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> principal bundle 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> principal bundle </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/3769/#Item_14" 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="bundles">Bundles</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/bundles">bundles</a></strong></p> <ul> <li> <p>(<a class="existingWikiWord" href="/nlab/show/parameterized+stable+homotopy+theory">stable</a>) <a class="existingWikiWord" href="/nlab/show/parameterized+homotopy+theory">parameterized homotopy theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+bundles+in+physics">fiber bundles in physics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory">Chern-Weil theory</a></p> </li> </ul> <h2 id="sidebar_context">Context</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/slice+topos">slice topos</a>, <a class="existingWikiWord" href="/nlab/show/slice+%28%E2%88%9E%2C1%29-topos">slice (∞,1)-topos</a></p> </li> <li> <p>(<a class="existingWikiWord" href="/nlab/show/dependent+linear+type+theory">linear</a>) <a class="existingWikiWord" href="/nlab/show/dependent+type+theory">dependent type theory</a></p> </li> </ul> <h2 id="sidebar_classes_of_bundles">Classes of bundles</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/covering+space">covering space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/retractive+space">retractive space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+bundle">fiber bundle</a>, <a class="existingWikiWord" href="/nlab/show/fiber+%E2%88%9E-bundle">fiber ∞-bundle</a></p> <p><a class="existingWikiWord" href="/nlab/show/numerable+bundle">numerable bundle</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/sphere+bundle">sphere bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/projective+bundle">projective bundle</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a>, <a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+2-bundle">principal 2-bundle</a>, <a class="existingWikiWord" href="/nlab/show/principal+3-bundle">principal 3-bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/circle+bundle">circle bundle</a>, <a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle n-bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orientation+bundle">orientation bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spinor+bundle">spinor bundle</a>, <a class="existingWikiWord" href="/nlab/show/stringor+bundle">stringor bundle</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/associated+bundle">associated bundle</a>, <a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundle">associated ∞-bundle</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/gerbe">gerbe</a>, <a class="existingWikiWord" href="/nlab/show/2-gerbe">2-gerbe</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-gerbe">∞-gerbe</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+coefficient+bundle">local coefficient bundle</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vector+bundle">vector bundle</a>, <a class="existingWikiWord" href="/nlab/show/2-vector+bundle">2-vector bundle</a>, <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-vector+bundle">(∞,1)-vector bundle</a></p> <p><a class="existingWikiWord" href="/nlab/show/real+vector+bundle">real</a>, <a class="existingWikiWord" href="/nlab/show/complex+vector+bundle">complex</a>/<a class="existingWikiWord" href="/nlab/show/holomorphic+vector+bundle">holomorphic</a>, <a class="existingWikiWord" href="/nlab/show/quaternionic+vector+bundle">quaternionic</a></p> <p><a class="existingWikiWord" href="/nlab/show/topological+vector+bundle">topological</a>, <a class="existingWikiWord" href="/nlab/show/differentiable+vector+bundle">differentiable</a>, <a class="existingWikiWord" href="/nlab/show/algebraic+vector+bundle">algebraic</a></p> <p><a class="existingWikiWord" href="/nlab/show/connection+on+a+vector+bundle">with connection</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/line+bundle">line bundle</a></p> <p><a class="existingWikiWord" href="/nlab/show/complex+line+bundle">complex</a>, <a class="existingWikiWord" href="/nlab/show/holomorphic+line+bundle">holomorphic</a>, <a class="existingWikiWord" href="/nlab/show/algebraic+line+bundle">algebraic</a></p> <p><a class="existingWikiWord" href="/nlab/show/cubical+structure+on+a+line+bundle">cubical structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory">Chern-Weil theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+K-theory">topological K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tensor+category">tensor category</a><a class="existingWikiWord" href="/nlab/show/Vect%28X%29">of vector bundles</a></p> <p>(<a class="existingWikiWord" href="/nlab/show/VectBund">VectBund</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/direct+sum+of+vector+bundles">direct sum</a>, <a class="existingWikiWord" href="/nlab/show/tensor+product+of+vector+bundles">tensor product</a>, <a class="existingWikiWord" href="/nlab/show/external+tensor+product+of+vector+bundles">external tensor product</a>, <a class="existingWikiWord" href="/nlab/show/inner+product+of+vector+bundles">inner product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dual+vector+bundle">dual vector bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+vector+bundle">stable vector bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/virtual+vector+bundle">virtual vector bundle</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bundle+of+spectra">bundle of spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/natural+bundle">natural bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+bundle">equivariant bundle</a></p> </li> </ul> <h2 id="sidebar_universal_bundles">Universal bundles</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+principal+bundle">universal principal bundle</a>, <a class="existingWikiWord" href="/nlab/show/universal+principal+%E2%88%9E-bundle">universal principal ∞-bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+vector+bundle">universal vector bundle</a>, <a class="existingWikiWord" href="/nlab/show/universal+complex+line+bundle">universal complex line bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/subobject+classifier">subobject classifier</a>, <a class="existingWikiWord" href="/nlab/show/object+classifier">object classifier</a></p> </li> </ul> <h2 id="sidebar_presentations">Presentations</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/bundle+gerbe">bundle gerbe</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/groupal+model+for+universal+principal+%E2%88%9E-bundles">groupal model for universal principal ∞-bundles</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/microbundle">microbundle</a></p> </li> </ul> <h2 id="sidebar_examples">Examples</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/empty+bundle">empty bundle</a>, <a class="existingWikiWord" href="/nlab/show/zero+bundle">zero bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/trivial+vector+bundle">trivial vector bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a>, <a class="existingWikiWord" href="/nlab/show/normal+bundle">normal bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tautological+line+bundle">tautological line bundle</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/basic+line+bundle+on+the+2-sphere">basic line bundle on the 2-sphere</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hopf+fibration">Hopf fibration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/canonical+line+bundle">canonical line bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/prequantum+circle+bundle">prequantum circle bundle</a>, <a class="existingWikiWord" href="/nlab/show/prequantum+circle+n-bundle">prequantum circle n-bundle</a></p> </li> </ul> <h2 id="sidebar_constructions">Constructions</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/clutching+construction">clutching construction</a></li> </ul> </div></div> <h4 id="cohomology">Cohomology</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a>, <a class="existingWikiWord" href="/nlab/show/coboundary">coboundary</a>, <a class="existingWikiWord" href="/nlab/show/coefficient">coefficient</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homology">homology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/chain">chain</a>, <a class="existingWikiWord" href="/nlab/show/cycle">cycle</a>, <a class="existingWikiWord" href="/nlab/show/boundary">boundary</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/characteristic+class">characteristic class</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+characteristic+class">universal characteristic class</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/secondary+characteristic+class">secondary characteristic class</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+characteristic+class">differential characteristic class</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+sequence">fiber sequence</a>/<a class="existingWikiWord" href="/nlab/show/long+exact+sequence+in+cohomology">long exact sequence in cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+%E2%88%9E-bundle">fiber ∞-bundle</a>, <a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a>, <a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundle">associated ∞-bundle</a>,</p> <p><a class="existingWikiWord" href="/nlab/show/twisted+%E2%88%9E-bundle">twisted ∞-bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group+extension">∞-group extension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/obstruction">obstruction</a></p> </li> </ul> <h3 id="special_and_general_types">Special and general types</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cochain+cohomology">cochain cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a>, <a class="existingWikiWord" href="/nlab/show/singular+cohomology">singular cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+cohomology">group cohomology</a>, <a class="existingWikiWord" href="/nlab/show/nonabelian+group+cohomology">nonabelian group cohomology</a>, <a class="existingWikiWord" href="/nlab/show/Lie+group+cohomology">Lie group cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Galois+cohomology">Galois cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/groupoid+cohomology">groupoid cohomology</a>, <a class="existingWikiWord" href="/nlab/show/nonabelian+groupoid+cohomology">nonabelian groupoid cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+%28Eilenberg-Steenrod%29+cohomology">generalized (Eilenberg-Steenrod) cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+cohomology+theory">cobordism cohomology theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integral+cohomology">integral cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K-theory">K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/elliptic+cohomology">elliptic cohomology</a>, <a class="existingWikiWord" href="/nlab/show/tmf">tmf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/taf">taf</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/de+Rham+cohomology">de Rham cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dolbeault+cohomology">Dolbeault cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/etale+cohomology">etale cohomology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/group+of+units">group of units</a>, <a class="existingWikiWord" href="/nlab/show/Picard+group">Picard group</a>, <a class="existingWikiWord" href="/nlab/show/Brauer+group">Brauer group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/crystalline+cohomology">crystalline cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/syntomic+cohomology">syntomic cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/motivic+cohomology">motivic cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+of+operads">cohomology of operads</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Hochschild cohomology</a>, <a class="existingWikiWord" href="/nlab/show/cyclic+cohomology">cyclic cohomology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/string+topology">string topology</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nonabelian+cohomology">nonabelian cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+principal+%E2%88%9E-bundle">universal principal ∞-bundle</a>, <a class="existingWikiWord" href="/nlab/show/groupal+model+for+universal+principal+%E2%88%9E-bundles">groupal model for universal principal ∞-bundles</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a>, <a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+groupoid">Atiyah Lie groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+2-bundle">principal 2-bundle</a>/<a class="existingWikiWord" href="/nlab/show/gerbe">gerbe</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+constant+%E2%88%9E-stack">covering ∞-bundle</a>/<a class="existingWikiWord" href="/nlab/show/local+system">local system</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-vector+bundle">(∞,1)-vector bundle</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-vector+bundle">(∞,n)-vector bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/orientation">orientation</a>, <a class="existingWikiWord" href="/nlab/show/Spin+structure">Spin structure</a>, <a class="existingWikiWord" href="/nlab/show/Spin%5Ec+structure">Spin^c structure</a>, <a class="existingWikiWord" href="/nlab/show/String+structure">String structure</a>, <a class="existingWikiWord" href="/nlab/show/Fivebrane+structure">Fivebrane structure</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+with+constant+coefficients">cohomology with constant coefficients</a> / <a class="existingWikiWord" href="/nlab/show/cohomology+with+a+local+system+of+coefficients">with a local system of coefficients</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebra+cohomology">∞-Lie algebra cohomology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Lie+algebra+cohomology">Lie algebra cohomology</a>, <a class="existingWikiWord" href="/nlab/show/nonabelian+Lie+algebra+cohomology">nonabelian Lie algebra cohomology</a>, <a class="existingWikiWord" href="/nlab/show/Lie+algebra+extensions">Lie algebra extensions</a>, <a class="existingWikiWord" href="/nlab/show/Gelfand-Fuks+cohomology">Gelfand-Fuks cohomology</a>,</li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gerstenhaber-Schack+cohomology">bialgebra cohomology</a></p> </li> </ul> <h3 id="special_notions">Special notions</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%C4%8Cech+cohomology">Čech cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hypercohomology">hypercohomology</a></p> </li> </ul> <h3 id="variants">Variants</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+cohomology">equivariant cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+theory">equivariant homotopy theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bredon+cohomology">Bredon cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+bundle">twisted bundle</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/twisted+K-theory">twisted K-theory</a>, <a class="existingWikiWord" href="/nlab/show/twisted+spin+structure">twisted spin structure</a>, <a class="existingWikiWord" href="/nlab/show/twisted+spin%5Ec+structure">twisted spin^c structure</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+differential+c-structures">twisted differential c-structures</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/twisted+differential+string+structure">twisted differential string structure</a>, <a class="existingWikiWord" href="/nlab/show/twisted+differential+fivebrane+structure">twisted differential fivebrane structure</a></li> </ul> </li> </ul> </li> <li> <p>differential cohomology</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential generalized (Eilenberg-Steenrod) cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+cobordism+cohomology">differential cobordism cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+K-theory">differential K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+elliptic+cohomology">differential elliptic cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/schreiber/show/differential+cohomology+in+a+cohesive+topos">differential cohomology in a cohesive topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory">Chern-Weil theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Weil+theory">∞-Chern-Weil theory</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relative+cohomology">relative cohomology</a></p> </li> </ul> <h3 id="extra_structure">Extra structure</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge+structure">Hodge structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orientation">orientation</a>, <a class="existingWikiWord" href="/nlab/show/orientation+in+generalized+cohomology">in generalized cohomology</a></p> </li> </ul> <h3 id="operations">Operations</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+operations">cohomology operations</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cup+product">cup product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connecting+homomorphism">connecting homomorphism</a>, <a class="existingWikiWord" href="/nlab/show/Bockstein+homomorphism">Bockstein homomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+integration">fiber integration</a>, <a class="existingWikiWord" href="/nlab/show/transgression">transgression</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+localization">cohomology localization</a></p> </li> </ul> <h3 id="theorems">Theorems</h3> <ul> <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> <li> <p><a class="existingWikiWord" href="/nlab/show/de+Rham+theorem">de Rham theorem</a>, <a class="existingWikiWord" href="/nlab/show/Poincare+lemma">Poincare lemma</a>, <a class="existingWikiWord" href="/nlab/show/Stokes+theorem">Stokes theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge+theory">Hodge theory</a>, <a class="existingWikiWord" href="/nlab/show/Hodge+theorem">Hodge theorem</a></p> <p><a class="existingWikiWord" href="/nlab/show/nonabelian+Hodge+theory">nonabelian Hodge theory</a>, <a class="existingWikiWord" href="/nlab/show/noncommutative+Hodge+theory">noncommutative Hodge theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brown+representability+theorem">Brown representability theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">hypercovering theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Eckmann-Hilton+duality">Eckmann-Hilton-Fuks duality</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/cohomology+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <ul> <li><a href='#InTopologicalSpaces'>In the category of topological spaces</a></li> <li><a href='#InA2Topos'>In a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(2,1)</annotation></semantics></math>-topos</a></li> <ul> <li><a href='#in_terms_of_fiber_sequences'>In terms of fiber sequences</a></li> <li><a href='#the_action_from_the_homotopy_pullback'>The <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>-action from the homotopy pullback</a></li> <li><a href='#UnwindingTheAbstractDefinition'>Unwinding the abstract description</a></li> </ul> <li><a href='#Other'>Other internalizations</a></li> <ul> <li><a href='#higher_generalizations'>Higher generalizations</a></li> </ul> </ul> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#gauge_groupoid'>Gauge groupoid</a></li> </ul> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#hopf_fibration'>Hopf fibration</a></li> <li><a href='#pullbacks_of_universal_bundles'>Pullbacks of universal bundles</a></li> <li><a href='#QuotientsByLieGroupActions'>Quotients by Lie group actions</a></li> <li><a href='#CosetSpaces'>Coset projections</a></li> </ul> <li><a href='#gauge_theory'>Gauge theory</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#References'>References</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#ReferencesInternalToMoreGeneralCategories'>Internal to more general categories</a></li> <li><a href='#ReferencesInRelationToLieGroupoids'>In relation to Lie groupoids</a></li> <li><a href='#examples_2'>Examples</a></li> <li><a href='#ReferencesExtensions'>Extensions</a></li> <li><a href='#automorphism_groups'>Automorphism groups</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <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> a <a class="existingWikiWord" href="/nlab/show/group">group</a> (<a class="existingWikiWord" href="/nlab/show/internalization">internal</a> to some <a class="existingWikiWord" href="/nlab/show/category">category</a>, traditionally that of <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a>) and <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> some other object, a <em><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>-principal bundle</em> over <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> – also called 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>-<a class="existingWikiWord" href="/nlab/show/torsor">torsor</a> over <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 a <a class="existingWikiWord" href="/nlab/show/bundle">bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">P \to X</annotation></semantics></math> equipped with 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>-<a class="existingWikiWord" href="/nlab/show/action">action</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo>:</mo><mi>P</mi><mo>×</mo><mi>G</mi><mo>→</mo><mi>P</mi></mrow><annotation encoding="application/x-tex">\rho : P \times G \to P</annotation></semantics></math> on <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> over <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>, such that</p> <ul> <li> <p>the action is <em>principal</em> meaning that</p> <ul> <li>the <a class="existingWikiWord" href="/nlab/show/shear+map">shear map</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>p</mi> <mn>1</mn></msub><mo>,</mo><mi>ρ</mi><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><mi>P</mi><mo>×</mo><mi>G</mi><mo>→</mo><mi>P</mi><msub><mo>×</mo> <mi>X</mi></msub><mi>P</mi></mrow><annotation encoding="application/x-tex">(p_1, \rho) \colon P \times G \to P \times_X P</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>, which in turn means that the action is <em><a class="existingWikiWord" href="/nlab/show/free+action">free</a></em> and <em><a class="existingWikiWord" href="/nlab/show/transitive+action">transitive</a></em> over <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>, hence that each <a class="existingWikiWord" href="/nlab/show/fiber">fiber</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">P \to X</annotation></semantics></math> looks like <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> once we choose a base point;</li> </ul> <p>and / or / equivalently (depending on technical details, see below)</p> <ul> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">P \to X</annotation></semantics></math> is isomorphic to the <a class="existingWikiWord" href="/nlab/show/quotient">quotient</a> map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>→</mo><mi>P</mi><mo stretchy="false">/</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">P \to P/G</annotation></semantics></math>.</li> </ul> </li> </ul> <p>and usually it is required that</p> <ul> <li>the bundle is <em>locally trivial</em> in that there is a <a class="existingWikiWord" href="/nlab/show/cover">cover</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>:</mo><mi>U</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\phi : U \to X</annotation></semantics></math> and an isomorphism 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>-actions between the <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ϕ</mi> <mo>*</mo></msup><mi>P</mi></mrow><annotation encoding="application/x-tex">\phi^* P</annotation></semantics></math> and the <em>trivial <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>-principal bundle</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>×</mo><mi>G</mi><mo>→</mo><mi>U</mi></mrow><annotation encoding="application/x-tex">U \times G \to U</annotation></semantics></math> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math>.</li> </ul> <p>A central property 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>-principal bundles over <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 that they are a geometric model of the degree-1 <a class="existingWikiWord" href="/nlab/show/nonabelian+cohomology">nonabelian cohomology</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> with coefficients in <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>. More precisely (subject to some technical details discussed below) there is a <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural isomorphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>G</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>G</mi><mi>Bund</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo>∼</mo></mrow><annotation encoding="application/x-tex"> H^1(X, G) \simeq G Bund(X)/{\sim} </annotation></semantics></math></div> <p>between the degree-1 <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>-<a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> and the <a class="existingWikiWord" href="/nlab/show/isomorphism+classes">isomorphism classes</a> 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>-principal bundles over <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>.</p> <p>The “naturality” of this relation is more pronounced when one refines it from cohomology to <a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a> <a class="existingWikiWord" href="/nlab/show/groupoids">groupoids</a>. This is discussed below in <a href="#SomeSection">some section</a>,</p> <h2 id="definition">Definition</h2> <p>We discuss first the definition of principal bundles</p> <ul> <li><a href="#InTopologicalSpaces">In the category of topological spaces</a></li> </ul> <p>This is historically and traditionally the default setup. But the theory exists in and is usefully regarded from a more abstract perspective, which, most naturally, is that of a <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-topos">(2,1)-topos</a>. This we introduce and discuss in detail in</p> <ul> <li><a href="#InA2Topos">In an (2,1)-topos</a>.</li> </ul> <p>Finally in</p> <ul> <li><a href="#OtherInternalizations">Other internalizations</a></li> </ul> <p>we discuss how the traditional setup and many other contexts are recovered from and illuminated by that abstract perspective.</p> <h3 id="InTopologicalSpaces">In the category of topological spaces</h3> <p>We discuss here principal bundles in the context <a class="existingWikiWord" href="/nlab/show/Top">Top</a> of <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a>. So the 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> here is a <em><a class="existingWikiWord" href="/nlab/show/topological+group">topological group</a></em>.</p> <p>This is the original and oldest branch of the theory. There is a modern established default of the definition, but many slight but crucial variants exists in the literature and are relevant in applications. We start with the modern default notion and then look into its variants.</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></p> <p>Let <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> be a <a class="existingWikiWord" href="/nlab/show/topological+group">topological group</a>.</p> <div class="num_defn" id="TrivialTopologicalPrincipalBundle"> <h6 id="definition_2">Definition</h6> <p>The <strong>trivial <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>-principal bundle</strong> on a <a class="existingWikiWord" href="/nlab/show/topological+space">topological 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> is the <a class="existingWikiWord" href="/nlab/show/product">product</a> space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">X \times G</annotation></semantics></math> equipped with</p> <ul> <li> <p>the <a class="existingWikiWord" href="/nlab/show/projection">projection</a> map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mn>1</mn></msub><mo>:</mo><mi>X</mi><mo>×</mo><mi>G</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">p_1 : X \times G \to X</annotation></semantics></math> ;</p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/action">action</a> 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> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">X \times G</annotation></semantics></math> by right multiplication 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> on itself.</p> </li> </ul> </div> <div class="num_defn"> <h6 id="definition_3">Definition</h6> <p>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>-<strong>principal bundle</strong> over a <a class="existingWikiWord" href="/nlab/show/topological+space">topological 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> is a topological space <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> equipped with</p> <ul> <li> <p>a <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>:</mo><mi>P</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">p : P \to X</annotation></semantics></math>;</p> </li> <li> <p>an <a class="existingWikiWord" href="/nlab/show/action">action</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo>:</mo><mi>P</mi><mo>×</mo><mi>G</mi><mo>→</mo><mi>P</mi></mrow><annotation encoding="application/x-tex">\rho : P \times G \to P</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> on <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> over <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>, hence fitting into a <a class="existingWikiWord" href="/nlab/show/coequalizer">coequalizing</a> <a class="existingWikiWord" href="/nlab/show/diagram">diagram</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><mi>P</mi><mo>×</mo><mi>G</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>p</mi> <mn>1</mn></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>ρ</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>P</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>p</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>X</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ P \times G \\ {}^{\mathllap{p_1}} \downarrow \downarrow^{\mathrlap{\rho}} \\ P \\ \downarrow^{\mathrlap{p}} \\ X } </annotation></semantics></math></div></li> </ul> <p>such that this is <em>locally trivial</em> in the sense that</p> <ul> <li>there exists a <a class="existingWikiWord" href="/nlab/show/cover">cover</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">U \to X</annotation></semantics></math> and a <a class="existingWikiWord" href="/nlab/show/continuous+map">continuous map</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><msub><mo stretchy="false">|</mo> <mi>U</mi></msub><mo>→</mo><mi>U</mi><mo>×</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">P|_U \to U \times G</annotation></semantics></math> from the <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> 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> to the cover to the trivial <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>-pricipal bundle on the cover, def. <a class="maruku-ref" href="#TrivialTopologicalPrincipalBundle"></a>, which is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> 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>-<a class="existingWikiWord" href="/nlab/show/actions">actions</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math></li> </ul> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>U</mi><mo>×</mo><mi>G</mi></mtd> <mtd><mover><mo>←</mo><mrow><msub><mo>≃</mo> <mi>G</mi></msub></mrow></mover></mtd> <mtd><mi>P</mi><msub><mo stretchy="false">|</mo> <mi>U</mi></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>P</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>p</mi> <mn>1</mn></msub></mrow></mpadded></msub><mo>↘</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>p</mi></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>U</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>X</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ U \times G& \stackrel{\simeq_G}{\leftarrow} & P|_U &\to& P \\ &{}_{\mathllap{p_1}}\searrow&\downarrow && \downarrow^{\mathrlap{p}} \\ &&U &\to& X } \,. </annotation></semantics></math></div></div> <p>In the <a href="#References">references</a> listed below, this appears for instance as (<a href="#Mitchell">Mitchell, section 2</a>, …)</p> <p>A central property of the above definition of principal bundle is</p> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">P \to X</annotation></semantics></math> 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>-principal bundle, it is naturally <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphic</a> to the <a class="existingWikiWord" href="/nlab/show/quotient">quotient</a> projection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>→</mo><mi>P</mi><mo stretchy="false">/</mo><mi>G</mi><mo>≃</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">P \to P/G \simeq X</annotation></semantics></math> of the <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>-<a class="existingWikiWord" href="/nlab/show/action">action</a>.</p> </div> <p> <div class='num_remark' id='CartanPrincipalBundles'> <h6>Remark</h6> <p><strong>(Cartan principal bundles)</strong></p> <p>Historically this quotient property of a free continuous action was sometimes taken as the very definition of “principal bundle” without requiring <a class="existingWikiWord" href="/nlab/show/local+triviality">local triviality</a>, e. g. in (<a href="#Cartan">Cartan, 1949-1950</a>), where this perspective is attributed to <a class="existingWikiWord" href="/nlab/show/Henri+Cartan">Henri Cartan</a>. A standard modern textbook following this tradition is (<a href="#Husemoeller">Husemöller</a>).</p> <p>Therefore in order to avoid ambiguous terminology in the following, we will now follow (<a href="#Palais61">Palais 61, Def. 1.1.2</a>) and refer to this alternative definition of principal bundle as that of <em>Cartan principal bundle</em>:</p> <p></p> </div> </p> <div class="num_defn"> <h6 id="definition_4">Definition</h6> <p>Let <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> be a <a class="existingWikiWord" href="/nlab/show/locally+compact+topological+group">locally compact topological group</a> 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> a <a class="existingWikiWord" href="/nlab/show/completely+regular+topological+space">completely regular topological space</a> equipped with a <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a> <a class="existingWikiWord" href="/nlab/show/action">action</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo>:</mo><mi>P</mi><mo>×</mo><mi>G</mi><mo>→</mo><mi>P</mi></mrow><annotation encoding="application/x-tex">\rho : P \times G \to P</annotation></semantics></math>. If <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> acts <em><a class="existingWikiWord" href="/nlab/show/free+action">freely</a></em> on <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>, (no element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>∈</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">g \in G</annotation></semantics></math> except the neutral element has any fixed points in <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> under the action) then the <a class="existingWikiWord" href="/nlab/show/coprojection">coprojection</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>→</mo><mi>P</mi><mo stretchy="false">/</mo><mi>G</mi><mo>=</mo><mo>:</mo><mi>X</mi></mrow><annotation encoding="application/x-tex"> P \to P/G =: X </annotation></semantics></math></div> <p>to the <a class="existingWikiWord" href="/nlab/show/quotient+topology">topological quotient</a> is called a <strong><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>-principal bundle in the wide sense</strong>. If furthermore the division map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>P</mi><msub><mo>×</mo> <mi>X</mi></msub><mi>P</mi><mo>→</mo><mi>G</mi></mrow><annotation encoding="application/x-tex"> P \times_X P \to G </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a>, then this is called a <strong>Cartan principal bundle</strong> (<a href="#Palais61">Palais 61, around theorem 1.1.3</a>), following (<a href="#Cartan">Cartan</a>).</p> <p>(…)</p> </div> <h3 id="InA2Topos">In a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(2,1)</annotation></semantics></math>-topos</h3> <p>It is no surprise that there is a good theory of principal bundles internal to every <a class="existingWikiWord" href="/nlab/show/topos">topos</a>. However, it turns out that the most “natural home” of the theory is the <a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theoretic</a> context of a <em><a class="existingWikiWord" href="/nlab/show/%282%2C1%29-topos">(2,1)-topos</a></em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>. This we discuss now, and then relate it to the traditional notion and to various other generalizations. More along these lines is at <em><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+principal+bundles">geometry of physics – principal bundles</a></em>.</p> <p>Notably the existence of <a class="existingWikiWord" href="/nlab/show/universal+principal+bundles">universal principal bundles</a> finds its fundamental “explanation” here, where they are seen to be but a presentation of the construction of the <a class="existingWikiWord" href="/nlab/show/homotopy+fiber">homotopy fiber</a> functor, which establishes the <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence of groupoids</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>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mo>≃</mo></mover><mi>G</mi><mi>Bund</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \mathbf{H}(X, \mathbf{B}G) \stackrel{\simeq}{\to} G Bund(X) \,, </annotation></semantics></math></div> <p>where on the left we have the <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a> of <a class="existingWikiWord" href="/nlab/show/cocycles">cocycles</a> with coefficients in the internal <a class="existingWikiWord" href="/nlab/show/delooping">delooping</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>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/group+object+in+an+%28infinity%2C1%29-category">group object</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>: the <a class="existingWikiWord" href="/nlab/show/moduli+stack">moduli stack</a> 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>-principal bundles.</p> <p>In this context, all of the non-natural aspects of the traditional theory of principal bundles disappear, for instance</p> <ul> <li> <p><em>every</em> <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>-principal bundle is locally trivial in a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(2,1)</annotation></semantics></math>-topos <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>;</p> </li> <li> <p>accordingly there is no mismatch between the various definitions anymore as in the context of topological spaces: the condition of principality becomes equivalent to the quotient space condition.</p> </li> </ul> <p>Moreover, all these facts are fairly direct consequences simply of the <a class="existingWikiWord" href="/nlab/show/Giraud+axioms">Giraud axioms</a> that characterize <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-toposes">(2,1)-toposes</a> in the first place.</p> <p>Conversely, the traditional theory nicely naturally embeds into a <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-topos">(2,1)-topos</a> – for instance that of <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-sheaves">(2,1)-sheaves</a> over the <a class="existingWikiWord" href="/nlab/show/site">site</a> <a class="existingWikiWord" href="/nlab/show/Top">Top</a> (or rather some <a class="existingWikiWord" href="/nlab/show/small+site">small</a> <a class="existingWikiWord" href="/nlab/show/dense+subsite">dense subsite</a> thereof) – and the higher topos theory helps to study it there.</p> <p>The failure of various definitions to match in the traditonal context becomes the fact that the <a class="existingWikiWord" href="/nlab/show/colimits">colimits</a> involved get “corrected” to <a class="existingWikiWord" href="/nlab/show/homotopy+colimits">homotopy colimits</a> after embedding into a <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-topos">(2,1)-topos</a>. For instance if an <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>-<a class="existingWikiWord" href="/nlab/show/action">action</a> on some object <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 not suitably free, then the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(2,1)</annotation></semantics></math>-topos theory still produces a healthy principal bundle by replacing the base space by a base <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a>/<a class="existingWikiWord" href="/nlab/show/stack">stack</a>. In fact, this way <em>every</em> action becomes principal over its <a class="existingWikiWord" href="/nlab/show/homotopy+colimit">homotopy quotient</a>. Notably the trivial <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>-action on the <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo></mrow><annotation encoding="application/x-tex">*</annotation></semantics></math> becomes principal over the <a class="existingWikiWord" href="/nlab/show/action+groupoid">action groupoid</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">*//G</annotation></semantics></math> and the resulting <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>-principal bundle is nothing but the <a class="existingWikiWord" href="/nlab/show/universal+principal+bundle">universal</a> one.</p> <p>As the notation suggests, thus formulating the theory in <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-topos">(2,1)-topos</a> theory immediately generalizes it to <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos+theory">(∞,1)-topos theory</a>. This is discussed at <em><a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a></em>.</p> <p>(…)</p> <blockquote> <p>The following is old material collected from elsewhere that is going to be rearranged….</p> </blockquote> <h4 id="in_terms_of_fiber_sequences">In terms of fiber sequences</h4> <p>This indicates the more fundamental way to <em>define</em> <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>-principal bundles in the first place:</p> <p>Recall (from <a class="existingWikiWord" href="/nlab/show/fiber+sequence">fiber sequence</a>) that for every <a class="existingWikiWord" href="/nlab/show/group">group</a> there is the one-object <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>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math>. Under the <a class="existingWikiWord" href="/nlab/show/Yoneda+embedding">Yoneda embedding</a> this <a class="existingWikiWord" href="/nlab/show/representable+functor">represents</a> a <a class="existingWikiWord" href="/nlab/show/simplicial+presheaf">prestack</a>. Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><mo stretchy="false">¯</mo></mover></mrow><annotation encoding="application/x-tex">\bar{\mathbf{B}G}</annotation></semantics></math> for the corresponding <a class="existingWikiWord" href="/nlab/show/stack">stack</a> obtained by <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-sheafification">stackification</a>. This is our <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mi>Bund</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G Bund(-)</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>G</mi><mi>Bund</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>=</mo><mover><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><mo stretchy="false">¯</mo></mover><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> G Bund(-) = \bar{\mathbf{B} G}(-) \,. </annotation></semantics></math></div> <p>This perspective in turn is by <a class="existingWikiWord" href="/nlab/show/category+theory">general abstract nonsense</a> equivalent to the following useful description:</p> <p>Let <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> be the suitable <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> internal to which one looks at <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>-principal bundles. For instance for topological bundles this would be <a class="existingWikiWord" href="/nlab/show/Top">Top</a>. For smooth bundles it would be the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-sheaves">(∞,1)-category of (∞,1)-sheaves</a> on <a class="existingWikiWord" href="/nlab/show/Diff">Diff</a>, etc.</p> <p>Then every element in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mi>Bund</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≃</mo><mover><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><mo stretchy="false">¯</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G Bund(X) \simeq \bar{\mathbf{B} G}(X)</annotation></semantics></math> is given by a morphism in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H(X,\mathbf{B}G)</annotation></semantics></math>, which may be thought of as an <a class="existingWikiWord" href="/nlab/show/anafunctor">anafunctor</a> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math> from the (categorially) <a class="existingWikiWord" href="/nlab/show/discrete+category">discrete category</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>; the <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>-principal bundle from the beginning of the above definition is just the <a class="existingWikiWord" href="/nlab/show/homotopy+limit">homotopy pullback</a> of the <a class="existingWikiWord" href="/nlab/show/point">point</a> along this map, i.e. the <a class="existingWikiWord" href="/nlab/show/fibration+sequence">homotopy fiber</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">X \to \mathbf{B}G</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>P</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ P &\to& {*} \\ \downarrow && \downarrow \\ X &\to& \mathbf{B} G } \,. </annotation></semantics></math></div> <p>This diagram, incidentally, directly tells us about another important property 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>-principal bundles: they all canonically trivialize when pulled back to their own total space <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>.</p> <p>This is what the homotopy commutativity of the above homotopy pullback diagram says: the cocycle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">X \to \mathbf{B}G</annotation></semantics></math> pulled back to the bundle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">P \to X</annotation></semantics></math> that it classifies becomes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>→</mo><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">P \to X \to \mathbf{B}G</annotation></semantics></math>, which is homotopic to the trivial cocycle (the one that factors through the <a class="existingWikiWord" href="/nlab/show/point">point</a>) on <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>.</p> <p>The homotopy pullback here is conveniently and traditionally computed as an ordinary pullback of a <a class="existingWikiWord" href="/nlab/show/fibration">fibrant replacement</a> of the pullback diagram. The canonical such fibrant replacement is obtained by replacing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">{*} \to \mathbf{B}G</annotation></semantics></math> by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>E</mi></mstyle><mi>G</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{E}G \to \mathbf{B}G</annotation></semantics></math>, with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>E</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{E}G</annotation></semantics></math> an object <a class="existingWikiWord" href="/nlab/show/weak+equivalence">weakly equivalent</a> to the point, called the <strong><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>-<a class="existingWikiWord" href="/nlab/show/universal+principal+bundle">universal principal bundle</a></strong>.</p> <p>With that the above homotopy pullback is computed as the ordinary <a class="existingWikiWord" href="/nlab/show/pullback">pullback</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><mi>P</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>E</mi></mstyle><mi>G</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ P &\to& \mathbf{E}G \\ \downarrow && \downarrow \\ X &\to& \mathbf{B} G } </annotation></semantics></math></div> <p>So every <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>-principal bundle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">P \to X</annotation></semantics></math> is the pullback along a classifying map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">X \to \mathbf{B}G</annotation></semantics></math> (in the right <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-categorical context, otherwise a span such as an <a class="existingWikiWord" href="/nlab/show/anafunctor">anafunctor</a>) of the <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>-<a class="existingWikiWord" href="/nlab/show/universal+principal+bundle">universal principal bundle</a>.</p> <h4 id="the_action_from_the_homotopy_pullback">The <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>-action from the homotopy pullback</h4> <p>Given the definition of the bundle <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> in terms of a <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">{*} \to \mathbf{B}G</annotation></semantics></math> we re-obtain the <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>-<a class="existingWikiWord" href="/nlab/show/action">action</a> on <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> as follows (with an eye towards its generalization to <a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundles">principal ∞-bundles</a>).</p> <p>Let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>⋯</mi><mi>G</mi><mo>×</mo><mi>G</mi><mover><mover><mover><mo>⟶</mo><mrow><msub><mi>d</mi> <mn>0</mn></msub></mrow></mover><mover><mo>⟶</mo><mrow><msub><mi>d</mi> <mn>1</mn></msub></mrow></mover></mover><mover><mo>⟶</mo><mrow><msub><mi>d</mi> <mn>2</mn></msub></mrow></mover></mover><mi>G</mi><mover><mover><mo>⟶</mo><mrow><msub><mi>d</mi> <mn>0</mn></msub></mrow></mover><mover><mo>⟶</mo><mrow><msub><mi>d</mi> <mn>1</mn></msub></mrow></mover></mover><mo>*</mo><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex"> \cdots G \times G \stackrel{\stackrel{d_2}{\longrightarrow}}{\stackrel{\stackrel{d_1}{\longrightarrow}}{\stackrel{d_0}{\longrightarrow}}} G \stackrel{\stackrel{d_1}{\longrightarrow}}{\stackrel{d_0}{\longrightarrow}} {*} \to \mathbf{B}G </annotation></semantics></math></div> <p>be the <a class="existingWikiWord" href="/nlab/show/quotient+object">effective</a> <a class="existingWikiWord" href="/nlab/show/groupoid+object+in+an+%28%E2%88%9E%2C1%29-category">groupoid object in an (∞,1)-category</a> that exhibits the <a class="existingWikiWord" href="/nlab/show/delooping">delooping</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>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</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>.</p> <p>Form the <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a> of the classifying morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">X \to \mathbf{B}G</annotation></semantics></math> along the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">d_0</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/simplicial+object">face maps</a> of this diagram. This yields a diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>⋯</mi></mtd> <mtd><mi>P</mi><mo>×</mo><mi>G</mi><mo>×</mo><mi>G</mi></mtd> <mtd><mover><mover><mo>⟶</mo><mo>⟶</mo></mover><mo>⟶</mo></mover></mtd> <mtd><mi>P</mi><mo>×</mo><mi>G</mi></mtd> <mtd><mover><mo>⟶</mo><mo>⟶</mo></mover></mtd> <mtd><mi>P</mi></mtd> <mtd><mover><mo>⟶</mo><mrow></mrow></mover></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>⋯</mi></mtd> <mtd><mi>G</mi><mo>×</mo><mi>G</mi></mtd> <mtd><mover><mover><mo>⟶</mo><mo>⟶</mo></mover><mo>⟶</mo></mover></mtd> <mtd><mi>G</mi></mtd> <mtd><mover><mo>⟶</mo><mo>⟶</mo></mover></mtd> <mtd><mo>*</mo></mtd> <mtd><mover><mo>⟶</mo><mrow></mrow></mover></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \cdots & P \times G \times G & \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}} & P \times G & \stackrel{\longrightarrow}{\longrightarrow} & P & \stackrel{}{\longrightarrow} & X \\ & \downarrow && \downarrow && \downarrow && \downarrow \\ \cdots & G \times G & \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}} & G & \stackrel{\longrightarrow}{\longrightarrow} & {*} & \stackrel{}{\longrightarrow} & \mathbf{B}G } </annotation></semantics></math></div> <p>where all squares formed by the lowest horizontal morphisms are <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a> squares, by construction, and where the remaining horizontal morphisms in the top row are induced by the universal property of the <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a> and the morphisms downstairs.</p> <p>The claim is that</p> <ul> <li> <p>the top row encodes the <a class="existingWikiWord" href="/nlab/show/action">action</a> 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> on <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> in that the action is the morphism indicated <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math> in</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>⋯</mi><mover><mover><mo>⟶</mo><mo>⟶</mo></mover><mo>⟶</mo></mover><mi>P</mi><mo>×</mo><mi>G</mi><mover><mover><mo>⟶</mo><mrow><msub><mi>p</mi> <mn>1</mn></msub></mrow></mover><mover><mo>⟶</mo><mi>ρ</mi></mover></mover><mi>P</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex"> \cdots \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}} P \times G \stackrel{\stackrel{\rho}{\longrightarrow}}{\stackrel{p_1}{\longrightarrow}} P \to X </annotation></semantics></math></div></li> <li> <p>and it exhibits <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>×</mo><msup><mi>G</mi> <mrow><mo>×</mo><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msup></mrow><annotation encoding="application/x-tex">P \times G^{\times (n-1)}</annotation></semantics></math> as the <a class="existingWikiWord" href="/nlab/show/groupoid+object+in+an+%28%E2%88%9E%2C1%29-category">groupoid object in an (∞,1)-category</a> being the <a class="existingWikiWord" href="/nlab/show/%C4%8Cech+nerve">Čech nerve</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">P \to X</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>⋯</mi></mtd> <mtd><mi>P</mi><msub><mo>×</mo> <mi>X</mi></msub><mi>P</mi><msub><mo>×</mo> <mi>X</mi></msub><mi>P</mi></mtd> <mtd><mover><mover><mo>⟶</mo><mo>⟶</mo></mover><mo>⟶</mo></mover></mtd> <mtd><mi>P</mi><msub><mo>×</mo> <mi>X</mi></msub><mi>P</mi></mtd> <mtd><mover><mo>⟶</mo><mo>⟶</mo></mover></mtd> <mtd><mi>P</mi></mtd> <mtd><mover><mo>⟶</mo><mrow></mrow></mover></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mo stretchy="false">↑</mo> <mo>≃</mo></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↑</mo> <mo>≃</mo></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↑</mo> <mo>≃</mo></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↑</mo> <mo>≃</mo></msup></mtd></mtr> <mtr><mtd><mi>⋯</mi></mtd> <mtd><mi>P</mi><mo>×</mo><mi>G</mi><mo>×</mo><mi>G</mi></mtd> <mtd><mover><mover><mo>⟶</mo><mo>⟶</mo></mover><mo>→</mo></mover></mtd> <mtd><mi>P</mi><mo>×</mo><mi>G</mi></mtd> <mtd><mover><mo>⟶</mo><mover><mo>⟶</mo><mi>ρ</mi></mover></mover></mtd> <mtd><mi>P</mi></mtd> <mtd><mover><mo>⟶</mo><mrow></mrow></mover></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>⋯</mi></mtd> <mtd><mi>G</mi><mo>×</mo><mi>G</mi></mtd> <mtd><mover><mover><mo>⟶</mo><mo>⟶</mo></mover><mo>→</mo></mover></mtd> <mtd><mi>G</mi></mtd> <mtd><mover><mo>⟶</mo><mo>⟶</mo></mover></mtd> <mtd><mo>*</mo></mtd> <mtd><mover><mo>⟶</mo><mrow></mrow></mover></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \cdots & P \times_X P \times_X P & \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}} & P \times_X P & \stackrel{\longrightarrow}{\longrightarrow} & P & \stackrel{}{\longrightarrow} & X \\ & \uparrow^{\simeq} && \uparrow^{\simeq} && \uparrow^{\simeq} && \uparrow^{\simeq} \\ \cdots & P \times G \times G & \stackrel{\to}{\stackrel{\longrightarrow}{\longrightarrow}} & P \times G & \stackrel{\stackrel{\rho}{\longrightarrow}}{\longrightarrow} & P & \stackrel{}{\longrightarrow} & X \\ & \downarrow && \downarrow && \downarrow && \downarrow \\ \cdots & G \times G & \stackrel{\to}{\stackrel{\longrightarrow}{\longrightarrow}} & G & \stackrel{\longrightarrow}{\longrightarrow} & {*} & \stackrel{}{\longrightarrow} & \mathbf{B}G } </annotation></semantics></math></div></li> </ul> <p>Here the second statement in particular encodes the familiar way to formulate <strong>principality of the action</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math>, in that it says that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>×</mo><mi>G</mi><mover><mo>→</mo><mrow><msub><mi>p</mi> <mn>1</mn></msub><mo>×</mo><mi>ρ</mi></mrow></mover><mi>P</mi><msub><mo>×</mo> <mi>X</mi></msub><mi>P</mi></mrow><annotation encoding="application/x-tex"> P \times G \stackrel{p_1 \times \rho}{\to} P \times_X P </annotation></semantics></math></div> <p>is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>.</p> <p>We now unwrap the first statement in gory detail to make clear that this abstract nonsense does reproduce the familiar definition of the action 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> on <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>.</p> <h4 id="UnwindingTheAbstractDefinition">Unwinding the abstract description</h4> <p>We now rederive the <a class="existingWikiWord" href="/nlab/show/action">action</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</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> on <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> given just the classifying map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">X \to \mathbf{B} G</annotation></semantics></math> by spelling out the details implied by the above abstract description.</p> <p>Whatever the precise context is (topological, smooth, etc.) we may assume that we are at least in a <a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">category of fibrant objects</a>. Then the classifying morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">X \to \mathbf{B}G</annotation></semantics></math> is represented by an <a class="existingWikiWord" href="/nlab/show/anafunctor">anafunctor</a>, namely a cocycle</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>C</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mi>g</mi></mover></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>W</mi><mo>∩</mo><mi>F</mi></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>X</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ C(U) &\stackrel{g}{\to} & \mathbf{B}G \\ \downarrow^{\mathrlap{\in W \cap F}} \\ X } </annotation></semantics></math></div> <p>in <a class="existingWikiWord" href="/nlab/show/%C4%8Cech+cohomology">Čech cohomology</a> coming from some <a class="existingWikiWord" href="/nlab/show/cover">cover</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><mi>X</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{U_i \to X\}</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/%C4%8Cech+nerve">Čech nerve</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C(U)</annotation></semantics></math> has</p> <ul> <li> <p>objects = <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>i</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo><mi>x</mi><mo>∈</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{(x,i) | x \in U_i\}</annotation></semantics></math></p> </li> <li> <p>morphisms = <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>i</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow></mrow></mover><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo><mi>x</mi><mo>∈</mo><msub><mi>U</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{ (x,i) \stackrel{}{\longrightarrow} (x,j) | x \in U_{i j}\}</annotation></semantics></math> .</p> </li> </ul> <p>The <a class="existingWikiWord" href="/nlab/show/functor">functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>:</mo><mi>C</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">g : C(U) \to \mathbf{B}G</annotation></semantics></math> sends</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>:</mo><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>i</mi><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>↦</mo><mo stretchy="false">(</mo><mo>•</mo><mover><mo>⟶</mo><mrow><msub><mi>g</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mover><mo>•</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> g : ((x,i) \to (x,j)) \mapsto (\bullet \stackrel{g_{i j}(x)}{\longrightarrow}\bullet ) </annotation></semantics></math></div> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>∈</mo><mi>Functions</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>,</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g_{i j} \in Functions(U_{i j}, G)</annotation></semantics></math> as described in detail at <a class="existingWikiWord" href="/nlab/show/%C4%8Cech+cohomology">Čech cohomology</a>.</p> <p>With <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>E</mi></mstyle><mi>G</mi><mo>=</mo><mo stretchy="false">{</mo><mi>g</mi><mover><mo>→</mo><mi>h</mi></mover><mi>g</mi><mi>h</mi><mo stretchy="false">|</mo><mi>g</mi><mo>,</mo><mi>h</mi><mo>∈</mo><mi>G</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\mathbf{E}G = \{g \stackrel{h}{\to} g h | g,h \in G \}</annotation></semantics></math> the fibrant replacement of the point, which we shall find it helpful to think of as given by</p> <ul> <li> <p>objects = <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo>{</mo><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mo>•</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mi>g</mi></msup><mo>↙</mo></mtd></mtr> <mtr><mtd><mo>•</mo></mtd></mtr></mtable></mrow><mo>}</mo></mrow></mrow><annotation encoding="application/x-tex">\left\{ \array{ && \bullet \\ & {}^g\swarrow \\ \bullet } \right\} </annotation></semantics></math></p> </li> <li> <p>morphisms = <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo>{</mo><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mo>•</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mi>g</mi></msup><mo>↙</mo></mtd> <mtd></mtd> <mtd><msup><mo>↘</mo> <mrow><mi>g</mi><mo>′</mo><mo>=</mo><mi>g</mi><mi>h</mi></mrow></msup></mtd></mtr> <mtr><mtd><mo>•</mo></mtd> <mtd></mtd> <mtd><mover><mo>⟶</mo><mi>h</mi></mover></mtd> <mtd></mtd> <mtd><mo>•</mo></mtd></mtr></mtable></mrow><mo>}</mo></mrow></mrow><annotation encoding="application/x-tex">\left\{ \array{ && \bullet \\ & {}^g\swarrow && \searrow^{g' = g h} \\ \bullet &&\stackrel{h}{\longrightarrow}&& \bullet } \right\} </annotation></semantics></math></p> </li> </ul> <p>we compute the <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a> as the homotopy fiber product given by the ordinary <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> (see <a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">category of fibrant objects</a> for details)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>P</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>E</mi></mstyle><mi>G</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>C</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mi>g</mi></mover></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ P &\to& \mathbf{E}G \\ \downarrow && \downarrow \\ C(U) &\stackrel{g}{\longrightarrow}& \mathbf{B}G } \,. </annotation></semantics></math></div> <p>So we read off that <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 the <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a> with</p> <ul> <li> <p>objects =</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>{</mo><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mo>•</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mi>g</mi></msup><mo>↙</mo></mtd></mtr> <mtr><mtd><mo>•</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>i</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mo>}</mo></mrow></mrow><annotation encoding="application/x-tex"> \left\{ \array{ && \bullet \\ & {}^{g}\swarrow \\ \bullet \\ (x,i) } \right\} </annotation></semantics></math></div></li> <li> <p>morphisms =</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>{</mo><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mo>•</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mi>g</mi></msup><mo>↙</mo></mtd> <mtd></mtd> <mtd><msup><mo>↘</mo> <mrow><mi>g</mi><mo>′</mo></mrow></msup></mtd></mtr> <mtr><mtd><mo>•</mo></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>g</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd></mtd> <mtd><mo>•</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>i</mi><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mrow></mrow></mover></mtd> <mtd></mtd> <mtd><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mo>}</mo></mrow></mrow><annotation encoding="application/x-tex"> \left\{ \array{ && \bullet \\ & {}^{g}\swarrow && \searrow^{g'} \\ \bullet &&\stackrel{g_{i j }(x)}{\to}&& \bullet \\ (x,i) &&\stackrel{}{\to}&& (x,j) } \right\} </annotation></semantics></math></div></li> </ul> <p>With <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> determined as an ordinary pullback of a replacement it is convenient for the following to realize it in turn as the pullback-up-to-2-cell in</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>P</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><msup><mo>⇓</mo> <mrow><msub><mi>η</mi> <mi>P</mi></msub></mrow></msup></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>C</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ P &\to& {*} \\ \downarrow &\Downarrow^{\eta_P}& \downarrow \\ C(U) &\to& \mathbf{B}G } \,. </annotation></semantics></math></div> <p>A moment reflection shows that the component of the <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>η</mi> <mi>P</mi></msub></mrow><annotation encoding="application/x-tex">\eta_P</annotation></semantics></math> here is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>η</mi> <mi>P</mi></msub><mo>:</mo><mrow><mo>(</mo><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mo>•</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mi>g</mi></msup><mo>↙</mo></mtd></mtr> <mtr><mtd><mo>•</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>i</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>↦</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><mo>•</mo><mover><mo>→</mo><mi>g</mi></mover><mo>•</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \eta_P : \left( \array{ && \bullet \\ & {}^{g}\swarrow \\ \bullet \\ (x,i) } \right) \;\;\; \mapsto \;\;\; (\bullet \stackrel{g}{\to}\bullet) </annotation></semantics></math></div> <p>At the same time recall from the discussion at <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> that the component of the transformation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>η</mi> <mi>G</mi></msub></mrow><annotation encoding="application/x-tex">\eta_G</annotation></semantics></math> in</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>G</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><msup><mo>⇓</mo> <mrow><msub><mi>η</mi> <mi>G</mi></msub></mrow></msup></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ G &\to& {*} \\ \downarrow &\Downarrow^{\eta_G}& \downarrow \\ {*} &\to& \mathbf{B}G } </annotation></semantics></math></div> <p>is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>η</mi> <mi>G</mi></msub><mo>:</mo><mi>g</mi><mo>↦</mo><mo stretchy="false">(</mo><mo>•</mo><mover><mo>→</mo><mi>g</mi></mover><mo>•</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \eta_G : g \mapsto (\bullet \stackrel{g}{\to} \bullet) \,. </annotation></semantics></math></div> <p>Taken together this shows that the universal morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>×</mo><mi>G</mi><mo>→</mo><mi>P</mi></mrow><annotation encoding="application/x-tex">P \times G \to P</annotation></semantics></math> induced from the commutativity of</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></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>P</mi><mo>×</mo><mi>G</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>P</mi></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>G</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><mi>C</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd><msup><mo>⇓</mo> <mrow><msub><mi>η</mi> <mi>P</mi></msub></mrow></msup></mtd> <mtd></mtd> <mtd><mo>*</mo></mtd> <mtd></mtd> <mtd><msup><mo>⇓</mo> <mrow><msub><mi>η</mi> <mi>G</mi></msub></mrow></msup></mtd> <mtd></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo>↙</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ &&&& P \times G \\ && & \swarrow && \searrow \\ && P &&&& G \\ & \swarrow && \searrow && \swarrow && \searrow \\ C(U) &&\Downarrow^{\eta_P}&& {*} &&\Downarrow^{\eta_G}&& {*} \\ &&\searrow&& \downarrow && \swarrow \\ &&&& \mathbf{B}G } </annotation></semantics></math></div> <p>and from the <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a> property of</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>P</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><msup><mo>⇓</mo> <mrow><msub><mi>η</mi> <mi>P</mi></msub></mrow></msup></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>C</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ P &\to& {*} \\ \downarrow &\Downarrow^{\eta_P}& \downarrow \\ C(U) &\to& \mathbf{B}G } </annotation></semantics></math></div> <p>is simply given by the composition of these two component maps</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>×</mo><mi>G</mi><mo>→</mo><mi>P</mi><mo>:</mo><mrow><mo>(</mo><mrow><mo>(</mo><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mo>•</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mi>g</mi></msup><mo>↙</mo></mtd></mtr> <mtr><mtd><mo>•</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>i</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>,</mo><mrow><mtable><mtr><mtd><mo>•</mo></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mrow><mi>g</mi><mo>′</mo></mrow></msup></mtd></mtr> <mtr><mtd><mo>•</mo></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>↦</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mo>(</mo><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mo>•</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mrow><mi>g</mi><mo>′</mo></mrow></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>•</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mi>g</mi></msup><mo>↙</mo></mtd></mtr> <mtr><mtd><mo>•</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>i</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> P \times G \to P : \left( \left( \array{ && \bullet \\ & {}^{g}\swarrow \\ \bullet \\ (x,i) } \right) \,, \array{ \bullet \\ \downarrow^{g'} \\ \bullet } \right) \; \;\; \; \mapsto \; \;\; \; \left( \array{ && \bullet \\ && \downarrow^{g'} \\ && \bullet \\ & {}^{g}\swarrow \\ \bullet \\ (x,i) } \right) \,. </annotation></semantics></math></div> <p>But this is manifestly the right (being both: from the right and correct :-) <a class="existingWikiWord" href="/nlab/show/action">action</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo>:</mo><mi>P</mi><mo>×</mo><mi>G</mi><mo>→</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">\rho : P \times G \to G</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> on <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>.</p> <h3 id="Other">Other internalizations</h3> <p>We discuss here aspects of formulating a theory of principal bundles in contexts different from those already discussed above.</p> <p>(…)</p> <h4 id="higher_generalizations">Higher generalizations</h4> <p>In <a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a> the notion of principal bundle has various <a class="existingWikiWord" href="/nlab/show/vertical+categorification">vertical categorification</a>s. See</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+2-bundle">principal 2-bundle</a>, <a class="existingWikiWord" href="/nlab/show/bundle+gerbe">bundle gerbe</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a></p> </li> </ul> <h2 id="properties">Properties</h2> <h3 id="gauge_groupoid">Gauge groupoid</h3> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">P \to X</annotation></semantics></math> 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>-principal bundle, its <strong><a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+groupoid">Atiyah Lie groupoid</a></strong> is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>P</mi><mo>×</mo><mi>P</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>G</mi><mover><munder><mo>→</mo><mrow><mi>p</mi><mo>∘</mo><msub><mi>p</mi> <mn>2</mn></msub></mrow></munder><mover><mo>→</mo><mrow><mi>p</mi><mo>∘</mo><msub><mi>p</mi> <mn>1</mn></msub></mrow></mover></mover><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> ((P \times P)/G \stackrel{\stackrel{p\circ p_1}{\to}}{\underset{p \circ p_2}{\to}} X ) </annotation></semantics></math></div> <p>with the evident composition operation.</p> <p>The principal bundle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">P \to X</annotation></semantics></math> is recovered from its Atiyah Lie groupoid, up to isomorphism, as the source fiber over any point.</p> <p>This is a classical statement due to Ehresmann … . See for instance (<a href="#Androulidakis">Androulidakis</a>).</p> <h2 id="examples">Examples</h2> <h3 id="hopf_fibration">Hopf fibration</h3> <p>A standard example of a nontrivial <a class="existingWikiWord" href="/nlab/show/circle+group">circle group</a>-principal bundle – a <a class="existingWikiWord" href="/nlab/show/circle+bundle">circle bundle</a> – is the <a class="existingWikiWord" href="/nlab/show/Hopf+fibration">Hopf fibration</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>3</mn></msup><mo>→</mo><msup><mi>S</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">S^3 \to S^2</annotation></semantics></math>, which has the structure of an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">S^1</annotation></semantics></math>-principal bundle in <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>s.</p> <h3 id="pullbacks_of_universal_bundles">Pullbacks of universal bundles</h3> <p>Generally, if we accept that we have a large supply of continuous maps between <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a>, we obtain 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>-principal bundle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup><mi>ℰ</mi><mi>G</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">f^* \mathcal{E}G \to X</annotation></semantics></math> on a space <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> for each continuous map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>ℬ</mi><mi>G</mi></mrow><annotation encoding="application/x-tex">f : X \to \mathcal{B}G</annotation></semantics></math> to the <a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a> 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>, by <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> of the <a class="existingWikiWord" href="/nlab/show/generalized+universal+bundle">universal bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℰ</mi><mi>G</mi><mo>→</mo><mi>ℬ</mi><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathcal{E}G \to \mathcal{B}G</annotation></semantics></math> along <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>.</p> <h3 id="QuotientsByLieGroupActions">Quotients by Lie group actions</h3> <p>We consider <a class="existingWikiWord" href="/nlab/show/actions">actions</a> by <a class="existingWikiWord" href="/nlab/show/topological+groups">topological groups</a> and <a class="existingWikiWord" href="/nlab/show/Lie+groups">Lie groups</a>.</p> <div class="num_prop" id="QuotientProjectionForCompactLieGroupActingFreelyOnManifoldIsPrincipa"> <h6 id="proposition_2">Proposition</h6> <p>For <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> a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/compact+Lie+group">compact Lie group</a> equipped with a <a class="existingWikiWord" href="/nlab/show/free+action">free</a> smooth <a class="existingWikiWord" href="/nlab/show/action">action</a> on <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>, then the <a class="existingWikiWord" href="/nlab/show/quotient">quotient</a> <a class="existingWikiWord" href="/nlab/show/projection">projection</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>⟶</mo><mi>X</mi><mo stretchy="false">/</mo><mi>G</mi></mrow><annotation encoding="application/x-tex"> X \longrightarrow X/G </annotation></semantics></math></div> <p>is 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>-<a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a> (hence in particular a <a class="existingWikiWord" href="/nlab/show/Serre+fibration">Serre fibration</a>).</p> </div> <p>This is originally due to (<a href="#Gleason50">Gleason 50</a>). See e.g. (<a href="#Cohen">Cohen, theorem 1.3</a>)</p> <div class="num_theorem" id="PalaisTheorem"> <h6 id="theorem">Theorem</h6> <p>Let <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> be a <a class="existingWikiWord" href="/nlab/show/completely+regular+topological+space">completely regular topological space</a> and let <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> be a <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a> equipped with a free <a class="existingWikiWord" href="/nlab/show/action">action</a> on <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>. Then the <a class="existingWikiWord" href="/nlab/show/quotient">quotient</a> map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>→</mo><mi>P</mi><mo stretchy="false">/</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">P \to P/G</annotation></semantics></math> is 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>-principal bundle – in that it is locally trivial – precisely if the division map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>P</mi><msub><mo>×</mo> <mrow><mi>P</mi><mo stretchy="false">/</mo><mi>G</mi></mrow></msub><mi>P</mi><mo>→</mo><mi>G</mi></mrow><annotation encoding="application/x-tex"> P \times_{P/G} P \to G </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a>.</p> </div> <p>This is (<a href="#Palais">Palais, theorem 4.1</a>).</p> <h3 id="CosetSpaces">Coset projections</h3> <p>We discuss principal bundles of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>→</mo><mi>P</mi><mo stretchy="false">/</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">P \to P/G</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>↪</mo><mi>P</mi></mrow><annotation encoding="application/x-tex">G \hookrightarrow P</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/subgroup">subgroup</a> of a <a class="existingWikiWord" href="/nlab/show/topological+group">topological group</a>, hence with base space a <a class="existingWikiWord" href="/nlab/show/coset">coset</a> space.</p> <div class="num_prop" id="QuotientProjectionForCompactLieSubgroupIsPrincipal"> <h6 id="proposition_3">Proposition</h6> <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> a <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo>⊂</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">H \subset G</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/compact+Lie+group">compact</a> <a class="existingWikiWord" href="/nlab/show/subgroup">subgroup</a>, then the <a class="existingWikiWord" href="/nlab/show/coset">coset</a> <a class="existingWikiWord" href="/nlab/show/quotient">quotient</a> <a class="existingWikiWord" href="/nlab/show/projection">projection</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>⟶</mo><mi>G</mi><mo stretchy="false">/</mo><mi>H</mi></mrow><annotation encoding="application/x-tex"> G \longrightarrow G/H </annotation></semantics></math></div> <p>is an <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>-<a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a> (hence in particular a <a class="existingWikiWord" href="/nlab/show/Serre+fibration">Serre fibration</a>).</p> </div> <p>This is a direct corollary of prop. <a class="maruku-ref" href="#QuotientProjectionForCompactLieGroupActingFreelyOnManifoldIsPrincipa"></a>. Originally this statement is due to (<a href="#Samelson41">Samelson 41</a>).</p> <div class="num_prop" id="ProjectionOfCosetsIsFiberBundleForClosedSubgroupsOfCompactLieGroup"> <h6 id="proposition_4">Proposition</h6> <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> a <a class="existingWikiWord" href="/nlab/show/compact+Lie+group">compact Lie group</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo>⊂</mo><mi>H</mi><mo>⊂</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">K \subset H \subset G</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/closed+subspace">closed</a> <a class="existingWikiWord" href="/nlab/show/subgroups">subgroups</a>, then the <a class="existingWikiWord" href="/nlab/show/projection">projection</a> map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>p</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>G</mi><mo stretchy="false">/</mo><mi>K</mi><mo>⟶</mo><mi>G</mi><mo stretchy="false">/</mo><mi>H</mi></mrow><annotation encoding="application/x-tex"> p \;\colon\; G/K \longrightarrow G/H </annotation></semantics></math></div> <p>is a locally trivial <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo stretchy="false">/</mo><mi>K</mi></mrow><annotation encoding="application/x-tex">H/K</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/fiber+bundle">fiber bundle</a> (hence in particular a <a class="existingWikiWord" href="/nlab/show/Serre+fibration">Serre fibration</a>).</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>Observe that the projection map in question is equivalently</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>G</mi><msub><mo>×</mo> <mi>H</mi></msub><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">/</mo><mi>K</mi><mo stretchy="false">)</mo><mo>⟶</mo><mi>G</mi><mo stretchy="false">/</mo><mi>H</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> G \times_H (H/K) \longrightarrow G/H \,, </annotation></semantics></math></div> <p>(where on the left we form the <a class="existingWikiWord" href="/nlab/show/Cartesian+product">Cartesian product</a> and then divide out the <a class="existingWikiWord" href="/nlab/show/diagonal+action">diagonal action</a> by <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>). This exhibits it as the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo stretchy="false">/</mo><mi>K</mi></mrow><annotation encoding="application/x-tex">H/K</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/fiber+bundle">fiber bundle</a> <a class="existingWikiWord" href="/nlab/show/associated+bundle">associated</a> to the <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>-<a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a> of corollary <a class="maruku-ref" href="#QuotientProjectionForCompactLieSubgroupIsPrincipal"></a>.</p> </div> <div class="num_prop"> <h6 id="proposition_5">Proposition</h6> <p>If <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/topological+group">topological group</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>↪</mo><mi>P</mi></mrow><annotation encoding="application/x-tex">G \hookrightarrow P</annotation></semantics></math> a closed <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie</a> subgroup, then the quotient map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>→</mo><mi>P</mi><mo stretchy="false">/</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">P \to P/G</annotation></semantics></math> is a locally trivial <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>-principal bundle.</p> </div> <p>This is a corollary of theorem <a class="maruku-ref" href="#PalaisTheorem"></a> (<a href="#Palais61">Palais 61</a>).</p> <p>Examples where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>→</mo><mi>P</mi><mo stretchy="false">/</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">P \to P/G</annotation></semantics></math> is <strong>not</strong> locally trivial are in (<a href="#Karube">Karube</a>), see also (<a href="#Mostert">Mostert</a>):</p> <div class="num_example"> <h6 id="example_counter_example">Example (counter example)</h6> <p>Let <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> be the <a class="existingWikiWord" href="/nlab/show/product">product</a> of infinitely many <a class="existingWikiWord" href="/nlab/show/circles">circles</a>, and let <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> be the product of their <a class="existingWikiWord" href="/nlab/show/order">order</a> 2 <a class="existingWikiWord" href="/nlab/show/subgroups">subgroups</a>. This cannot have local <a class="existingWikiWord" href="/nlab/show/section">section</a> because <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 class="existingWikiWord" href="/nlab/show/locally+connected+topological+space">locally connected</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> is not. Therefore <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 not even locally <a class="existingWikiWord" href="/nlab/show/homeomorphism">homeomorphic</a> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>P</mi><mo stretchy="false">/</mo><mi>G</mi><mo stretchy="false">)</mo><mo>×</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">(P / G) \times G</annotation></semantics></math>.</p> </div> <h2 id="gauge_theory">Gauge theory</h2> <p>In <a class="existingWikiWord" href="/nlab/show/physics">physics</a>, principal bundles <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">with connection</a> and their higher categorical analogs model <a class="existingWikiWord" href="/nlab/show/gauge+fields">gauge fields</a>. See at <em><a class="existingWikiWord" href="/nlab/show/fiber+bundles+in+physics">fiber bundles in physics</a></em>.</p> <p>In fact, the history of the development of the theory of principal bundles and <a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a> is closely related. In the early 1930s Dirac and Hopf independently introduced <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(1)</annotation></semantics></math>-principal bundles: Dirac, somewhat implicitly, in his study of the <a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a> as a background for <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">quantum mechanics</a>, Hopf in terms of the fibration named after him. However, from there it took apparently many years for the first publication to appear that explicitly states that these two considerations are aspects of the same phenomenon.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/slice+theorem">slice theorem</a></p> </li> <li> <p><strong>(<a class="existingWikiWord" href="/nlab/show/formally+principal+bundle">formally</a>) principal bundle</strong> / (<a class="existingWikiWord" href="/nlab/show/pseudo-torsor">pseudo</a>-)<a class="existingWikiWord" href="/nlab/show/torsor">torsor</a> / <a class="existingWikiWord" href="/nlab/show/groupoid+principal+bundle">groupoid principal bundle</a> / <a class="existingWikiWord" href="/nlab/show/associated+bundle">associated bundle</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+principal+bundle">universal principal bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/caloron+correspondence">caloron correspondence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Galois+cover">Galois cover</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bibundle">bibundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/groupoid+principal+bundle">groupoid principal bundle</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+2-bundle">principal 2-bundle</a> / <a class="existingWikiWord" href="/nlab/show/gerbe">gerbe</a> / <a class="existingWikiWord" href="/nlab/show/bundle+gerbe">bundle gerbe</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+3-bundle">principal 3-bundle</a> / <a class="existingWikiWord" href="/nlab/show/2-gerbe">2-gerbe</a> / <a class="existingWikiWord" href="/nlab/show/bundle+2-gerbe">bundle 2-gerbe</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a> / <a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundle">associated ∞-bundle</a> / <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-gerbe">∞-gerbe</a></p> <p><a class="existingWikiWord" href="/nlab/show/universal+principal+%E2%88%9E-bundle">universal principal ∞-bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/noncommutative+principal+bundle">noncommutative principal bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/n-vector+bundle">n-vector bundle</a></p> </li> </ul> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/gauge+field">gauge field</a></strong>: models and components</p> <table><thead><tr><th><strong><a class="existingWikiWord" href="/nlab/show/physics">physics</a></strong></th><th><strong><a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a></strong></th><th><strong><a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential cohomology</a></strong></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/gauge+field">gauge field</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connection on a bundle</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a> in <a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential cohomology</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/instanton+sector">instanton</a>/<a class="existingWikiWord" href="/nlab/show/charge">charge</a> sector</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a> in underlying <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/gauge+potential">gauge potential</a></td><td style="text-align: left;">local connection <a class="existingWikiWord" href="/nlab/show/differential+form">differential form</a></td><td style="text-align: left;">local connection <a class="existingWikiWord" href="/nlab/show/differential+form">differential form</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/field+strength">field strength</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/curvature">curvature</a></td><td style="text-align: left;">underlying <a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a> in <a class="existingWikiWord" href="/nlab/show/de+Rham+cohomology">de Rham cohomology</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/gauge+transformation">gauge transformation</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/equivalence">equivalence</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/coboundary">coboundary</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/minimal+coupling">minimal coupling</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/covariant+derivative">covariant derivative</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted cohomology</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/BRST+complex">BRST complex</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Lie+algebroid">Lie algebroid</a> of <a class="existingWikiWord" href="/nlab/show/moduli+stack">moduli stack</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Lie+algebroid">Lie algebroid</a> of <a class="existingWikiWord" href="/nlab/show/moduli+stack">moduli stack</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/extended+Lagrangian">extended Lagrangian</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/universal+Chern-Simons+n-bundle">universal Chern-Simons n-bundle</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/universal+characteristic+map">universal characteristic map</a></td></tr> </tbody></table> </div> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/horizontal+differential+form">horizontal differential form</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/basic+differential+form">basic differential form</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+de+Rham+cohomology">equivariant de Rham cohomology</a></p> </li> </ul> <h2 id="References">References</h2> <h3 id="general">General</h3> <p>An original reference on the notion of a principal bundle as a quotient map by a free continuous action of a topological group is</p> <ul> <li id="Cartan"><em>Séminaire <a class="existingWikiWord" href="/nlab/show/Henri+Cartan">Henri Cartan</a></em> 1949-1950 <em>Topologie algébrique</em> (<a href="http://www.numdam.org/volume/SHC_1948-1949__1">numdam:SHC_1948-1949__1</a>)</li> </ul> <p>some of which is recollected in (<a href="#Palais61">Palais 61</a>).</p> <p>See also the references at <em><a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a></em>.</p> <p>Textbook accounts:</p> <ul> <li id="Steenrod51"> <p><a class="existingWikiWord" href="/nlab/show/Norman+Steenrod">Norman Steenrod</a>, section I.7 of <em>The topology of fibre bundles</em>, Princeton Mathematical Series 14, Princeton Univ. Press, 1951 (<a href="https://www.jstor.org/stable/j.ctt1bpm9t5">jstor:j.ctt1bpm9t5</a>)</p> </li> <li id="Husemoeller"> <p><a class="existingWikiWord" href="/nlab/show/Dale+Husem%C3%B6ller">Dale Husemöller</a>, <em>Fiber bundles</em>, Springer (1994) (<a href="https://doi.org/10.1007/978-1-4757-2261-1">doi:10.1007/978-1-4757-2261-1</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dale+Husemoeller">Dale Husemoeller</a>, <a class="existingWikiWord" href="/nlab/show/Michael+Joachim">Michael Joachim</a>, <a class="existingWikiWord" href="/nlab/show/Branislav+Jurco">Branislav Jurco</a>, <a class="existingWikiWord" href="/nlab/show/Martin+Schottenloher">Martin Schottenloher</a>, <em><a class="existingWikiWord" href="/nlab/show/Basic+Bundle+Theory+and+K-Cohomology+Invariants">Basic Bundle Theory and K-Cohomology Invariants</a></em>, Lecture Notes in Physics, Springer 2008 (<a href="http://www.mathematik.uni-muenchen.de/~schotten/Texte/978-3-540-74955-4_Book_LNP726corr1.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Ralph+Cohen">Ralph Cohen</a>, <em>The topology of fiber bundles</em>, Stanford University (2017) (<a href="http://math.stanford.edu/~ralph/fiber.pdf">pdf</a>, <a href="https://www.ams.org/open-math-notes/omn-view-listing?listingId=110706">OMN:201707.110706</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Loring+Tu">Loring Tu</a>, Parts I-II in: <em>Introductory Lectures on Equivariant Cohomology</em>, Annals of Mathematics Studies <strong>204</strong>, AMS 2020 (<a href="https://press.princeton.edu/books/hardcover/9780691191744/introductory-lectures-on-equivariant-cohomology">ISBN:9780691191744</a>)</p> </li> </ul> <p>With an eye towards application in <a class="existingWikiWord" href="/nlab/show/mathematical+physics">mathematical physics</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Mikio+Nakahara">Mikio Nakahara</a>, Chapter 9 of: <em><a class="existingWikiWord" href="/nlab/show/Geometry%2C+Topology+and+Physics">Geometry, Topology and Physics</a></em>, IOP 2003 (<a href="https://doi.org/10.1201/9781315275826">doi:10.1201/9781315275826</a>, <a href="http://alpha.sinp.msu.ru/~panov/LibBooks/GRAV/(Graduate_Student_Series_in_Physics)Mikio_Nakahara-Geometry,_Topology_and_Physics,_Second_Edition_(Graduate_Student_Series_in_Physics)-Institute_of_Physics_Publishing(2003).pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gerd+Rudolph">Gerd Rudolph</a>, <a class="existingWikiWord" href="/nlab/show/Matthias+Schmidt">Matthias Schmidt</a>, Section 1.1: <em>Differential Geometry and Mathematical Physics Part II. Fibre Bundles, Topology and Gauge Fields</em>, Springer 2017 (<a href="https://link.springer.com/book/10.1007/978-94-024-0959-8">doi:10.1007/978-94-024-0959-8</a>)</p> </li> </ul> <p>For principal bundles in the smooth context see most textbooks on <em><a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a></em>, for instance</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Werner+Greub">Werner Greub</a>, <a class="existingWikiWord" href="/nlab/show/Stephen+Halperin">Stephen Halperin</a>, <a class="existingWikiWord" href="/nlab/show/Ray+Vanstone">Ray Vanstone</a>, <em><a class="existingWikiWord" href="/nlab/show/Connections%2C+Curvature%2C+and+Cohomology">Connections, Curvature, and Cohomology</a></em> Academic Press (1973)</li> </ul> <p>also around section 3.1 of</p> <ul> <li id="Moerdijk"><a class="existingWikiWord" href="/nlab/show/Ieke+Moerdijk">Ieke Moerdijk</a>, <em>On the classification of regular Lie groupoids</em> (<a href="http://igitur-archive.library.uu.nl/math/2007-0201-202453/moerdijk_02_on_the_classification.pdf">pdf</a>)</li> </ul> <p>Questions related to the existence <a class="existingWikiWord" href="/nlab/show/slice+theorem">slices</a> of <a class="existingWikiWord" href="/nlab/show/topological+G-spaces">topological G-spaces</a>, of sections 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>-bundles and conditions for <a class="existingWikiWord" href="/nlab/show/proper+map">properness</a> of some related maps are treated in</p> <ul> <li id="Palais61"><a class="existingWikiWord" href="/nlab/show/Richard+Palais">Richard Palais</a>, <em>On the Existence of Slices for Actions of Non-Compact Lie Groups</em>, Annals of Mathematics <p>Second Series, Vol. 73, No. 2 (Mar., 1961), pp. 295-323 (<a href="https://www.jstor.org/stable/1970335">jstor:1970335</a>, <a href="https://doi.org/10.2307/1970335">doi:10.2307/1970335</a>, <a href="http://vmm.math.uci.edu/ExistenceOfSlices.pdf">pdf</a>)</p> </li> </ul> <p>Lecture notes on principal bundles include</p> <ul> <li id="Mitchell"><a class="existingWikiWord" href="/nlab/show/Stephen+A.+Mitchell">Stephen A. Mitchell</a>, <em>Notes on principal bundles and classifying spaces</em> (2011) [<a href="https://sites.math.washington.edu/~mitchell/Notes/prin.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/MitchellPrincipalBundles.pdf" title="pdf">pdf</a>]</li> </ul> <h3 id="ReferencesInternalToMoreGeneralCategories">Internal to more general categories</h3> <p>Discussion of <a class="existingWikiWord" href="/nlab/show/fiber+bundles">fiber bundles</a> <a class="existingWikiWord" href="/nlab/show/internalization">internal</a> to <a class="existingWikiWord" href="/nlab/show/finitely+complete+categories">finitely complete categories</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Anders+Kock">Anders Kock</a>, <em>Fibre bundles in general categories</em>, Journal of Pure and Applied Algebra <strong>56</strong> 3 (1989) 233-245 [<a href="https://doi.org/10.1016/0022-4049(89)90059-5">doi:10.1016/0022-4049(89)90059-5</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Anders+Kock">Anders Kock</a>, <em>Generalized fibre bundles</em>, in: Categorical Algebra and its Applications, Lecture Notes in Mathematics <strong>1348</strong> (2006) 194-207 [<a href="https://doi.org/10.1007/BFb0081359">doi:10.1007/BFb0081359</a>]</p> </li> <li id="SatiSchreiber21"> <p><a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, Part 2 of: <em><a class="existingWikiWord" href="/schreiber/show/Equivariant+principal+infinity-bundles">Equivariant principal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mn>∞</mn> </mrow> <annotation encoding="application/x-tex">\infty</annotation> </semantics> </math>-bundles</a></em> [<a href="https://arxiv.org/abs/2112.13654">arXiv:2112.13654</a>]</p> <blockquote> <p>(in the generality of <a class="existingWikiWord" href="/nlab/show/equivariant+principal+bundles">equivariant principal bundles</a>)</p> </blockquote> </li> </ul> <p>See also:</p> <ul> <li> <p>C. Townsend, <em>Principal bundles as Frobenius adjunctions with application to geometric morphisms</em>, Math. Proc. Camb. Phil. Soc. 159(03) (2015), 433-444 <a href="http://www.christophertownsend.org/Documents/Principal.pdf">pdf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tomasz+Brzezinski">Tomasz Brzezinski</a>, <em>On synthetic interpretation of quantum principal bundles</em>, AJSE D - Mathematics 35(1D): 13-27, 2010 <a href="http://uk.arxiv.org/abs/0912.0213">arxiv:0912.0213</a></p> </li> </ul> <h3 id="ReferencesInRelationToLieGroupoids">In relation to Lie groupoids</h3> <p>Discussion of <a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+groupoids">Atiyah Lie groupoids</a> associated to principal bundles and the reconstruction of principal bundles from their Atiyah Lie groupoids is due to</p> <ul> <li>Ehresmann, …</li> </ul> <p>Further discussion along these lines is for instance in</p> <ul id="Androulidakis"> <li><a class="existingWikiWord" href="/nlab/show/Iakovos+Androulidakis">Iakovos Androulidakis</a>, <em>Classification of extensions of principal bundles and transitive Lie groupoids with prescribed kernel and cokernel</em> (<a href="http://arxiv.org/abs/math/0402007">arXiv:math/0402007</a>)</li> </ul> <h3 id="examples_2">Examples</h3> <p>Discussion of topological quotients of groups <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>→</mo><mi>G</mi><mo stretchy="false">/</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">G \to G/H</annotation></semantics></math> as principal <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>-bundles is in</p> <ul> <li id="Gleason50"> <p><a class="existingWikiWord" href="/nlab/show/Andrew+Gleason">Andrew Gleason</a>, <em>Spaces with a compact Lie group of transformations</em>, Proc. of A.M.S 1, (1950), 35 - 43.</p> </li> <li> <p><a href="#Palais61">Palais 61</a></p> </li> </ul> <p>Explicit examples and counter examples of coset principal bundles are discussed in</p> <ul id="Karube"> <li>Takashi Karube, <em>On the local cross-sections in locally compact groups</em>, J. Math. Soc. Japan 10 (1958) 343–347 (<a href="http://projecteuclid.org/euclid.jmsj/1261148669">Euclid</a>)</li> </ul> <ul id="Mostert"> <li>Paul Mostert, <em>Local cross sections in locally compact groups</em> (<a href="http://www.pages.drexel.edu/~gln22/Local%20Cross%20Sections%20in%20Locally%20Compact%20Groups.pdf">pdf</a>)</li> </ul> <p>Relations between classes of continuous and of smooth principal bundles are discussed in</p> <ul> <li>Christoph Müller, <a class="existingWikiWord" href="/nlab/show/Christoph+Wockel">Christoph Wockel</a>, <em>Equivalences of smooth and continuous principal bundles with infinite-dimensional structure groups</em>, Advances in Geometry. Volume 9, Issue 4, Pages 605–626 (2009)</li> </ul> <h3 id="ReferencesExtensions">Extensions</h3> <p>Extensions of principal bundles are discussed for instance in</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Kirill+Mackenzie">Kirill Mackenzie</a>, <em>On extensions of principal bundles</em>, Annals of Global Analysis and Geometry Volume 6, Number 2 (1988),</p> </li> <li> <p>I. Androulidakis, <em>Classification of extensions of principal bundles and transitive Lie groupoids with prescribed kernel and cokernel</em>, J. Math. Phys. 45, 3995 (2004); (<a href="http://users.uoa.gr/~iandroul/A-gpdextnclass.pdf">pdf</a>)</p> </li> </ul> <h3 id="automorphism_groups">Automorphism groups</h3> <p>The <a class="existingWikiWord" href="/nlab/show/automorphism+groups">automorphism groups</a> of principal bundles are discussed for instance in</p> <ul> <li>M.C. Abbati, R. Cirelli, A. Mania, P. Michor <em>The Lie group of automorphisms of a principal bundle</em> 1989 JGP <strong>6</strong> 215 (<a href="http://www.mat.univie.ac.at/~michor/AutomorphismPrincipleBundle.pdf">pdf</a>, <a href="http://dx.doi.org/10.1016/0393-0440%2889%2990015-6">doi</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on February 13, 2025 at 18:20:09. See the <a href="/nlab/history/principal+bundle" 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/principal+bundle" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/3769/#Item_14">Discuss</a><span class="backintime"><a href="/nlab/revision/principal+bundle/69" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/principal+bundle" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/principal+bundle" accesskey="S" class="navlink" id="history" rel="nofollow">History (69 revisions)</a> <a href="/nlab/show/principal+bundle/cite" style="color: black">Cite</a> <a href="/nlab/print/principal+bundle" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/principal+bundle" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>