">(</mo><msub><mi>C</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>C</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^3(C_2, C_2)</annotation></semantics></math>, so <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 not a product of <a class="existingWikiWord" href="/nlab/show/Eilenberg-MacLane+space"> Eilenberg-MacLane spaces</a>. However the crossed module is an algebraic model and so one one can do algebraic constructions with it. It gives in some ways a better feel for the space than the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-invariant. The <a class="existingWikiWord" href="/nlab/show/higher+homotopy+van+Kampen+theorem">higher homotopy van Kampen theorem</a> implies that the above <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> gives the 2-type of the <a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a> of the map of <a class="existingWikiWord" href="/nlab/show/classifying+space"> classifying spaces</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>BC</mi> <mn>2</mn></msub><mo>→</mo><msub><mi>BC</mi> <mn>4</mn></msub></mrow><annotation encoding="application/x-tex">BC_2 \to BC_4</annotation></semantics></math>.</p> </li> <li> <p>Suppose <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mover><mo>→</mo><mi>i</mi></mover><mi>E</mi><mover><mo>→</mo><mi>p</mi></mover><mi>B</mi></mrow><annotation encoding="application/x-tex">F\stackrel{i}{\to}E\stackrel{p}{\to}B</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/fibration+sequence">fibration sequence</a></p> <p>of <a class="existingWikiWord" href="/nlab/show/pointed+object">pointed spaces</a>, thus <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/fibration">fibration</a> in the <a class="existingWikiWord" href="/nlab/show/model+structure+on+topological+spaces">topological sense</a> (lifting of paths and homotopies of paths will suffice), <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>=</mo><msup><mi>p</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F = p^{-1}(b_0)</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>b</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">b_0</annotation></semantics></math> is the basepoint of <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>. The fibre <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 pointed at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">f_0</annotation></semantics></math>, say, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">f_0</annotation></semantics></math> is taken as the basepoint of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> as well.</p> <p>There is an induced map on <a class="existingWikiWord" href="/nlab/show/homotopy+group"> homotopy groups</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></mover><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_1(F) \stackrel{\pi_1(i)}{\longrightarrow} \pi_1(E)</annotation></semantics></math></div> <p>and if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math> is a loop in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> based at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">f_0</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math> a loop in <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> based at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">f_0</annotation></semantics></math>, then the composite path corresponding to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mi>b</mi><msup><mi>a</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">a b a^{-1}</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/homotopy">homotopic</a> to one wholly within <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>. To see this, note that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>a</mi><mi>b</mi><msup><mi>a</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p(a b a^{-1})</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/null+homotopy">null homotopic</a>. Pick a <a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a> in <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> between it and the constant map, then lift that homotopy back up to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> to one starting at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mi>b</mi><msup><mi>a</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">a b a^{-1}</annotation></semantics></math>. This homotopy is the required one and its other end gives a well defined element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mrow></mrow> <mi>a</mi></msup><mi>b</mi><mo>∈</mo><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">{}^a b \in \pi_1(F)</annotation></semantics></math> (abusing notation by confusing paths and their homotopy classes). With this action <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo><mo>,</mo><mi>π</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\pi_1(F), \pi(E), \pi_1(i))</annotation></semantics></math> is a crossed module. This will not be proved here, but is not that difficult. (Of course, secretly, this example is ‘really’ the same as the previous one since a fibration of <a class="existingWikiWord" href="/nlab/show/simplicial+group"> simplicial groups</a> is just morphism that is an <a class="existingWikiWord" href="/nlab/show/epimorphism">epimorphism</a> in each degree, and the <a class="existingWikiWord" href="/nlab/show/fibration+sequence">fibre</a> is thus just a <a class="existingWikiWord" href="/nlab/show/simplicial+group">normal simplicial subgroup</a>. What is fun is that this generalises to ‘higher dimensions’.)</p> </li> <li> <p>A particular case of this last example can be obtained from the inclusion of a subspace <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">A\to X</annotation></semantics></math> into a pointed space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>x</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,x_0)</annotation></semantics></math>, (where we assume <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mn>0</mn></msub><mo>∈</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">x_0\in A</annotation></semantics></math>). We can replace this inclusion by a homotopic fibration, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>A</mi><mo>¯</mo></mover><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\overline{A}\to X</annotation></semantics></math> in ‘the standard way’, and then find that the fundamental group of its fibre is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo>,</mo><msub><mi>x</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_2(X,A,x_0)</annotation></semantics></math>.</p> </li> </ul> <p>A deep theorem of J.H.C. Whitehead is that the crossed module</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>δ</mi><mo>:</mo><msub><mi>π</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>A</mi><mo>∪</mo><mo stretchy="false">{</mo><msubsup><mi>e</mi> <mi>λ</mi> <mn>2</mn></msubsup><msub><mo stretchy="false">}</mo> <mrow><mi>λ</mi><mo>∈</mo><mi>Λ</mi></mrow></msub><mo>,</mo><mi>A</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\delta: \pi_2(A \cup \{e^2_\lambda\}_{\lambda \in \Lambda},A,x) \to \pi_1(A,x)</annotation></semantics></math></div> <p>is the <a class="existingWikiWord" href="/nlab/show/free+crossed+module">free crossed module</a> on the characteristic maps of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>-cells. One utility of this is that it enables the expression of nonabelian chains and boundaries ideas in dimensions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>: thus for the standard picture of a Klein Bottle formed by identifications from a square <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math> the formula</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>δ</mi><mi>σ</mi><mo>=</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo>−</mo><mi>a</mi><mo>+</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">\delta \sigma = a+b-a +b </annotation></semantics></math></div> <p>makes sense with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math> a generator of a free crossed module; in the usual abelian chain theory we can write only <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∂</mo><mi>σ</mi><mo>=</mo><mn>2</mn><mi>b</mi></mrow><annotation encoding="application/x-tex">\partial \sigma =2b</annotation></semantics></math>, thus losing information.</p> <p>Whitehead’s proof of this theorem used knot theory and transversality. The theorem is also a consequence of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>-dimensional Seifert-van Kampen Theorem, proved by Brown and Higgins, which states that the functor</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\Pi_2</annotation></semantics></math>: (pairs of pointed spaces) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\to</annotation></semantics></math> (crossed modules)</p> <p>preserves certain colimits (see reference below).</p> <p>This last example was one of the first investigated by Whitehead and his proof appears also in a little book by <a class="existingWikiWord" href="/nlab/show/Hilton">Hilton</a>; see also <a class="existingWikiWord" href="/nlab/show/Nonabelian+algebraic+topology">Nonabelian algebraic topology</a>, however the more general result of Brown and Higgins determines also the group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo>∪</mo><mi>CA</mi><mo>,</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_2(X \cup CA,X,x)</annotation></semantics></math> as a crossed <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_1(X,x)</annotation></semantics></math> module, and then Whitehead’s result is the case with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is a wedge of circles.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+crossed+module">Lie crossed module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/internal+crossed+module">internal crossed module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group">group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-group">2-group</a>, <strong>crossed module</strong>, <a class="existingWikiWord" href="/nlab/show/differential+crossed+module">differential crossed module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/3-group">3-group</a>, <a class="existingWikiWord" href="/nlab/show/2-crossed+module">2-crossed module</a> / <a class="existingWikiWord" href="/nlab/show/crossed+square">crossed square</a>, <a class="existingWikiWord" href="/nlab/show/differential+2-crossed+module">differential 2-crossed module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/n-group">n-group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a>, <a class="existingWikiWord" href="/nlab/show/simplicial+group">simplicial group</a>, <a class="existingWikiWord" href="/nlab/show/crossed+complex">crossed complex</a>, <a class="existingWikiWord" href="/nlab/show/hypercrossed+complex">hypercrossed complex</a></p> </li> </ul> <h2 id="References">References</h2> <p>The second axiom for a crossed module first appeared as footnote 35 on p. 422 of Whitehead’s paper:</p> <ul> <li id="Whitehead41"><a class="existingWikiWord" href="/nlab/show/J.H.C.+Whitehead">J.H.C. Whitehead</a>, <em>On adding relations to homotopy groups</em>, Ann. of Math. <strong>42</strong> 2 (1941) 409–428 [<a href="https://doi.org/10.2307/1968907">doi:10.2307/1968907</a>];</li> </ul> <p>A key result on “Free crossed modules”.</p> <ul> <li id="Whitehead49"><a class="existingWikiWord" href="/nlab/show/J.+H.+C.+Whitehead">J. H. C. Whitehead</a>, Section 16 of: <em>Combinatorial Homotopy II</em>, Bull. Amer. Math. Soc., {\bf 55} (1949) 453-496 [<a href="https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society/volume-55/issue-5/Combinatorial-homotopy-II/bams/1183513797.full">euclid</a>]</li> </ul> <p>An exposition of this proof is in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/R.+Brown">R. Brown</a>, <em>On the second relative homotopy group of an adjunction space: an exposition of a theorem of J.H.C. Whitehead</em>, J. London Math. Soc._ (2) 22 (1980) 146-152 (<a href="https://doi.org/10.1112/jlms/s2-22.1.146">doi:10.1112/jlms/s2-22.1.146</a>)</li> </ul> <p>see also</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Peter+J.+Hilton">Peter J. Hilton</a>, <em>An Introduction to Homotopy Theory</em>, Cambridge University Press (1953) [<a href="https://doi.org/10.1017/CBO9780511666278">doi:10.1017/CBO9780511666278</a>]</li> </ul> <p>Note that the geometric core of the proof uses <a class="existingWikiWord" href="/nlab/show/knot+theory">knot theory</a> and transversality arguments which come from the “previous paper” of Whitehead:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/J.+H.+C.+Whitehead">J. H. C. Whitehead</a>, <em>On adding relations to homotopy groups</em>, Annals of Math., <strong>42</strong> (1941) 400-428 [<a href="https://doi.org/10.2307/1968907">doi:10.2307/1968907</a>]</li> </ul> <p>Textbook account of crossed modules and their relation to <a class="existingWikiWord" href="/nlab/show/strict+2-groups">strict 2-groups</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Saunders+MacLane">Saunders MacLane</a>, §XII.8 of: <em><a class="existingWikiWord" href="/nlab/show/Categories+for+the+Working+Mathematician">Categories for the Working Mathematician</a></em>, Graduate Texts in Mathematics <strong>5</strong> Springer (second ed. 1997) [<a href="https://link.springer.com/book/10.1007/978-1-4757-4721-8">doi:10.1007/978-1-4757-4721-8</a>]</li> </ul> <p>further exposition:</p> <ul> <li id="BaezLauda03"><a class="existingWikiWord" href="/nlab/show/John+Baez">John Baez</a>, <a class="existingWikiWord" href="/nlab/show/Aaron+Lauda">Aaron Lauda</a>, pp. 45 of: <em>HDA V: 2-Groups</em>, Theory and Applications of Categories <strong>12</strong> (2004) 423-491. [<a href="http://arxiv.org/abs/math.QA/0307200">arXiv:math.QA/0307200</a>]</li> </ul> <p>The following paper</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ronnie+Brown">Ronnie Brown</a>, <a class="existingWikiWord" href="/nlab/show/Philip+Higgins">Philip Higgins</a>, <em>On the connections between the second relative homotopy groups of some related spaces</em>, <em>Proc. London Math. Soc.</em> (3) 36 (1978) 193-212.</li> </ul> <p>showed that the theorem of Whitehead on free crossed modules from CH II Sec 16 is a special case of a 2-dimensional Van Kampen type Theorem for the homotopy crossed modules (over groupoids) of open unions of “connected” triples <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo>,</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">(X,A,S</annotation></semantics></math> of spaces where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> is a set of base points. However the proof of the main theorem uses the relation of crossed modules not to cat<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mo></mo><mn>1</mn></msup></mrow><annotation encoding="application/x-tex">^1</annotation></semantics></math>-groups but to “double groupoids with connections”, also proved with Spencer. Full details and references are in Part I of:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ronnie+Brown">Ronnie Brown</a>, <a class="existingWikiWord" href="/nlab/show/Philip+Higgins">Philip Higgins</a>, and R. Sivera, <em><a class="existingWikiWord" href="/nlab/show/Nonabelian+Algebraic+Topology">Nonabelian Algebraic Topology</a>: Filtered spaces, Crossed Complexes, Cubical Homotopy Groupoids</em>, EMS Tracts in Mathematics, Vol. 15, (2011).</li> </ul> <p>See also</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Ronnie+Brown">Ronnie Brown</a>, <em>Groupoids and crossed objects in algebraic topology</em>, Homology, Homotopy and Applications, 1 (1999) 1-78.</p> </li> <li id="Janelidze_03"> <p><a class="existingWikiWord" href="/nlab/show/George+Janelidze">George Janelidze</a>, <em>Internal crossed modules</em>, Georgian Mathematical Journal <strong>10</strong> (2003) pp 99-114. (<a href="https://eudml.org/doc/51553">EuDML</a>)</p> </li> </ul> <p>On crossed modules in other algebraic contexts:</p> <ul> <li>A. S-T. Lue, <em>Cohomology of groups relative to a variety</em>, J. Algebra 69 (1) (1981) 155–174.</li> </ul> <p>Discussion of crossed modules <a class="existingWikiWord" href="/nlab/show/internalization">internal to</a> the <a class="existingWikiWord" href="/nlab/show/category">category</a> of <a class="existingWikiWord" href="/nlab/show/Lie-Rinehart+algebras">Lie-Rinehart algebras</a>:</p> <ul> <li>Joel Couchman, <em>Crossed modules and internal categories of Lie-Rinehart algebras</em>, Master’s thesis, Macquarie University (2017) [<a href="https://doi.org/10.25949/19439645.v1">doi:10.25949/19439645.v1</a>, <a class="existingWikiWord" href="/nlab/files/Couchman-CrossedModules.pdf" title="pdf">pdf</a>]</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on July 3, 2024 at 12:25:41. See the <a href="/nlab/history/crossed+module" 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/crossed+module" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/3860/#Item_16">Discuss</a><span class="backintime"><a href="/nlab/revision/crossed+module/65" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/crossed+module" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/crossed+module" accesskey="S" class="navlink" id="history" rel="nofollow">History (65 revisions)</a> <a href="/nlab/show/crossed+module/cite" style="color: black">Cite</a> <a href="/nlab/print/crossed+module" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/crossed+module" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>