CINXE.COM
algebraic topology in nLab
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0 plus SVG 1.1//EN" "http://www.w3.org/2002/04/xhtml-math-svg/xhtml-math-svg-flat.dtd" > <html xmlns="http://www.w3.org/1999/xhtml"> <head> <title> algebraic topology 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[/*><!--*/ .toc ul {margin: 0; padding: 0;} .toc ul ul {margin: 0; padding: 0 0 0 10px;} .toc li > p {margin: 0} .toc ul li {list-style-type: none; position: relative;} .toc div {border-top:1px dotted #ccc;} .rightHandSide h2 {font-size: 1.5em;color:#008B26} table.plaintable { border-collapse:collapse; margin-left:30px; border:0; } .plaintable td {border:1px solid #000; padding: 3px;} .plaintable th {padding: 3px;} .plaintable caption { font-weight: bold; font-size:1.1em; text-align:center; margin-left:30px; } /* Query boxes for questioning and answering mechanism */ div.query{ background: #f6fff3; border: solid #ce9; border-width: 2px 1px; padding: 0 1em; margin: 0 1em; max-height: 20em; overflow: auto; } /* Standout boxes for putting important text */ div.standout{ background: #fff1f1; border: solid black; border-width: 2px 1px; padding: 0 1em; margin: 0 1em; overflow: auto; } /* Icon for links to n-category arXiv documents (commented out for now i.e. disabled) a[href*="http://arxiv.org/"] { background-image: url(../files/arXiv_icon.gif); background-repeat: no-repeat; background-position: right bottom; padding-right: 22px; } */ /* Icon for links to n-category cafe posts (disabled) a[href*="http://golem.ph.utexas.edu/category"] { background-image: url(../files/n-cafe_5.gif); background-repeat: no-repeat; background-position: right bottom; padding-right: 25px; } */ /* Icon for links to pdf files (disabled) a[href$=".pdf"] { background-image: url(../files/pdficon_small.gif); background-repeat: no-repeat; background-position: right bottom; padding-right: 25px; } */ /* Icon for links to pages, etc. -inside- pdf files (disabled) a[href*=".pdf#"] { background-image: url(../files/pdf_entry.gif); background-repeat: no-repeat; background-position: right bottom; padding-right: 25px; } */ a.existingWikiWord { color: #226622; } a.existingWikiWord:visited { color: #226622; } a.existingWikiWord[title] { border: 0px; color: #aa0505; text-decoration: none; } a.existingWikiWord[title]:visited { border: 0px; color: #551111; text-decoration: none; } a[href^="http://"] { border: 0px; color: #003399; } a[href^="http://"]:visited { border: 0px; color: #330066; } a[href^="https://"] { border: 0px; color: #003399; } a[href^="https://"]:visited { border: 0px; color: #330066; } div.dropDown .hide { display: none; } div.dropDown:hover .hide { display:block; } div.clickDown .hide { display: none; } div.clickDown:focus { outline:none; } div.clickDown:focus .hide, div.clickDown:hover .hide { display: block; } div.clickDown .clickToReveal, div.clickDown:focus .clickToHide { display:block; } div.clickDown:focus .clickToReveal, div.clickDown .clickToHide { display:none; } div.clickDown .clickToReveal:after { content: "A(Hover to reveal, click to "hold")"; font-size: 60%; } div.clickDown .clickToHide:after { content: "A(Click to hide)"; font-size: 60%; } div.clickDown .clickToHide, div.clickDown .clickToReveal { white-space: pre-wrap; } .un_theorem, .num_theorem, .un_lemma, .num_lemma, .un_prop, .num_prop, .un_cor, .num_cor, .un_defn, .num_defn, .un_example, .num_example, .un_note, .num_note, .un_remark, .num_remark { margin-left: 1em; } span.theorem_label { margin-left: -1em; } .proof span.theorem_label { margin-left: 0em; } :target { background-color: #BBBBBB; border-radius: 5pt; } /*]]>*/--></style> <script src="/javascripts/prototype.js?1660229990" type="text/javascript"></script> <script src="/javascripts/effects.js?1660229990" type="text/javascript"></script> <script src="/javascripts/dragdrop.js?1660229990" type="text/javascript"></script> <script src="/javascripts/controls.js?1660229990" type="text/javascript"></script> <script src="/javascripts/application.js?1660229990" type="text/javascript"></script> <script src="/javascripts/page_helper.js?1660229990" type="text/javascript"></script> <script src="/javascripts/thm_numbering.js?1660229990" type="text/javascript"></script> <script type="text/x-mathjax-config"> <!--//--><![CDATA[//><!-- MathJax.Ajax.config.path["Contrib"] = "/MathJax"; MathJax.Hub.Config({ MathML: { useMathMLspacing: true }, "HTML-CSS": { scale: 90, extensions: ["handle-floats.js"] } }); MathJax.Hub.Queue( function () { var fos = document.getElementsByTagName('foreignObject'); for (var i = 0; i < fos.length; i++) { MathJax.Hub.Typeset(fos[i]); } }); //--><!]]> </script> <script type="text/javascript"> <!--//--><![CDATA[//><!-- window.addEventListener("DOMContentLoaded", function () { var div = document.createElement('div'); var math = document.createElementNS('http://www.w3.org/1998/Math/MathML', 'math'); document.body.appendChild(div); div.appendChild(math); // Test for MathML support comparable to WebKit version https://trac.webkit.org/changeset/203640 or higher. div.setAttribute('style', 'font-style: italic'); var mathml_unsupported = !(window.getComputedStyle(div.firstChild).getPropertyValue('font-style') === 'normal'); div.parentNode.removeChild(div); if (mathml_unsupported) { // MathML does not seem to be supported... var s = document.createElement('script'); s.src = "https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.7/MathJax.js?config=MML_HTMLorMML-full"; document.querySelector('head').appendChild(s); } else { document.head.insertAdjacentHTML("beforeend", '<style>svg[viewBox] {max-width: 100%}</style>'); } }); //--><!]]> </script> <link href="https://ncatlab.org/nlab/atom_with_headlines" rel="alternate" title="Atom with headlines" type="application/atom+xml" /> <link href="https://ncatlab.org/nlab/atom_with_content" rel="alternate" title="Atom with full content" type="application/atom+xml" /> <script type="text/javascript"> document.observe("dom:loaded", function() { generateThmNumbers(); }); </script> </head> <body> <div id="Container"> <div id="Content"> <h1 id="pageName"> <span style="float: left; margin: 0.5em 0.25em -0.25em 0"> <svg xmlns="http://www.w3.org/2000/svg" width="1.872em" height="1.8em" viewBox="0 0 190 181"> <path fill="#226622" d="M72.8 145c-1.6 17.3-15.7 10-23.6 20.2-5.6 7.3 4.8 15 11.4 15 11.5-.2 19-13.4 26.4-20.3 3.3-3 8.2-4 11.2-7.2a14 14 0 0 0 2.9-11.1c-1.4-9.6-12.4-18.6-16.9-27.2-5-9.6-10.7-27.4-24.1-27.7-17.4-.3-.4 26 4.7 30.7 2.4 2.3 5.4 4.1 7.3 6.9 1.6 2.3 2.1 5.8-1 7.2-5.9 2.6-12.4-6.3-15.5-10-8.8-10.6-15.5-23-26.2-31.8-5.2-4.3-11.8-8-18-3.7-7.3 4.9-4.2 12.9.2 18.5a81 81 0 0 0 30.7 23c3.3 1.5 12.8 5.6 10 10.7-2.5 5.2-11.7 3-15.6 1.1-8.4-3.8-24.3-21.3-34.4-13.7-3.5 2.6-2.3 7.6-1.2 11.1 2.8 9 12.2 17.2 20.9 20.5 17.3 6.7 34.3-8 50.8-12.1z"/> <path fill="#a41e32" d="M145.9 121.3c-.2-7.5 0-19.6-4.5-26-5.4-7.5-12.9-1-14.1 5.8-1.4 7.8 2.7 14.1 4.8 21.3 3.4 12 5.8 29-.8 40.1-3.6-6.7-5.2-13-7-20.4-2.1-8.2-12.8-13.2-15.1-1.9-2 9.7 9 21.2 12 30.1 1.2 4 2 8.8 6.4 10.3 6.9 2.3 13.3-4.7 17.7-8.8 12.2-11.5 36.6-20.7 43.4-36.4 6.7-15.7-13.7-14-21.3-7.2-9.1 8-11.9 20.5-23.6 25.1 7.5-23.7 31.8-37.6 38.4-61.4 2-7.3-.8-29.6-13-19.8-14.5 11.6-6.6 37.6-23.3 49.2z"/> <path fill="#193c78" d="M86.3 47.5c0-13-10.2-27.6-5.8-40.4 2.8-8.4 14.1-10.1 17-1 3.8 11.6-.3 26.3-1.8 38 11.7-.7 10.5-16 14.8-24.3 2.1-4.2 5.7-9.1 11-6.7 6 2.7 7.4 9.2 6.6 15.1-2.2 14-12.2 18.8-22.4 27-3.4 2.7-8 6.6-5.9 11.6 2 4.4 7 4.5 10.7 2.8 7.4-3.3 13.4-16.5 21.7-16 14.6.7 12 21.9.9 26.2-5 1.9-10.2 2.3-15.2 3.9-5.8 1.8-9.4 8.7-15.7 8.9-6.1.1-9-6.9-14.3-9-14.4-6-33.3-2-44.7-14.7-3.7-4.2-9.6-12-4.9-17.4 9.3-10.7 28 7.2 35.7 12 2 1.1 11 6.9 11.4 1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> </span> <span class="webName">nLab</span> algebraic topology </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/6882/#Item_23" 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="homotopy_theory">Homotopy theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>, <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+theory">(∞,1)-category theory</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a></strong></p> <p>flavors: <a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable</a>, <a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+theory">equivariant</a>, <a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rational</a>, <a class="existingWikiWord" href="/nlab/show/p-adic+homotopy+theory">p-adic</a>, <a class="existingWikiWord" href="/nlab/show/proper+homotopy+theory">proper</a>, <a class="existingWikiWord" href="/nlab/show/geometric+homotopy+theory">geometric</a>, <a class="existingWikiWord" href="/nlab/show/cohesive+homotopy+theory">cohesive</a>, <a class="existingWikiWord" href="/nlab/show/directed+homotopy+theory">directed</a>…</p> <p>models: <a class="existingWikiWord" href="/nlab/show/topological+homotopy+theory">topological</a>, <a class="existingWikiWord" href="/nlab/show/simplicial+homotopy+theory">simplicial</a>, <a class="existingWikiWord" href="/nlab/show/localic+homotopy+theory">localic</a>, …</p> <p>see also <strong><a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a></strong></p> <p><strong>Introductions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Topology+--+2">Introduction to Basic Homotopy Theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Homotopy+Theory">Introduction to Abstract Homotopy Theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+homotopy+types">geometry of physics – homotopy types</a></p> </li> </ul> <p><strong>Definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a>, <a class="existingWikiWord" href="/nlab/show/higher+homotopy">higher homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pi-algebra">Pi-algebra</a>, <a class="existingWikiWord" href="/nlab/show/spherical+object+and+Pi%28A%29-algebra">spherical object and Pi(A)-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coherent+category+theory">homotopy coherent category theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopical+category">homotopical category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">category of fibrant objects</a>, <a class="existingWikiWord" href="/nlab/show/cofibration+category">cofibration category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Waldhausen+category">Waldhausen category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ho%28Top%29">Ho(Top)</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/homotopy+category+of+an+%28%E2%88%9E%2C1%29-category">homotopy category of an (∞,1)-category</a></li> </ul> </li> </ul> <p><strong>Paths and cylinders</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cylinder+object">cylinder object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/right+homotopy">right homotopy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/path+object">path object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cocone">mapping cocone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+universal+bundle">universal bundle</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interval+object">interval object</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+localization">homotopy localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infinitesimal+interval+object">infinitesimal interval object</a></p> </li> </ul> </li> </ul> <p><strong>Homotopy groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+group+of+a+topos">fundamental group of a topos</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brown-Grossman+homotopy+group">Brown-Grossman homotopy group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/categorical+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">categorical homotopy groups in an (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">geometric homotopy groups in an (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid">fundamental ∞-groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+groupoid">fundamental groupoid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/path+groupoid">path groupoid</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+in+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+of+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%28%E2%88%9E%2C1%29-category">fundamental (∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+category">fundamental category</a></li> </ul> </li> </ul> <p><strong>Basic facts</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+group+of+the+circle+is+the+integers">fundamental group of the circle is the integers</a></li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+theorem+of+covering+spaces">fundamental theorem of covering spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freudenthal+suspension+theorem">Freudenthal suspension theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Blakers-Massey+theorem">Blakers-Massey theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+homotopy+van+Kampen+theorem">higher homotopy van Kampen theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nerve+theorem">nerve theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitehead%27s+theorem">Whitehead's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hurewicz+theorem">Hurewicz theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Galois+theory">Galois theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis</a>-theorem</p> </li> </ul> </div></div> <h4 id="higher_algebra">Higher algebra</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a></strong></p> <p><a class="existingWikiWord" href="/nlab/show/universal+algebra">universal algebra</a></p> <h2 id="algebraic_theories">Algebraic theories</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/algebraic+theory">algebraic theory</a> / <a class="existingWikiWord" href="/nlab/show/2-algebraic+theory">2-algebraic theory</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-algebraic+theory">(∞,1)-algebraic theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monad">monad</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-monad">(∞,1)-monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/operad">operad</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-operad">(∞,1)-operad</a></p> </li> </ul> <h2 id="algebras_and_modules">Algebras and modules</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+a+monad">algebra over a monad</a></p> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-algebra+over+an+%28%E2%88%9E%2C1%29-monad">∞-algebra over an (∞,1)-monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+an+algebraic+theory">algebra over an algebraic theory</a></p> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-algebra+over+an+%28%E2%88%9E%2C1%29-algebraic+theory">∞-algebra over an (∞,1)-algebraic theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+an+operad">algebra over an operad</a></p> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-algebra+over+an+%28%E2%88%9E%2C1%29-operad">∞-algebra over an (∞,1)-operad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/action">action</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/representation">representation</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-representation">∞-representation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/module">module</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-module">∞-module</a></p> </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> </li> </ul> <h2 id="higher_algebras">Higher algebras</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+%28%E2%88%9E%2C1%29-category">monoidal (∞,1)-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28%E2%88%9E%2C1%29-category">symmetric monoidal (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoid+in+an+%28%E2%88%9E%2C1%29-category">monoid in an (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/commutative+monoid+in+an+%28%E2%88%9E%2C1%29-category">commutative monoid in an (∞,1)-category</a></p> </li> </ul> </li> <li> <p>symmetric monoidal (∞,1)-category of spectra</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/smash+product+of+spectra">smash product of spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+smash+product+of+spectra">symmetric monoidal smash product of spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ring+spectrum">ring spectrum</a>, <a class="existingWikiWord" href="/nlab/show/module+spectrum">module spectrum</a>, <a class="existingWikiWord" href="/nlab/show/algebra+spectrum">algebra spectrum</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+algebra">A-∞ algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+ring">A-∞ ring</a>, <a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+space">A-∞ space</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/C-%E2%88%9E+algebra">C-∞ algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+ring">E-∞ ring</a>, <a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+algebra">E-∞ algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-module">∞-module</a>, <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-module+bundle">(∞,1)-module bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/multiplicative+cohomology+theory">multiplicative cohomology theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebra">L-∞ algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/deformation+theory">deformation theory</a></li> </ul> </li> </ul> <h2 id="model_category_presentations">Model category presentations</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+T-algebras">model structure on simplicial T-algebras</a> / <a class="existingWikiWord" href="/nlab/show/homotopy+T-algebra">homotopy T-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+operads">model structure on operads</a></p> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+algebras+over+an+operad">model structure on algebras over an operad</a></p> </li> </ul> <h2 id="geometry_on_formal_duals_of_algebras">Geometry on formal duals of algebras</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+geometry">derived geometry</a></p> </li> </ul> <h2 id="theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Deligne+conjecture">Deligne conjecture</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/delooping+hypothesis">delooping hypothesis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+Dold-Kan+correspondence">monoidal Dold-Kan correspondence</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/higher+algebra+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="algebraic_topology">Algebraic topology</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a></strong> – application of <a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a> and <a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a> to the study of (<a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable</a>) <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy</a></p> <p><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></p> <p><a class="existingWikiWord" href="/nlab/show/homology">homology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+sequence">spectral sequence</a></p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <ul> <li><a href='#the_idea_of_functorial_invariants'>The idea of functorial invariants</a></li> </ul> <li><a href='#overview_of_methods'>Overview of methods</a></li> <li><a href='#related_entries'>Related entries</a></li> <li><a href='#References'>References</a></li> <ul> <li><a href='#prehistory'>Pre-history</a></li> <li><a href='#ReferencesTopologicalHomotopyTheory'>Topological homotopy theory</a></li> <li><a href='#ReferencesAlegbraicTopology'>Algebraic topology</a></li> <li><a href='#ReferencesAbstractHomotopyTheory'>Abstract homotopy theory</a></li> <li><a href='#ReferencesSimplicialHomotopyTheory'>Simplicial homotopy theory</a></li> <li><a href='#ReferencesBasicInfinityCategoryTheory'>Basic <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>-category theory</a></li> <li><a href='#ReferencesBasicHomotopyTypeTheory'>Basic homotopy type theory</a></li> <li><a href='#ReferencesOutlook'>Outlook</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p><em>Algebraic topology</em> refers to the application of methods of <a class="existingWikiWord" href="/nlab/show/algebra">algebra</a> to problems in <a class="existingWikiWord" href="/nlab/show/topology">topology</a>. More specifically, the method of algebraic topology is to assign <a class="existingWikiWord" href="/nlab/show/homeomorphism">homeomorphism</a>/<a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a>-<a class="existingWikiWord" href="/nlab/show/invariants">invariants</a> to <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a>, or more systematically, to the construction and applications of <a class="existingWikiWord" href="/nlab/show/functors">functors</a> from some <a class="existingWikiWord" href="/nlab/show/category">category</a> of topological objects (e.g. <a class="existingWikiWord" href="/nlab/show/Hausdorff+spaces">Hausdorff spaces</a>, topological <a class="existingWikiWord" href="/nlab/show/fibre+bundles">fibre bundles</a>) to some algebraic category (e.g. <a class="existingWikiWord" href="/nlab/show/abelian+groups">abelian groups</a>, <a class="existingWikiWord" href="/nlab/show/modules">modules</a> over the <a class="existingWikiWord" href="/nlab/show/Steenrod+algebra">Steenrod algebra</a>). Landing in an algebraic category aids to the computability, but typically loses some information (say getting from a topological spaces with a continuum or more points to rather discrete algebraic structures).</p> <h3 id="the_idea_of_functorial_invariants">The idea of functorial invariants</h3> <p>The basic idea of the functorial method for the problem of existence of morphisms is the following: If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">F:A\to B</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/functor">functor</a> (we present here a general statement, but in the above context <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is a category of topological objects and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> some category of algebraic objects) and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>:</mo><mi>D</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">d:D\to A</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>∘</mo><mi>d</mi></mrow><annotation encoding="application/x-tex">F\circ d</annotation></semantics></math> is a diagram in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>. If one can fill certain additional arrow <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> in the diagram <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math> making the extended diagram commutative, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(f)</annotation></semantics></math> is a morphism between the corresponding vertices in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> extending <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>∘</mo><mi>d</mi></mrow><annotation encoding="application/x-tex">F\circ d</annotation></semantics></math> to a commutative diagram. Thus if we prove that there is no morphism extending <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>∘</mo><mi>d</mi></mrow><annotation encoding="application/x-tex">F\circ d</annotation></semantics></math> then there was no morphism extending <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math> in the first place. Therefore, the functorial method is very suitable to prove <em>negative</em> existence for morphisms. Sometimes, however, there is a theorem showing that some set of invariants completely characterizes a problem hence being able to show positive existence or uniqueness for maps or spaces. For the uniqueness for morphisms, it is enough to show that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> is faithful and that there is at most one solution for the existence problem in the target category. Faithful functors in this context are rare, but it is sufficient for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> to be faithful on some subcategory <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mi>p</mi></msub></mrow><annotation encoding="application/x-tex">A_p</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> containing at least all morphisms which are the possible candidates for the solution of the particular existence problem for morphisms.</p> <h2 id="overview_of_methods">Overview of methods</h2> <p>The archetypical example is the classification of <a class="existingWikiWord" href="/nlab/show/surfaces">surfaces</a> via their <a class="existingWikiWord" href="/nlab/show/Euler+characteristic">Euler characteristic</a>. But as this example already shows, algebraic topology tends to be less about <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a> themselves as rather about the <a class="existingWikiWord" href="/nlab/show/homotopy+types">homotopy types</a> which they <a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">present</a>. Therefore the topological invariants in question are typically homotopy invariants of spaces with some exceptions, like the <a class="existingWikiWord" href="/nlab/show/shape+theory">shape invariants</a> for spaces with bad local behaviour.</p> <p>Hence modern algebraic topology is to a large extent the application of algebraic methods to <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>.</p> <p>A general and powerful such method is the assignment of <a class="existingWikiWord" href="/nlab/show/homology">homology</a> and <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> <a class="existingWikiWord" href="/nlab/show/groups">groups</a> to topological spaces, such that these <a class="existingWikiWord" href="/nlab/show/abelian+groups">abelian groups</a> depend only on the <a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a>. The simplest such are <a class="existingWikiWord" href="/nlab/show/ordinary+homology">ordinary homology</a> and <a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a> groups, given by <a class="existingWikiWord" href="/nlab/show/singular+simplicial+complexes">singular simplicial complexes</a>. This way algebraic topology makes use of tools of <a class="existingWikiWord" href="/nlab/show/homological+algebra">homological algebra</a>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/axiom">axiomatization</a> of the properties of such <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> group assignments is what led to the formulation of the trinity of concepts of <em><a class="existingWikiWord" href="/nlab/show/category">category</a></em>, <em><a class="existingWikiWord" href="/nlab/show/functor">functor</a></em> and <em><a class="existingWikiWord" href="/nlab/show/natural+transformations">natural transformations</a></em>, and algebraic topology has come to make intensive use of <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a>.</p> <p>In particular this leads to the formulation of <a class="existingWikiWord" href="/nlab/show/generalized+%28Eilenberg-Steenrod%29+cohomology">generalized (Eilenberg-Steenrod) cohomology</a> theories which detect more information about classes of homotopy types. By the <a class="existingWikiWord" href="/nlab/show/Brown+representability+theorem">Brown representability theorem</a> such are represented by <a class="existingWikiWord" href="/nlab/show/spectra">spectra</a> (generalizing <a class="existingWikiWord" href="/nlab/show/chain+complexes">chain complexes</a>), hence <a class="existingWikiWord" href="/nlab/show/stable+homotopy+types">stable homotopy types</a>, and this way algebraic topology comes to use and be about <a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable homotopy theory</a>.</p> <p>Still finer invariants of <a class="existingWikiWord" href="/nlab/show/homotopy+types">homotopy types</a> are detected by further refinements of these “algebraic” structures, for instance to <a class="existingWikiWord" href="/nlab/show/multiplicative+cohomology+theories">multiplicative cohomology theories</a>, to <a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+theory">equivariant homotopy theory</a>/<a class="existingWikiWord" href="/nlab/show/equivariant+stable+homotopy+theory">equivariant stable homotopy theory</a> and so forth. The construction and analysis of these requires the intimate combination of algebra and homotopy theory to <a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a> and <a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a>, notably embodied in the <a class="existingWikiWord" href="/nlab/show/universal+algebra">universal</a> higher algebra of <a class="existingWikiWord" href="/nlab/show/operads">operads</a>.</p> <p>The central tool for breaking down all this <a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebraic</a> data into computable pieces are <a class="existingWikiWord" href="/nlab/show/spectral+sequences">spectral sequences</a>, which are maybe the main heavy-lifting workhorses of algebraic topology.</p> <h2 id="related_entries">Related entries</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/topology">topology</a>, <a class="existingWikiWord" href="/nlab/show/differential+topology">differential topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homology">homology</a>/<a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>, <a class="existingWikiWord" href="/nlab/show/shape+theory">shape theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rational homotopy theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nonabelian+algebraic+topology">nonabelian algebraic topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+data+analysis">topological data analysis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+lifting+property">homotopy lifting property</a>, <a class="existingWikiWord" href="/nlab/show/Hurewicz+fibration">Hurewicz fibration</a>, <a class="existingWikiWord" href="/nlab/show/Hurewicz+connection">Hurewicz connection</a>, <a class="existingWikiWord" href="/nlab/show/Serre+fibration">Serre fibration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+extension+property">homotopy extension property</a>, <a class="existingWikiWord" href="/nlab/show/Hurewicz+cofibration">Hurewicz cofibration</a>, <a class="existingWikiWord" href="/nlab/show/deformation+retract">deformation retract</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/suspension">suspension</a>, <a class="existingWikiWord" href="/nlab/show/loop+space">loop space</a>, <a class="existingWikiWord" href="/nlab/show/mapping+cylinder">mapping cylinder</a>, <a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a>, <a class="existingWikiWord" href="/nlab/show/mapping+cocylinder">mapping cocylinder</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a>, <a class="existingWikiWord" href="/nlab/show/spectrum">spectrum</a>, <a class="existingWikiWord" href="/nlab/show/Brown+representability+theorem">Brown representability theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a>, <a class="existingWikiWord" href="/nlab/show/fundamental+groupoid">fundamental groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a>, <a class="existingWikiWord" href="/nlab/show/Eckmann-Hilton+duality">Eckmann-Hilton duality</a>, <a class="existingWikiWord" href="/nlab/show/H-space">H-space</a>, <a class="existingWikiWord" href="/nlab/show/Whitehead+product">Whitehead product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+K-theory">topological K-theory</a>, <a class="existingWikiWord" href="/nlab/show/complex+cobordism">complex cobordism</a>, <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/CW+complex">CW complex</a>, <a class="existingWikiWord" href="/nlab/show/CW+approximation">CW approximation</a>, <a class="existingWikiWord" href="/nlab/show/simplicial+complex">simplicial complex</a>, <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+category">model category</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+on+topological+spaces">model structure on topological spaces</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fibration+sequence">fibration sequence</a>, <a class="existingWikiWord" href="/nlab/show/cofibration+sequence">cofibration sequence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freudenthal+suspension+theorem">Freudenthal suspension theorem</a>, <a class="existingWikiWord" href="/nlab/show/Whitehead+theorem">Whitehead theorem</a></p> </li> </ul> <h2 id="References">References</h2> <div> <p>The following lists basic references on <em><a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a></em>, <em><a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a></em> and some <em><a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-category+theory"><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>-category theory</a></em> and <em><a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a></em>, but see these entries for more pointers.</p> <h3 id="prehistory">Pre-history</h3> <p>Historical article at the origin of all these subjects:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Henri+Poincar%C3%A9">Henri Poincaré</a>, <em><a class="existingWikiWord" href="/nlab/show/Analysis+Situs">Analysis Situs</a></em>, Journal de l’École Polytechnique. (2). 1: 1–123 (1895) (<a href="https://gallica.bnf.fr/ark:/12148/bpt6k4337198/f7">gallica:12148/bpt6k4337198/f7</a>, Engl: <a href="https://www.maths.ed.ac.uk/~v1ranick/papers/poincare2009.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Stillwell_AnalysisSitus.pdf" title="pdf">pdf</a>)</li> </ul> <p>On early developments from there, such as the eventual understanding of the notion of higher <a class="existingWikiWord" href="/nlab/show/homotopy+groups">homotopy groups</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Peter+Hilton">Peter Hilton</a>, <em>Subjective History of Homology and Homotopy Theory</em>, Mathematics Magazine <strong>61</strong> 5 (1988) 282-291 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://doi.org/10.2307/2689545">doi:10.2307/2689545</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></li> </ul> <h3 id="ReferencesTopologicalHomotopyTheory">Topological homotopy theory</h3> <p>Textbook accounts of <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> of <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a> (i.e. via “<a class="existingWikiWord" href="/nlab/show/point-set+topology">point-set topology</a>”):</p> <ul> <li id="Hilton53"> <p><a class="existingWikiWord" href="/nlab/show/Peter+J.+Hilton">Peter J. Hilton</a>, <em>An introduction to homotopy theory</em>, Cambridge University Press 1953 (<a href="https://doi.org/10.1017/CBO9780511666278">doi:10.1017/CBO9780511666278</a>)</p> </li> <li id="SzeTsen59"> <p><a class="existingWikiWord" href="/nlab/show/Sze-Tsen+Hu">Sze-Tsen Hu</a>, <em>Homotopy Theory</em>, Academic Press 1959 (<a href="https://www.maths.ed.ac.uk/~v1ranick/papers/hu2.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Robert+E.+Mosher">Robert E. Mosher</a>, <a class="existingWikiWord" href="/nlab/show/Martin+C.+Tangora">Martin C. Tangora</a>, <em>Cohomology operations and applications in homotopy theory</em>, Harper & Row, 1968, reprinted by <a href="https://store.doverpublications.com/0486466647.html">Dover 2008</a> <a href="https://www.google.com/books/edition/Cohomology_Operations_and_Applications_i/wu79f-7V_6AC">GoogleBooks</a></p> </li> <li id="Homotopietheorie"> <p><a class="existingWikiWord" href="/nlab/show/Tammo+tom+Dieck">Tammo tom Dieck</a>, <a class="existingWikiWord" href="/nlab/show/Klaus+Heiner+Kamps">Klaus Heiner Kamps</a>, <a class="existingWikiWord" href="/nlab/show/Dieter+Puppe">Dieter Puppe</a>, <em>Homotopietheorie</em>, Lecture Notes in Mathematics <strong>157</strong> Springer 1970 (<a href="https://link.springer.com/book/10.1007/BFb0059721">doi:10.1007/BFb0059721</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brayton+Gray">Brayton Gray</a>, <em>Homotopy Theory: An Introduction to Algebraic Topology</em>, Academic Press (1975) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://www.sciencedirect.com/bookseries/pure-and-applied-mathematics/vol/64/suppl/C">978-0-12-296050-5</a>, <a href="https://www.maths.ed.ac.uk/~v1ranick/papers/gray.pdf">pdf</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/George+W.+Whitehead">George W. Whitehead</a>, <em>Elements of Homotopy Theory</em>, Springer 1978 (<a href="https://link.springer.com/book/10.1007/978-1-4612-6318-0">doi:10.1007/978-1-4612-6318-0</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Ioan+Mackenzie+James">Ioan Mackenzie James</a>, <em>General Topology and Homotopy Theory</em>, Springer 1984 (<a href="https://doi.org/10.1007/978-1-4613-8283-6">doi:10.1007/978-1-4613-8283-6</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Renzo+A.+Piccinini">Renzo A. Piccinini</a>, <em>Lectures on Homotopy Theory</em>, Mathematics Studies <strong>171</strong>, North Holland 1992 (<a href="https://www.sciencedirect.com/bookseries/north-holland-mathematics-studies/vol/171/suppl/C">ISBN:978-0-444-89238-6</a>)</p> </li> <li id="Bredon93"> <p><a class="existingWikiWord" href="/nlab/show/Glen+Bredon">Glen Bredon</a>, Chapter VII of: <em>Topology and Geometry</em>, Graduate texts in mathematics <strong>139</strong>, Springer 1993 (<a href="https://link.springer.com/book/10.1007/978-1-4757-6848-0">doi:10.1007/978-1-4757-6848-0</a>, <a href="http://virtualmath1.stanford.edu/~ralph/math215b/Bredon.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hans-Joachim+Baues">Hans-Joachim Baues</a>, <em>Homotopy types</em>, in <a class="existingWikiWord" href="/nlab/show/Ioan+Mackenzie+James">Ioan Mackenzie James</a> (ed.) <em><a class="existingWikiWord" href="/nlab/show/Handbook+of+Algebraic+Topology">Handbook of Algebraic Topology</a></em>, North Holland, 1995 (<a href="https://www.elsevier.com/books/handbook-of-algebraic-topology/james/978-0-444-81779-2">ISBN:9780080532981</a>, <a href="https://doi.org/10.1016/B978-0-444-81779-2.X5000-7">doi:10.1016/B978-0-444-81779-2.X5000-7</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Nicolas+Bourbaki">Nicolas Bourbaki</a>, <em>Topologie Algébrique</em>, Chapitres 1 à 4, Springer (1998, 2016) [ISBN 978-3-662-49361-8, <a href="https://doi.org/10.1007/978-3-662-49361-8">doi:10.1007/978-3-662-49361-8</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Marcelo+Aguilar">Marcelo Aguilar</a>, <a class="existingWikiWord" href="/nlab/show/Samuel+Gitler">Samuel Gitler</a>, <a class="existingWikiWord" href="/nlab/show/Carlos+Prieto">Carlos Prieto</a>, <em>Algebraic topology from a homotopical viewpoint</em>, Springer (2008) (<a href="https://link.springer.com/book/10.1007/b97586">doi:10.1007/b97586</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Jeffrey+Strom">Jeffrey Strom</a>, <em>Modern classical homotopy theory</em>, Graduate Studies in Mathematics <strong>127</strong>, American Mathematical Society (2011) [<a href="http://www.ams.org/books/gsm/127">ams:gsm/127</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Martin+Arkowitz">Martin Arkowitz</a>, <em>Introduction to Homotopy Theory</em>, Springer (2011) [<a href="https://doi.org/10.1007/978-1-4419-7329-0">doi:10.1007/978-1-4419-7329-0</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Anatoly+Fomenko">Anatoly Fomenko</a>, <a class="existingWikiWord" href="/nlab/show/Dmitry+Fuchs">Dmitry Fuchs</a>: <em>Homotopical Topology</em>, Graduate Texts in Mathematics <strong>273</strong>, Springer (2016) [<a href="https://doi.org/10.1007/978-3-319-23488-5">doi:10.1007/978-3-319-23488-5</a>, <a href="https://www.cimat.mx/~gil/docencia/2020/topologia_diferencial/[Fomenko,Fuchs]Homotopical_Topology(2016).pdf">pdf</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dai+Tamaki">Dai Tamaki</a>, <em>Fiber Bundles and Homotopy</em>, World Scientific (2021) [<a href="https://doi.org/10.1142/12308">doi:10.1142/12308</a>]</p> <blockquote> <p>(motivated from <a class="existingWikiWord" href="/nlab/show/classifying+spaces">classifying spaces</a> for <a class="existingWikiWord" href="/nlab/show/principal+bundles">principal bundles</a>/<a class="existingWikiWord" href="/nlab/show/fiber+bundles">fiber bundles</a>)</p> </blockquote> </li> </ul> <h3 id="ReferencesAlegbraicTopology">Algebraic topology</h3> <p>On <a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a>:</p> <p>Monographs:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Samuel+Eilenberg">Samuel Eilenberg</a>, <a class="existingWikiWord" href="/nlab/show/Norman+Steenrod">Norman Steenrod</a>, <em>Foundations of Algebraic Topology</em>, Princeton University Press 1952 (<a href="https://www.maths.ed.ac.uk/~v1ranick/papers/eilestee.pdf">pdf</a>, <a href="https://press.princeton.edu/books/hardcover/9780691653297/foundations-of-algebraic-topology">ISBN:9780691653297</a>)</p> </li> <li id="Godement58"> <p><a class="existingWikiWord" href="/nlab/show/Roger+Godement">Roger Godement</a>, <em>Topologie algébrique et theorie des faisceaux</em>, Actualités Sci. Ind. <strong>1252</strong>, Hermann, Paris (1958) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://www.editions-hermann.fr/livre/topologie-algebrique-et-theorie-des-faisceaux-roger-godement">webpage</a>, <a class="existingWikiWord" href="/nlab/files/Godement-TopologieAlgebrique.pdf" title="pdf">pdf</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></p> </li> <li id="Spanier66"> <p><a class="existingWikiWord" href="/nlab/show/Edwin+Spanier">Edwin Spanier</a>, <em>Algebraic topology</em>, McGraw Hill (1966), Springer (1982) (<a href="https://link.springer.com/book/10.1007/978-1-4684-9322-1">doi:10.1007/978-1-4684-9322-1</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/William+S.+Massey">William S. Massey</a>, <em>Algebraic Topology: An Introduction</em>, Harcourt Brace & World 1967, reprinted in: Graduate Texts in Mathematics, Springer 1977 (<a href="https://link.springer.com/book/9780387902715">ISBN:978-0-387-90271-5</a>)</p> </li> <li id="Maunder70"> <p><a class="existingWikiWord" href="/nlab/show/C.+R.+F.+Maunder">C. R. F. Maunder</a>, <em>Algebraic Topology</em>, Cambridge University Press, Cambridge (1970, 1980) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://www.maths.ed.ac.uk/~v1ranick/papers/maunder.pdf">pdf</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></p> </li> <li id="Switzer75"> <p><a class="existingWikiWord" href="/nlab/show/Robert+Switzer">Robert Switzer</a>, <em>Algebraic Topology - Homotopy and Homology</em>, Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, Vol. 212, Springer-Verlag, New York, N. Y., 1975 (<a href="https://link.springer.com/book/10.1007/978-3-642-61923-6">doi:10.1007/978-3-642-61923-6</a>)</p> </li> <li id="Giblin77"> <p>P. J. Giblin: <em>Graphs, Surfaces and Homology – An Introduction to Algebraic Topology</em>, Chapman and Hall (1977) [<a href="https://doi.org/10.1007/978-94-009-5953-8">doi:10.1007/978-94-009-5953-8</a>]</p> <blockquote> <p>(focus on <a class="existingWikiWord" href="/nlab/show/graphs">graphs</a> and <a class="existingWikiWord" href="/nlab/show/surfaces">surfaces</a>)</p> </blockquote> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Raoul+Bott">Raoul Bott</a>, <a class="existingWikiWord" href="/nlab/show/Loring+Tu">Loring Tu</a>, <em><a class="existingWikiWord" href="/nlab/show/Differential+Forms+in+Algebraic+Topology">Differential Forms in Algebraic Topology</a></em>, Graduate Texts in Mathematics <strong>82</strong>, Springer (1982) [<a href="https://link.springer.com/book/10.1007/978-1-4757-3951-0">doi:10.1007/978-1-4757-3951-0</a>]</p> <blockquote> <p>(with focus on <a class="existingWikiWord" href="/nlab/show/differential+forms">differential forms</a>, <a class="existingWikiWord" href="/nlab/show/differential+topology">differential topology</a>)</p> </blockquote> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/James+Munkres">James Munkres</a>, <em>Elements of Algebraic Topology</em>, Addison-Wesley (1984) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://people.dm.unipi.it/benedett/MUNKRES-ETA.pdf">pdf</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Joseph+J.+Rotman">Joseph J. Rotman</a>, <em>An Introduction to Algebraic Topology</em>, Graduate Texts in Mathematics <strong>119</strong> (1988) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://doi.org/10.1007/978-1-4612-4576-6">doi:10.1007/978-1-4612-4576-6</a>]</p> </li> <li id="Bredon93"> <p><a class="existingWikiWord" href="/nlab/show/Glen+Bredon">Glen Bredon</a>, <em>Topology and Geometry</em>, Graduate texts in mathematics <strong>139</strong>, Springer 1993 (<a href="https://link.springer.com/book/10.1007/978-1-4757-6848-0">doi:10.1007/978-1-4757-6848-0</a>, <a href="http://virtualmath1.stanford.edu/~ralph/math215b/Bredon.pdf">pdf</a>)</p> </li> <li id="Dold95"> <p><a class="existingWikiWord" href="/nlab/show/Albrecht+Dold">Albrecht Dold</a>, <em>Lectures on Algebraic Topology</em>, Springer 1995 (<a href="https://www.springer.com/gp/book/9783540586609">doi:10.1007/978-3-642-67821-9</a>, <a href="https://link.springer.com/content/pdf/bfm%3A978-3-642-67821-9%2F1.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/William+Fulton">William Fulton</a>, <em>Algebraic Topology – A First Course</em>, Graduate Texts in Mathematics <strong>153</strong>, Springer (1995) [<a href="https://doi.org/10.1007/978-1-4612-4180-5">doi:10.1007/978-1-4612-4180-5</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Peter+May">Peter May</a>, <em><a class="existingWikiWord" href="/nlab/show/A+concise+course+in+algebraic+topology">A concise course in algebraic topology</a></em>, University of Chicago Press 1999 (<a href="https://www.press.uchicago.edu/ucp/books/book/chicago/C/bo3777031.html">ISBN: 9780226511832</a>, <a href="http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tammo+tom+Dieck">Tammo tom Dieck</a>, <em>Topologie</em>, De Gruyter 2000 (<a href="https://doi.org/10.1515/9783110802542">doi:10.1515/9783110802542</a>)</p> </li> <li id="Hatcher02"> <p><a class="existingWikiWord" href="/nlab/show/Allen+Hatcher">Allen Hatcher</a>, <em>Algebraic Topology</em>, Cambridge University Press (2002) [<a href="https://www.cambridge.org/gb/academic/subjects/mathematics/geometry-and-topology/algebraic-topology-1?format=PB=9780521795401">ISBN:9780521795401</a>, <a href="https://pi.math.cornell.edu/~hatcher/AT/ATpage.html">webpage</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dai+Tamaki">Dai Tamaki</a>, <a class="existingWikiWord" href="/nlab/show/Akira+Kono">Akira Kono</a>, <em>Generalized Cohomology</em>, Translations of Mathematical Monographs, American Mathematical Society, 2006 (<a href="https://bookstore.ams.org/mmono-230">ISBN: 978-0-8218-3514-2</a>)</p> </li> <li id="tomDieck2008"> <p><a class="existingWikiWord" href="/nlab/show/Tammo+tom+Dieck">Tammo tom Dieck</a>, <em>Algebraic topology</em>, European Mathematical Society, Zürich (2008) (<a href="https://www.ems-ph.org/books/book.php?proj_nr=86">doi:10.4171/048</a>, <a href="https://www.maths.ed.ac.uk/~v1ranick/papers/diecktop.pdf">pdf</a>)</p> </li> <li id="Warner05"> <p><a class="existingWikiWord" href="/nlab/show/Garth+Warner">Garth Warner</a>: <em>Topics in Topology and Homotopy Theory</em>, EPrint Collection, University of Washington (2005) [<a href="http://hdl.handle.net/1773/2641">hdl:1773/2641</a>, <a href="https://sites.math.washington.edu//~warner/TTHT_Warner.pdf">pdf</a>, <a href="https://arxiv.org/abs/2007.02467">arXiv:2007.02467</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Peter+May">Peter May</a>, <a class="existingWikiWord" href="/nlab/show/Kate+Ponto">Kate Ponto</a>, <em><a class="existingWikiWord" href="/nlab/show/More+concise+algebraic+topology">More concise algebraic topology</a></em>, University of Chicago Press (2012) (<a href="https://press.uchicago.edu/ucp/books/book/chicago/M/bo12322308.html">ISBN:9780226511795</a>, <a href="https://www.math.uchicago.edu/~may/TEAK/KateBookFinal.pdf">pdf</a>)</p> </li> <li> <p>Clark Bray, Adrian Butcher, Simon Rubinstein-Salzedo: <em>Algebraic Topology</em>, Springer (2021) [<a href="https://doi.org/10.1007/978-3-030-70608-1">doi:10.1007/978-3-030-70608-1</a>, <a href="https://link.springer.com/content/pdf/10.1007/978-3-030-70608-1.pdf">pdf</a>]</p> </li> </ul> <p>On <a class="existingWikiWord" href="/nlab/show/constructive+mathematics">constructive</a> methods (<a class="existingWikiWord" href="/nlab/show/constructive+algebraic+topology">constructive algebraic topology</a>):</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Julio+Rubio">Julio Rubio</a>, <a class="existingWikiWord" href="/nlab/show/Francis+Sergeraert">Francis Sergeraert</a>, <em>Constructive Algebraic Topology</em>, Bulletin des Sciences Mathématiques <p><strong>126</strong> 5 (2002) 389-412 [<a href="https://doi.org/10.1016/S0007-4497(02)01119-3">doi:10.1016/S0007-4497(02)01119-3</a>, <a href="https://arxiv.org/abs/math/0111243">arXiv:math/0111243</a>]</p> </li> </ul> <p>Lecture notes:</p> <ul> <li id="HopkinsMathew"> <p><a class="existingWikiWord" href="/nlab/show/Michael+Hopkins">Michael Hopkins</a> (notes by <a class="existingWikiWord" href="/nlab/show/Akhil+Mathew">Akhil Mathew</a>), <em>algebraic topology – Lectures</em> (<a href="http://people.fas.harvard.edu/~amathew/ATnotes.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Friedhelm+Waldhausen">Friedhelm Waldhausen</a>, <em>Algebraische Topologie</em> I (<a href="https://www.math.uni-bielefeld.de/~fw/at.pdf">pdf</a>) , II (<a href="https://www.math.uni-bielefeld.de/~fw/at_II.pdf">pdf</a>), III (<a href="https://www.math.uni-bielefeld.de/~fw/at_III.pdf">pdf</a>) (<a href="https://www.math.uni-bielefeld.de/~fw/">web</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/James+F.+Davis">James F. Davis</a> and <a class="existingWikiWord" href="/nlab/show/Paul+Kirk">Paul Kirk</a>, <em>Lecture notes in algebraic topology</em> (<a href="http://www.indiana.edu/~jfdavis/teaching/m623/book.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gereon+Quick">Gereon Quick</a>, <em><a href="https://folk.ntnu.no/gereonq/Math231br.html">Advanced algebraic topology</a></em>, 2014</p> </li> </ul> <p>Survey of various subjects in algebraic topology:</p> <ul> <li id="James95"><a class="existingWikiWord" href="/nlab/show/Ioan+Mackenzie+James">Ioan Mackenzie James</a>, <em><a class="existingWikiWord" href="/nlab/show/Handbook+of+Algebraic+Topology">Handbook of Algebraic Topology</a></em> 1995</li> </ul> <p>Survey with relation to <a class="existingWikiWord" href="/nlab/show/differential+topology">differential topology</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Sergei+Novikov">Sergei Novikov</a>, <em>Topology I – General survey</em>, in: Encyclopedia of Mathematical Sciences Vol. 12, Springer 1986 (<a href="https://link.springer.com/book/10.1007/978-3-662-10579-5">doi:10.1007/978-3-662-10579-5</a>, <a href="https://web.math.rochester.edu/people/faculty/doug/otherpapers/novikovsurv.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Jean+Dieudonn%C3%A9">Jean Dieudonné</a>, <em>A History of Algebraic and Differential Topology, 1900 - 1960</em>, Modern Birkhäuser Classics 2009 (<a href="https://www.springer.com/de/book/9780817649067">ISBN:978-0-8176-4907-4</a>)</p> </li> </ul> <p>With focus on <a class="existingWikiWord" href="/nlab/show/ordinary+homology">ordinary homology</a>, <a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a> and <a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Jean+Gallier">Jean Gallier</a>, <a class="existingWikiWord" href="/nlab/show/Jocelyn+Quaintance">Jocelyn Quaintance</a>, <em>Homology, Cohomology, and Sheaf Cohomology for Algebraic Topology, Algebraic Geometry, and Differential Geometry</em>, World Scientific (2022) [<a href="https://doi.org/10.1142/12495">doi:10.1142/12495</a>, <a href="https://www.cis.upenn.edu/~jean/gbooks/sheaf-coho.html">webpage</a>]</li> </ul> <p>Some interactive 3D demos:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Neil+Strickland">Neil Strickland</a>, <em>Interactive pages for Algebraic Topology</em>, <a href="http://neil-strickland.staff.shef.ac.uk/courses/MAS435/demos/">web site</a></li> </ul> <p>Further pointers:</p> <ul> <li><a href="http://mathoverflow.net/questions/18041/algebraic-topology-beyond-the-basicsany-texts-bridging-the-gap">a thread on AlgTop literature at MathOverflow</a></li> </ul> <h3 id="ReferencesAbstractHomotopyTheory">Abstract homotopy theory</h3> <p>On <a class="existingWikiWord" href="/nlab/show/localization">localization</a> at <a class="existingWikiWord" href="/nlab/show/weak+equivalences">weak equivalences</a> to <a class="existingWikiWord" href="/nlab/show/homotopy+categories">homotopy categories</a>:</p> <ul> <li id="Brown65"><a class="existingWikiWord" href="/nlab/show/Edgar+Brown">Edgar Brown</a>, <em>Abstract homotopy theory</em>, Trans. AMS 119 no. 1 (1965) (<a href="https://doi.org/10.1090/S0002-9947-1965-0182970-6">doi:10.1090/S0002-9947-1965-0182970-6</a>)</li> </ul> <p>On <a class="existingWikiWord" href="/nlab/show/localization">localization</a> via <a class="existingWikiWord" href="/nlab/show/calculus+of+fractions">calculus of fractions</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Pierre+Gabriel">Pierre Gabriel</a>, <a class="existingWikiWord" href="/nlab/show/Michel+Zisman">Michel Zisman</a>, <em><a class="existingWikiWord" href="/nlab/show/Calculus+of+fractions+and+homotopy+theory">Calculus of fractions and homotopy theory</a></em>, <em>Ergebnisse der Mathematik und ihrer Grenzgebiete</em>, Band 35. Springer, New York (1967)</li> </ul> <p>On <a class="existingWikiWord" href="/nlab/show/homotopy+category+of+a+model+category">localization via</a> <a class="existingWikiWord" href="/nlab/show/model+category">model category</a>-theory:</p> <ul> <li id="Quillen67"> <p><a class="existingWikiWord" href="/nlab/show/Daniel+Quillen">Daniel Quillen</a>, <em>Homotopical algebra</em>, Lecture Notes in Mathematics 43, Berlin, New York, 1967</p> </li> <li id="Hovey99"> <p><a class="existingWikiWord" href="/nlab/show/Mark+Hovey">Mark Hovey</a>, <em><a class="existingWikiWord" href="/nlab/show/Model+Categories">Model Categories</a></em>, Mathematical Surveys and Monographs, Volume 63, AMS (1999) (<a href="https://bookstore.ams.org/surv-63-s">ISBN:978-0-8218-4361-1</a>, <a href="https://doi.org/http://dx.doi.org/10.1090/surv/063">doi:10.1090/surv/063</a>, <a href="https://people.math.rochester.edu/faculty/doug/otherpapers/hovey-model-cats.pdf">pdf</a>, <a href="http://books.google.co.uk/books?id=Kfs4uuiTXN0C&printsec=frontcover">Google books</a>)</p> </li> <li id="Hirschhorn02"> <p><a class="existingWikiWord" href="/nlab/show/Philip+Hirschhorn">Philip Hirschhorn</a>, <em>Model Categories and Their Localizations</em>, AMS Math. Survey and Monographs Vol 99 (2002) (<a href="https://bookstore.ams.org/surv-99-s/">ISBN:978-0-8218-4917-0</a>, <a href="http://www.gbv.de/dms/goettingen/360115845.pdf">pdf toc</a>, <a href="http://www.maths.ed.ac.uk/~aar/papers/hirschhornloc.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/William+G.+Dwyer">William G. Dwyer</a>, <a class="existingWikiWord" href="/nlab/show/Philip+S.+Hirschhorn">Philip S. Hirschhorn</a>, <a class="existingWikiWord" href="/nlab/show/Daniel+M.+Kan">Daniel M. Kan</a>, <a class="existingWikiWord" href="/nlab/show/Jeffrey+H.+Smith">Jeffrey H. Smith</a>, <em><a class="existingWikiWord" href="/nlab/show/Homotopy+Limit+Functors+on+Model+Categories+and+Homotopical+Categories">Homotopy Limit Functors on Model Categories and Homotopical Categories</a></em>, Mathematical Surveys and Monographs 113 (2004) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://bookstore.ams.org/surv-113-s">ISBN: 978-1-4704-1340-8</a>, <a href="http://dodo.pdmi.ras.ru/~topology/books/dhks.pdf">pdf</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></p> </li> </ul> <p>On <a class="existingWikiWord" href="/nlab/show/localization">localization</a> (especially of categories of <a class="existingWikiWord" href="/nlab/show/simplicial+sheaves">simplicial sheaves</a>/<a class="existingWikiWord" href="/nlab/show/simplicial+presheaves">simplicial presheaves</a>) via <a class="existingWikiWord" href="/nlab/show/categories+of+fibrant+objects">categories of fibrant objects</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kenneth+S.+Brown">Kenneth S. Brown</a>, <em><a class="existingWikiWord" href="/nlab/files/BrownAbstractHomotopyTheory.pdf" title="Abstract Homotopy Theory and Generalized Sheaf Cohomology">Abstract Homotopy Theory and Generalized Sheaf Cohomology</a></em>, Transactions of the American Mathematical Society <strong>186</strong> (1973) 419-458 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="http://www.jstor.org/stable/1996573">jstor:1996573</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math>.</li> </ul> <p>See also:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Klaus+Heiner+Kamps">Klaus Heiner Kamps</a>, <a class="existingWikiWord" href="/nlab/show/Tim+Porter">Tim Porter</a>, <em>Abstract Homotopy and Simple Homotopy Theory</em>, World Scientific 1997 (<a href="https://doi.org/10.1142/2215">doi:10.1142/2215</a>, <a href="http://books.google.de/books?id=7JYKxInRMdAC&dq=Porter+Kamps&printsec=frontcover&source=bl&ots=uuyl_tIjs4&sig=Lt8I92xQBZ4DNKVXD0x76WkcxCE&hl=de&sa=X&oi=book_result&resnum=3&ct=result#PPP1,M1">GoogleBooks</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Haynes+Miller">Haynes Miller</a> (ed.), <em><a class="existingWikiWord" href="/nlab/show/Handbook+of+Homotopy+Theory">Handbook of Homotopy Theory</a></em>, 2019</p> </li> </ul> <p>Lecture notes:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/William+Dwyer">William Dwyer</a>, <em>Homotopy theory and classifying spaces</em>, Copenhagen, June 2008 (<a href="http://www.math.ku.dk/~jg/homotopical2008/Dwyer.CopenhagenNotes.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Dwyer_HomotopyTheoryOfClassifyingSpaces.pdf" title="pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Jesper+Michael+M%C3%B8ller">Jesper Michael Møller</a>, <em>Homotopy theory for beginners</em>, 2015 (<a href="http://www.math.ku.dk/~moller/e01/algtopI/comments.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Moller_HomotopyTheory.pdf" title="pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/nlab/show/Introduction+to+Homotopy+Theory">Introduction to Homotopy Theory</a></em> (2016)</p> </li> <li id="Martins20"> <p><a class="existingWikiWord" href="/nlab/show/Yuri+Ximenes+Martins">Yuri Ximenes Martins</a>, <em>Introduction to Abstract Homotopy Theory</em> (<a href="https://arxiv.org/abs/2008.05302">arXiv:2008.05302</a>)</p> </li> </ul> <p>Introduction, from <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a> to (mostly <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-category+theory">abstract</a>, <a class="existingWikiWord" href="/nlab/show/simplicial+homotopy+theory">simplicial</a>) homotopy theory:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Emily+Riehl">Emily Riehl</a>, <em><a class="existingWikiWord" href="/nlab/show/Categorical+Homotopy+Theory">Categorical Homotopy Theory</a></em>, Cambridge University Press, 2014 (<a href="http://www.math.jhu.edu/~eriehl/cathtpy.pdf">pdf</a>, <a href="https://doi.org/10.1017/CBO9781107261457">doi:10.1017/CBO9781107261457</a>)</p> </li> <li id="Richter19"> <p><a class="existingWikiWord" href="/nlab/show/Birgit+Richter">Birgit Richter</a>, <em>From categories to homotopy theory</em>, Cambridge Studies in Advanced Mathematics 188, Cambridge University Press 2020 (<a href="https://doi.org/10.1017/9781108855891">doi:10.1017/9781108855891</a>, <a href="https://www.math.uni-hamburg.de/home/richter/catbook.html">book webpage</a>, <a href="https://www.math.uni-hamburg.de/home/richter/bookdraft.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+categories+and+toposes">geometry of physics – categories and toposes</a></em></p> </li> </ul> <p>See also:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/William+Dwyer">William Dwyer</a>, <a class="existingWikiWord" href="/nlab/show/Philip+Hirschhorn">Philip Hirschhorn</a>, <a class="existingWikiWord" href="/nlab/show/Daniel+Kan">Daniel Kan</a>, <a class="existingWikiWord" href="/nlab/show/Jeff+Smith">Jeff Smith</a>, <em>Homotopy Limit Functors on Model Categories and Homotopical Categories</em>, volume 113 of <em>Mathematical Surveys and Monographs</em>, American Mathematical Society (2004) (there exists <a href="http://dodo.pdmi.ras.ru/~topology/books/dhks.pdf">this</a> pdf copy of what seems to be a preliminary version of this book)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Zhen+Lin+Low">Zhen Lin Low</a>, <em><a class="existingWikiWord" href="/nlab/show/Notes+on+homotopical+algebra">Notes on homotopical algebra</a></em>, 2015</p> </li> <li id="MunsonVolic15"> <p><a class="existingWikiWord" href="/nlab/show/Brian+Munson">Brian Munson</a>, <a class="existingWikiWord" href="/nlab/show/Ismar+Volic">Ismar Volic</a>, <em>Cubical homotopy theory</em>, Cambridge University Press, 2015 (<a href="http://palmer.wellesley.edu/~ivolic/pdf/Papers/CubicalHomotopyTheory.pdf">pdf</a>, <a href="https://doi.org/10.1017/CBO9781139343329">doi:10.1017/CBO9781139343329</a>)</p> <blockquote> <p>(with emphasis on <a class="existingWikiWord" href="/nlab/show/cubical+objects">cubical objects</a> such as in <a class="existingWikiWord" href="/nlab/show/n-excisive+functors">n-excisive functors</a> and <a class="existingWikiWord" href="/nlab/show/Goodwillie+calculus">Goodwillie calculus</a>)</p> </blockquote> </li> </ul> <h3 id="ReferencesSimplicialHomotopyTheory">Simplicial homotopy theory</h3> <p>On <a class="existingWikiWord" href="/nlab/show/simplicial+homotopy+theory">simplicial homotopy theory</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Peter+May">Peter May</a>, <em>Simplicial objects in algebraic topology</em>, University of Chicago Press 1967 (<a href="https://press.uchicago.edu/ucp/books/book/chicago/S/bo5956688.html">ISBN:9780226511818</a>, <a href="http://www.math.uchicago.edu/~may/BOOKS/Simp.djvu">djvu</a>, <a class="existingWikiWord" href="/nlab/files/May_SimplicialObjectsInAlgebraicTopology.pdf" title="pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Edward+B.+Curtis">Edward B. Curtis</a>, <em>Simplicial homotopy theory</em>, Advances in Mathematics 6 (1971) 107–209 (<a href="https://doi.org/10.1016/0001-8708(71)90015-6">doi:10.1016/0001-8708(71)90015-6</a>, <a href="http://www.ams.org/mathscinet-getitem?mr=279808">MR279808</a>)</p> </li> <li id="JoyalTierney05"> <p><a class="existingWikiWord" href="/nlab/show/Andr%C3%A9+Joyal">André Joyal</a>, <a class="existingWikiWord" href="/nlab/show/Myles+Tierney">Myles Tierney</a> <em>Notes on simplicial homotopy theory</em>, Lecture at <em><a href="https://lists.lehigh.edu/pipermail/algtop-l/2007q4/000017.html">Advanced Course on Simplicial Methods in Higher Categories</a></em>, CRM 2008 (<a class="existingWikiWord" href="/nlab/files/JoyalTierneyNotesOnSimplicialHomotopyTheory.pdf" title="pdf">pdf</a>)</p> </li> <li id="JoyalTierney05"> <p><a class="existingWikiWord" href="/nlab/show/Andr%C3%A9+Joyal">André Joyal</a>, <a class="existingWikiWord" href="/nlab/show/Myles+Tierney">Myles Tierney</a>, <em>An introduction to simplicial homotopy theory</em>, 2009 (<a href="http://hopf.math.purdue.edu/cgi-bin/generate?/Joyal-Tierney/JT-chap-01">web</a>, <a class="existingWikiWord" href="/nlab/files/JoyalTierneySimplicialHomotopyTheory.pdf" title="pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Paul+Goerss">Paul Goerss</a>, <a class="existingWikiWord" href="/nlab/show/Kirsten+Schemmerhorn">Kirsten Schemmerhorn</a>, <em>Model categories and simplicial methods</em>, Notes from lectures given at the University of Chicago, August 2004, in: <em>Interactions between Homotopy Theory and Algebra</em>, Contemporary Mathematics 436, AMS 2007 (<a href="http://arxiv.org/abs/math.AT/0609537">arXiv:math.AT/0609537</a>, <a href="http://dx.doi.org/10.1090/conm/436">doi:10.1090/conm/436</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Francis+Sergeraert">Francis Sergeraert</a>, <em>Introduction to Combinatorial Homotopy Theory</em>, 2008 (<a href="https://www-fourier.ujf-grenoble.fr/~%20sergerar/Papers/Trieste-Lecture-Notes.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/SergeraertCombinatorialHomotopyTheory.pdf" title="pdf">pdf</a>)</p> </li> <li id="GoerssJardine09"> <p><a class="existingWikiWord" href="/nlab/show/Paul+Goerss">Paul Goerss</a>, <a class="existingWikiWord" href="/nlab/show/J.+F.+Jardine">J. F. Jardine</a>, Section V.4 of: <em><a class="existingWikiWord" href="/nlab/show/Simplicial+homotopy+theory">Simplicial homotopy theory</a></em>, Progress in Mathematics, Birkhäuser (1999) Modern Birkhäuser Classics (2009) (<a href="https://link.springer.com/book/10.1007/978-3-0346-0189-4">doi:10.1007/978-3-0346-0189-4</a>, <a href="http://web.archive.org/web/19990208220238/http://www.math.uwo.ca/~jardine/papers/simp-sets/">webpage</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Garth+Warner">Garth Warner</a>: <em>Categorical Homotopy Theory</em>, EPrint Collection, University of Washington (2012) [<a href="http://hdl.handle.net/1773/19589">hdl:1773/19589</a>, <a href="https://digital.lib.washington.edu/researchworks/bitstreams/0082c74f-f4e0-4578-a44e-d57a0ea29112/download">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Waner-CategoricalHomotopy.pdf" title="pdf">pdf</a>]</p> </li> </ul> <h3 id="ReferencesBasicInfinityCategoryTheory">Basic <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>-category theory</h3> <p>On <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+theory">(∞,1)-category theory</a> and <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos+theory">(∞,1)-topos theory</a>:</p> <ul> <li id="Joyal08"> <p><a class="existingWikiWord" href="/nlab/show/Andr%C3%A9+Joyal">André Joyal</a>, <em>The theory of quasicategories and its applications</em> lectures at <em><a href="https://lists.lehigh.edu/pipermail/algtop-l/2007q4/000017.html">Advanced Course on Simplicial Methods in Higher Categories</a></em>, CRM 2008 (<a href="http://mat.uab.cat/~kock/crm/hocat/advanced-course/Quadern45-2.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/JoyalTheoryOfQuasiCategories.pdf" title="pdf">pdf</a>)</p> </li> <li id="Joyal08"> <p><a class="existingWikiWord" href="/nlab/show/Andr%C3%A9+Joyal">André Joyal</a>, <em>Notes on Logoi</em>, 2008 (<a href="http://www.math.uchicago.edu/~may/IMA/JOYAL/Joyal.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/JoyalOnLogoi2008.pdf" title="pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, <em><a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">Higher Topos Theory</a></em></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Denis-Charles+Cisinski">Denis-Charles Cisinski</a>, <em>Higher category theory and homotopical algebra</em> (<a href="http://www.mathematik.uni-regensburg.de/cisinski/CatLR.pdf">pdf</a>)</p> </li> </ul> <h3 id="ReferencesBasicHomotopyTypeTheory">Basic homotopy type theory</h3> <p>On <a class="existingWikiWord" href="/nlab/show/synthetic+homotopy+theory">synthetic homotopy theory</a> in <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a>:</p> <p>Exposition:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Dan+Licata">Dan Licata</a>: <em>Homotopy theory in type theory</em> (2013) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="http://dlicata.web.wesleyan.edu/pubs/bll13homotopy/bll13homotopy.pdf">pdf slides</a>, <a class="existingWikiWord" href="/nlab/files/Licata-HomotopyInTypeTheory.pdf" title="pdf">pdf</a>, <a href="https://homotopytypetheory.org/2013/03/08/homotopy-theory-in-homotopy-type-theory-introduction">blog entry 1</a>, <a href="https://homotopytypetheory.org/2013/05/20/homotopy-theory-in-type-theory-progress-report/">blog entry 2</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></p> </li> <li id="Shulman17"> <p><a class="existingWikiWord" href="/nlab/show/Mike+Shulman">Mike Shulman</a>, <em>The logic of space</em>, in: <a class="existingWikiWord" href="/nlab/show/Gabriel+Catren">Gabriel Catren</a>, <a class="existingWikiWord" href="/nlab/show/Mathieu+Anel">Mathieu Anel</a> (eds.), <em><a class="existingWikiWord" href="/nlab/show/New+Spaces+for+Mathematics+and+Physics">New Spaces for Mathematics and Physics</a></em>, Cambridge University Press (2021) 322-404 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://arxiv.org/abs/1703.03007">arXiv:1703.03007</a>, <a href="https://doi.org/10.1017/9781108854429.009">doi:10.1017/9781108854429.009</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></p> </li> </ul> <p>Textbook accounts:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Univalent+Foundations+Project">Univalent Foundations Project</a>: <em><a class="existingWikiWord" href="/nlab/show/Homotopy+Type+Theory+--+Univalent+Foundations+of+Mathematics">Homotopy Type Theory – Univalent Foundations of Mathematics</a></em> (2013) (<a href="http://homotopytypetheory.org/book/">webpage</a>, <a href="http://hottheory.files.wordpress.com/2013/03/hott-online-323-g28e4374.pdf">pdf</a>)</p> </li> <li id="Rijke19"> <p><a class="existingWikiWord" href="/nlab/show/Egbert+Rijke">Egbert Rijke</a>, <em><a class="existingWikiWord" href="/nlab/show/Introduction+to+Homotopy+Type+Theory">Introduction to Homotopy Type Theory</a></em> (2019) (<a href="http://www.andrew.cmu.edu/user/erijke/hott/">web</a>, <a href="http://www.andrew.cmu.edu/user/erijke/hott/hott_intro.pdf">pdf</a>, <a href="https://github.com/EgbertRijke/HoTT-Intro">GitHub</a>)</p> </li> </ul> <p>For more see also at <em><a href="homotopy+type+theory#HomotopyTheoryInHomotopyTyepTheoryReferences">homotopy theory formalized in homotopy type theory</a></em>.</p> <h3 id="ReferencesOutlook">Outlook</h3> <p>Indications of open questions and possible future directions in <a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a> and (<a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable</a>) <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Mark+Hovey">Mark Hovey</a>, <em><a href="https://www-users.cse.umn.edu/~tlawson/hovey/">Algebraic Topology Problem List</a></em></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tyler+Lawson">Tyler Lawson</a>, <em>The future</em>, Talbot lectures 2013 (<a href="http://math.mit.edu/conferences/talbot/2013/19-Lawson-thefuture.pdf">pdf</a>)</p> </li> <li id="ProblemsInHomotopyTheoryWiki"> <p><em>Problems in homotopy theory</em> (<a href="http://topology-octopus.herokuapp.com/problemsinhomotopytheory/show/HomePage">wiki</a>)</p> </li> </ul> <p>More regarding the sociology of the field (such as its <a class="existingWikiWord" href="/nlab/show/folklore">folklore</a> results):</p> <ul> <li id="Barwick17"><a class="existingWikiWord" href="/nlab/show/Clark+Barwick">Clark Barwick</a>, <em>The future of homotopy theory</em>, 2017 (<a href="http://www.maths.ed.ac.uk/~cbarwick/papers/future.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/BarwickFutureOfHomotopyTheory.pdf" title="pdf">pdf</a>)</li> </ul> </div></body></html> </div> <div class="revisedby"> <p> Last revised on May 22, 2022 at 17:40:14. See the <a href="/nlab/history/algebraic+topology" 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/algebraic+topology" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/6882/#Item_23">Discuss</a><span class="backintime"><a href="/nlab/revision/algebraic+topology/46" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/algebraic+topology" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/algebraic+topology" accesskey="S" class="navlink" id="history" rel="nofollow">History (46 revisions)</a> <a href="/nlab/show/algebraic+topology/cite" style="color: black">Cite</a> <a href="/nlab/print/algebraic+topology" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/algebraic+topology" id="view_source" rel="nofollow">Source</a> </div> </div> <!-- Content --> </div> <!-- Container --> </body> </html>