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vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Concepts"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Concepts</span> </div> </a> <button aria-controls="toc-Concepts-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Concepts subsection</span> </button> <ul id="toc-Concepts-sublist" class="vector-toc-list"> <li id="toc-Spaces_and_maps" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Spaces_and_maps"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Spaces and maps</span> </div> </a> <ul id="toc-Spaces_and_maps-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Homotopy" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Homotopy"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Homotopy</span> </div> </a> <ul id="toc-Homotopy-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-CW_complex" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#CW_complex"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>CW complex</span> </div> </a> <ul id="toc-CW_complex-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Cofibration_and_fibration" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Cofibration_and_fibration"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Cofibration and fibration</span> </div> </a> <ul id="toc-Cofibration_and_fibration-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lifting_property" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Lifting_property"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.5</span> <span>Lifting property</span> </div> </a> <ul id="toc-Lifting_property-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Loop_and_suspension" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Loop_and_suspension"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.6</span> <span>Loop and suspension</span> </div> </a> <ul id="toc-Loop_and_suspension-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Classifying_spaces_and_homotopy_operations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Classifying_spaces_and_homotopy_operations"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.7</span> <span>Classifying spaces and homotopy operations</span> </div> </a> <ul id="toc-Classifying_spaces_and_homotopy_operations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Spectrum_and_generalized_cohomology" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Spectrum_and_generalized_cohomology"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.8</span> <span>Spectrum and generalized cohomology</span> </div> </a> <ul id="toc-Spectrum_and_generalized_cohomology-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ring_spectrum_and_module_spectrum" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ring_spectrum_and_module_spectrum"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.9</span> <span>Ring spectrum and module spectrum</span> </div> </a> <ul id="toc-Ring_spectrum_and_module_spectrum-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Key_theorems" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Key_theorems"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Key theorems</span> </div> </a> <ul id="toc-Key_theorems-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Obstruction_theory_and_characteristic_class" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Obstruction_theory_and_characteristic_class"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Obstruction theory and characteristic class</span> </div> </a> <ul id="toc-Obstruction_theory_and_characteristic_class-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Localization_and_completion_of_a_space" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Localization_and_completion_of_a_space"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Localization and completion of a space</span> </div> </a> <ul id="toc-Localization_and_completion_of_a_space-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Specific_theories" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Specific_theories"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Specific theories</span> </div> </a> <ul id="toc-Specific_theories-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Homotopy_hypothesis" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Homotopy_hypothesis"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Homotopy hypothesis</span> </div> </a> <ul id="toc-Homotopy_hypothesis-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Abstract_homotopy_theory" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Abstract_homotopy_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Abstract homotopy theory</span> </div> </a> <button aria-controls="toc-Abstract_homotopy_theory-sublist" class="cdx-button cdx-button--weight-quiet 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class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" 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It allows adding a reason in the summary.">undo</a></span></strong></div><div id="mw-diff-ntitle2"><a href="/wiki/User:Dave-okanagan" class="mw-userlink" title="User:Dave-okanagan" data-mw-revid="1258166327"><bdi>Dave-okanagan</bdi></a> <span class="mw-usertoollinks">(<a href="/wiki/User_talk:Dave-okanagan" class="mw-usertoollinks-talk" title="User talk:Dave-okanagan">talk</a> | <a href="/wiki/Special:Contributions/Dave-okanagan" class="mw-usertoollinks-contribs" title="Special:Contributions/Dave-okanagan">contribs</a>)</span><div class="mw-diff-usermetadata"><div class="mw-diff-userroles"><a href="/wiki/Wikipedia:Extended_confirmed_editors" class="mw-redirect" title="Wikipedia:Extended confirmed editors">Extended confirmed users</a></div><div class="mw-diff-usereditcount"><span>6,966</span> edits</div></div></div><div id="mw-diff-ntitle3"> <span class="comment comment--without-parentheses">All Refs WORK - though techically Not coded properly in the article???) (please check ref 15 added new url pdf pg 215) (DEL Read url=Ex Lnk</span></div><div id="mw-diff-ntitle5"></div><div id="mw-diff-ntitle4"> </div></td> </tr><tr> <td colspan="2" class="diff-lineno">Line 1:</td> <td colspan="2" class="diff-lineno">Line 1:</td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>{{in use}}</div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>{{Short description|Branch of mathematics}}</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>{{Short description|Branch of mathematics}}</div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>In [[mathematics]], '''homotopy theory''' is a systematic study of situations in which [[Map (mathematics)|maps]] can come with [[homotopy|homotopies]] between them. It originated as a topic in [[algebraic topology]], but nowadays is learned as an independent discipline.</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>In [[mathematics]], '''homotopy theory''' is a systematic study of situations in which [[Map (mathematics)|maps]] can come with [[homotopy|homotopies]] between them. It originated as a topic in [[algebraic topology]], but nowadays is learned as an independent discipline.</div></td> </tr> <tr> <td colspan="2" class="diff-lineno">Line 192:</td> <td colspan="2" class="diff-lineno">Line 191:</td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>== References ==</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>== References ==</div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>{{reflist}}</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>{{reflist}}</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><br /></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>* {{cite book|first1=Raoul|last1=Bott|first2=Loring W.|last2=Tu|title=Differential Forms in Algebraic Topology|publisher=Springer|year=1995|isbn=978-<del class="diffchange diffchange-inline">0387906133</del>}}</div></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>* {{cite book<ins class="diffchange diffchange-inline"> </ins>|first1=Raoul<ins class="diffchange diffchange-inline"> </ins>|last1=Bott<ins class="diffchange diffchange-inline"> </ins>|first2=Loring W.<ins class="diffchange diffchange-inline"> </ins>|last2=Tu<ins class="diffchange diffchange-inline"> </ins>|title=Differential Forms in Algebraic Topology<ins class="diffchange diffchange-inline"> </ins>|publisher=Springer<ins class="diffchange diffchange-inline"> </ins>|year=1995<ins class="diffchange diffchange-inline"> </ins>|isbn=978-<ins class="diffchange diffchange-inline">038790613-3 </ins>}}</div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>*May<del class="diffchange diffchange-inline">,</del> J. <del class="diffchange diffchange-inline">[</del>http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf A Concise Course in Algebraic Topology]</div></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>*<ins class="diffchange diffchange-inline"> {{cite web |last=</ins>May <ins class="diffchange diffchange-inline">|first=</ins>J. <ins class="diffchange diffchange-inline">Peter |url=</ins>http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf <ins class="diffchange diffchange-inline">|title=</ins>A Concise Course in Algebraic Topology<ins class="diffchange diffchange-inline"> |website=[[University of Chicago</ins>]<ins class="diffchange diffchange-inline">] }}</ins></div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>* J. P. May and K. Ponto, More Concise Algebraic Topology: Localization, completion, and model categories</div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><br /></td> </tr> <tr> <td class="diff-marker"><a class="mw-diff-movedpara-left" title="Paragraph was moved. Click to jump to new location." href="#movedpara_14_0_rhs">⚫</a></td> <td class="diff-deletedline diff-side-deleted"><div><a name="movedpara_9_0_lhs"></a>* {{cite book|<del class="diffchange diffchange-inline">author</del>=George William Whitehead|author-link=George W. Whitehead|title=Elements of homotopy theory|url=https://books.google.com/books?id=wlrvAAAAMAAJ|access-date=September 6, 2011|edition=3rd|series=Graduate Texts in Mathematics|volume=61|year=1978|publisher=Springer-Verlag|location=New York-Berlin|isbn=978-0-387-90336-1|pages=xxi+744|mr=0516508 }}</div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>* {{cite book |first1=J. Peter |last1=May |first2=Kate |last2=Ponto |title=More Concise Algebraic Topology: Localization, completion, and model categories |publisher=[[University of Chicago Press]] |isbn=978-022651178-8 |url=https://www.maths.ed.ac.uk/~v1ranick/papers/mayponto.pdf |via=[[University of Edinburgh]] |page=215 }}</div></td> </tr> <tr> <td class="diff-marker"><a class="mw-diff-movedpara-left" title="Paragraph was moved. Click to jump to new location." href="#movedpara_16_0_rhs">⚫</a></td> <td class="diff-deletedline diff-side-deleted"><div><a name="movedpara_11_0_lhs"></a>*Ronald Brown<del class="diffchange diffchange-inline">,</del> <del class="diffchange diffchange-inline">''[</del>http://arquivo.pt/wayback/20160514115224/http://www.bangor.ac.uk/r.brown/topgpds.html Topology and groupoids<del class="diffchange diffchange-inline">]''</del> <del class="diffchange diffchange-inline">(</del>2006<del class="diffchange diffchange-inline">)</del> Booksurge LLC <del class="diffchange diffchange-inline">{{ISBN</del>|1-4196-2722-8}}<del class="diffchange diffchange-inline">.</del></div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><br /></td> </tr> <tr> <td class="diff-marker"><a class="mw-diff-movedpara-left" title="Paragraph was moved. Click to jump to new location." href="#movedpara_16_1_rhs">⚫</a></td> <td class="diff-deletedline diff-side-deleted"><div><a name="movedpara_13_0_lhs"></a>* https://ncatlab.org/nlab/show/homotopical+algebra</div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker"><a class="mw-diff-movedpara-right" title="Paragraph was moved. Click to jump to old location." href="#movedpara_9_0_lhs">⚫</a></td> <td class="diff-addedline diff-side-added"><div><a name="movedpara_14_0_rhs"></a>* {{cite book<ins class="diffchange diffchange-inline"> </ins>|<ins class="diffchange diffchange-inline">first</ins>=George William <ins class="diffchange diffchange-inline">|last=</ins>Whitehead<ins class="diffchange diffchange-inline"> </ins>|author-link=George W. Whitehead<ins class="diffchange diffchange-inline"> </ins>|title=Elements of homotopy theory<ins class="diffchange diffchange-inline"> </ins>|url=https://books.google.com/books?id=wlrvAAAAMAAJ<ins class="diffchange diffchange-inline"> </ins>|access-date=September 6, 2011<ins class="diffchange diffchange-inline"> </ins>|edition=3rd<ins class="diffchange diffchange-inline"> </ins>|series=Graduate Texts in Mathematics<ins class="diffchange diffchange-inline"> </ins>|volume=61<ins class="diffchange diffchange-inline"> </ins>|year=1978<ins class="diffchange diffchange-inline"> </ins>|publisher=Springer-Verlag<ins class="diffchange diffchange-inline"> </ins>|location=New York-Berlin<ins class="diffchange diffchange-inline"> </ins>|isbn=978-0-387-90336-1<ins class="diffchange diffchange-inline"> </ins>|pages=xxi+744<ins class="diffchange diffchange-inline"> </ins>|mr=0516508 }}</div></td> </tr> <tr> <td class="diff-marker"><a class="mw-diff-movedpara-left" title="Paragraph was moved. Click to jump to new location." href="#movedpara_16_3_rhs">⚫</a></td> <td class="diff-deletedline diff-side-deleted"><div><a name="movedpara_15_0_lhs"></a>* Homotopy Theories and Model Categories <del class="diffchange diffchange-inline">by </del>W.G. Dwyer <del class="diffchange diffchange-inline">and </del>J. Spalinski <del class="diffchange diffchange-inline">in [</del>https://books.google.com/books?id=xoM5DxQZihQC&printsec=copyright#v=onepage&q&f=false <del class="diffchange diffchange-inline"> </del>Handbook of Algebraic Topology<del class="diffchange diffchange-inline">]</del> <del class="diffchange diffchange-inline">edited by</del> I.M. James</div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker"><a class="mw-diff-movedpara-right" title="Paragraph was moved. Click to jump to old location." href="#movedpara_11_0_lhs">⚫</a></td> <td class="diff-addedline diff-side-added"><div><a name="movedpara_16_0_rhs"></a>*<ins class="diffchange diffchange-inline"> {{cite web |first=</ins>Ronald <ins class="diffchange diffchange-inline">|last=</ins>Brown <ins class="diffchange diffchange-inline">|url=</ins>http://arquivo.pt/wayback/20160514115224/http://www.bangor.ac.uk/r.brown/topgpds.html <ins class="diffchange diffchange-inline">|title=</ins>Topology and groupoids <ins class="diffchange diffchange-inline">|year=</ins>2006 <ins class="diffchange diffchange-inline">|publisher=</ins>Booksurge LLC |<ins class="diffchange diffchange-inline">isbn=</ins>1-4196-2722-8<ins class="diffchange diffchange-inline"> </ins>}}</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker"><a class="mw-diff-movedpara-right" title="Paragraph was moved. Click to jump to old location." href="#movedpara_13_0_lhs">⚫</a></td> <td class="diff-addedline diff-side-added"><div><a name="movedpara_16_1_rhs"></a>* <ins class="diffchange diffchange-inline">{{cite web |title=Homotopical algebra |website=nLab |url=</ins>https://ncatlab.org/nlab/show/homotopical+algebra<ins class="diffchange diffchange-inline"> }}</ins></div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><br /></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker"><a class="mw-diff-movedpara-right" title="Paragraph was moved. Click to jump to old location." href="#movedpara_15_0_lhs">⚫</a></td> <td class="diff-addedline diff-side-added"><div><a name="movedpara_16_3_rhs"></a>* <ins class="diffchange diffchange-inline">{{cite book |chapter=</ins>Homotopy Theories and Model Categories <ins class="diffchange diffchange-inline">|first1=</ins>W.G. <ins class="diffchange diffchange-inline">|last1=</ins>Dwyer <ins class="diffchange diffchange-inline">|first2=</ins>J. <ins class="diffchange diffchange-inline">|last2=</ins>Spalinski <ins class="diffchange diffchange-inline">|url=</ins>https://books.google.com/books?id=xoM5DxQZihQC&printsec=copyright#v=onepage&q&f=false <ins class="diffchange diffchange-inline">|title=</ins>Handbook of Algebraic Topology <ins class="diffchange diffchange-inline">|isbn=0-444-81779-4</ins> <ins class="diffchange diffchange-inline">|editor-first=</ins>I.M. <ins class="diffchange diffchange-inline">|editor-last=</ins>James<ins class="diffchange diffchange-inline"> }}</ins></div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><br /></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>* {{cite web |first=Allen |last=Hatcher |url=http://pi.math.cornell.edu/~hatcher/AT/ATpage.html |title=Algebraic topology }}</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>* {{cite web |first=Allen |last=Hatcher |url=http://pi.math.cornell.edu/~hatcher/AT/ATpage.html |title=Algebraic topology }}</div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>* {{cite journal |<del class="diffchange diffchange-inline">last1</del>=Milnor |<del class="diffchange diffchange-inline">first1</del>=John |title=On spaces having the homotopy type of 𝐶𝑊-complex |journal=Transactions of the American Mathematical Society |year=1959 |volume=90 |issue=2 |pages=272–280 |doi=10.1090/S0002-9947-1959-0100267-4 |s2cid=123048606 |language=en |issn=0002-9947 }}</div></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>* {{cite journal |<ins class="diffchange diffchange-inline">last</ins>=Milnor |<ins class="diffchange diffchange-inline">first</ins>=John |title=On spaces having the homotopy type of 𝐶𝑊-complex |journal=<ins class="diffchange diffchange-inline">[[</ins>Transactions of the American Mathematical Society<ins class="diffchange diffchange-inline">]]</ins> |year=1959 |volume=90 |issue=2 |pages=272–280 |doi=10.1090/S0002-9947-1959-0100267-4 |s2cid=123048606 |language=en |issn=0002-9947 }}</div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>* Edwin Spanier<del class="diffchange diffchange-inline">,</del> Algebraic topology</div></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>* <ins class="diffchange diffchange-inline">{{cite book |first=</ins>Edwin <ins class="diffchange diffchange-inline">|last=</ins>Spanier <ins class="diffchange diffchange-inline">|title=</ins>Algebraic topology<ins class="diffchange diffchange-inline"> |isbn=978-0-387-94426-5 }}</ins></div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>* Dennis Sullivan<del class="diffchange diffchange-inline">.</del> Genetics of homotopy theory and the Adams conjecture<del class="diffchange diffchange-inline">.</del> <del class="diffchange diffchange-inline">Ann.</del> of <del class="diffchange diffchange-inline">Math.</del> <del class="diffchange diffchange-inline">(</del>2<del class="diffchange diffchange-inline">),</del> 100<del class="diffchange diffchange-inline">:</del>1–79<del class="diffchange diffchange-inline">,</del> 1974.</div></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>* <ins class="diffchange diffchange-inline">{{cite journal |first=</ins>Dennis <ins class="diffchange diffchange-inline">|last=</ins>Sullivan <ins class="diffchange diffchange-inline">|title=</ins>Genetics of homotopy theory and the Adams conjecture <ins class="diffchange diffchange-inline">|journal=[[Annals</ins> of <ins class="diffchange diffchange-inline">Mathematics]]</ins> <ins class="diffchange diffchange-inline">|series=</ins>2 <ins class="diffchange diffchange-inline">|volume=</ins>100<ins class="diffchange diffchange-inline"> |issue=1 |pages=</ins>1–79<ins class="diffchange diffchange-inline"> |date=July</ins> 1974<ins class="diffchange diffchange-inline"> |url=https://math</ins>.<ins class="diffchange diffchange-inline">univ-cotedazur.fr/~cazanave/Gdt/ImJ/Sullivan.pdf |via=Math - [[Côte d'Azur University]] }}</ins></div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><br /></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>== Further reading ==</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>== Further reading ==</div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>*<del class="diffchange diffchange-inline">[</del>http://www.math.univ-toulouse.fr/~dcisinsk/1097.pdf Cisinski<del class="diffchange diffchange-inline">'s</del> <del class="diffchange diffchange-inline">notes</del>]</div></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>*<ins class="diffchange diffchange-inline"> {{cite web |url=</ins>http://www.math.univ-toulouse.fr/~dcisinsk/1097.pdf <ins class="diffchange diffchange-inline">|first=Denis-Charles |last=</ins>Cisinski <ins class="diffchange diffchange-inline">|title=Higher Categories And Topos Theory(in french) |date=March 2015 |website=Math - [[University of Toulouse</ins>]<ins class="diffchange diffchange-inline">] }}</ins></div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>* {{cite web |last=Porter |first=Timothy |date=February 12, 2010 |title=Abstract Homotopy Theory: The Interaction Of Category Theory And Homotopy Theory: A Revised Version Of The 2001 Article |website=nLab |url=http://ncatlab.org/nlab/files/Abstract-Homotopy.pdf }}</div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>*http://ncatlab.org/nlab/files/Abstract-Homotopy.pdf</div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>*<del class="diffchange diffchange-inline">[</del>https://uregina.ca/~franklam/Math527/Math527.html Math 527 - Homotopy Theory Spring 2013, Section F1], lectures by Martin Frankland</div></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>*<ins class="diffchange diffchange-inline"> {{cite web |url=</ins>https://uregina.ca/~franklam/Math527/Math527.html <ins class="diffchange diffchange-inline">|title=</ins>Math 527 - Homotopy Theory Spring 2013, Section F1<ins class="diffchange diffchange-inline"> |publisher=[[University of Illinois Urbana-Champaign</ins>]<ins class="diffchange diffchange-inline">] |via=[[University of Regina]] }}</ins>, lectures by Martin Frankland</div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>* D. Quillen<del class="diffchange diffchange-inline">,</del> Homotopical algebra<del class="diffchange diffchange-inline">,</del> Lectures Notes in Math<del class="diffchange diffchange-inline">. vol.</del> 43<del class="diffchange diffchange-inline">,</del> Springer Verlag<del class="diffchange diffchange-inline">,</del> 1967<del class="diffchange diffchange-inline">.</del></div></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>* <ins class="diffchange diffchange-inline">{{cite book |first=</ins>D. <ins class="diffchange diffchange-inline">|last=</ins>Quillen <ins class="diffchange diffchange-inline">|title=</ins>Homotopical algebra <ins class="diffchange diffchange-inline">|series=</ins>Lectures Notes in Math <ins class="diffchange diffchange-inline">|volume=</ins>43 <ins class="diffchange diffchange-inline">|publisher=</ins>Springer Verlag <ins class="diffchange diffchange-inline">|year=</ins>1967<ins class="diffchange diffchange-inline"> |isbn=978-3-540-03914-3 }}</ins></div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>* https://ncatlab.org/nlab/show/homotopy+theory</div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>== External links ==</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>== External links ==</div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>{{cite web|title=<del class="diffchange diffchange-inline">homotopy</del> theory |url=https://ncatlab.org/nlab/show/homotopy+theory |website=ncatlab.org}}</div></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div><ins class="diffchange diffchange-inline">* </ins>{{cite web<ins class="diffchange diffchange-inline"> </ins>|title=<ins class="diffchange diffchange-inline">Homotopy</ins> theory |url=https://ncatlab.org/nlab/show/homotopy+theory |website=ncatlab.org<ins class="diffchange diffchange-inline"> </ins>}}</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><br /></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>{{Authority control}}</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>{{Authority control}}</div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> </table><hr class='diff-hr' id='mw-oldid' /> <h2 class='diff-currentversion-title'>Latest revision as of 13:52, 18 November 2024</h2> <div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Branch of mathematics</div> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, <b>homotopy theory</b> is a systematic study of situations in which <a href="/wiki/Map_(mathematics)" title="Map (mathematics)">maps</a> can come with <a href="/wiki/Homotopy" title="Homotopy">homotopies</a> between them. It originated as a topic in <a href="/wiki/Algebraic_topology" title="Algebraic topology">algebraic topology</a>, but nowadays is learned as an independent discipline. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Applications_to_other_fields_of_mathematics">Applications to other fields of mathematics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Homotopy_theory&action=edit&section=1" title="Edit section: Applications to other fields of mathematics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Besides algebraic topology, the theory has also been used in other areas of mathematics such as: </p> <ul><li><a href="/wiki/Algebraic_geometry" title="Algebraic geometry">Algebraic geometry</a> (e.g., <a href="/wiki/A1_homotopy_theory" class="mw-redirect" title="A1 homotopy theory">A<sup>1</sup> homotopy theory</a>)</li> <li><a href="/wiki/Category_theory" title="Category theory">Category theory</a> (specifically the study of <a href="/wiki/Higher_category_theory" title="Higher category theory">higher categories</a>)</li></ul> <div class="mw-heading mw-heading2"><h2 id="Concepts">Concepts</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Homotopy_theory&action=edit&section=2" title="Edit section: Concepts"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Spaces_and_maps">Spaces and maps</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Homotopy_theory&action=edit&section=3" title="Edit section: Spaces and maps"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In homotopy theory and algebraic topology, the word "space" denotes a <a href="/wiki/Topological_space" title="Topological space">topological space</a>. In order to avoid <a href="/wiki/Pathological_(mathematics)" title="Pathological (mathematics)">pathologies</a>, one rarely works with arbitrary spaces; instead, one requires spaces to meet extra constraints, such as being <a href="/wiki/Category_of_compactly_generated_weak_Hausdorff_spaces" title="Category of compactly generated weak Hausdorff spaces">compactly generated weak Hausdorff</a> or a <a href="/wiki/CW_complex" title="CW complex">CW complex</a>. </p><p>In the same vein as above, a "<a href="/wiki/Map_(mathematics)" title="Map (mathematics)">map</a>" is a continuous function, possibly with some extra constraints. </p><p>Often, one works with a <a href="/wiki/Pointed_space" title="Pointed space">pointed space</a>—that is, a space with a "distinguished point", called a basepoint. A pointed map is then a map which preserves basepoints; that is, it sends the basepoint of the domain to that of the codomain. In contrast, a free map is one which needn't preserve basepoints. </p><p>The Cartesian product of two pointed spaces <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>,</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X,Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8705438171d938b7f59cd1bfa5b7d99b6afa5cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.787ex; height:2.509ex;" alt="{\displaystyle X,Y}"></span> are not naturally pointed. A substitute is the <a href="/wiki/Smash_product" title="Smash product">smash product</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\wedge Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>∧</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\wedge Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6edc6915b42026ef5d46c585f7e44955f2d15ecf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.336ex; height:2.176ex;" alt="{\displaystyle X\wedge Y}"></span> which is characterized by the <a href="/wiki/Adjoint_functor" class="mw-redirect" title="Adjoint functor">adjoint relation</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Map} (X\wedge Y,Z)=\operatorname {Map} (X,\operatorname {Map} (Y,Z))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Map</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>∧</mo> <mi>Y</mi> <mo>,</mo> <mi>Z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>Map</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Map</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>Y</mi> <mo>,</mo> <mi>Z</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Map} (X\wedge Y,Z)=\operatorname {Map} (X,\operatorname {Map} (Y,Z))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7c9565c7ff0768fabf7bb5cb2c57d19b905fa96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.836ex; height:2.843ex;" alt="{\displaystyle \operatorname {Map} (X\wedge Y,Z)=\operatorname {Map} (X,\operatorname {Map} (Y,Z))}"></span>,</dd></dl> <p>that is, a smash product is an analog of a <a href="/wiki/Tensor_product" title="Tensor product">tensor product</a> in abstract algebra (see <a href="/wiki/Tensor-hom_adjunction" title="Tensor-hom adjunction">tensor-hom adjunction</a>). Explicitly, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\wedge Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>∧</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\wedge Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6edc6915b42026ef5d46c585f7e44955f2d15ecf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.336ex; height:2.176ex;" alt="{\displaystyle X\wedge Y}"></span> is the quotient of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\times Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>×</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\times Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1613c1ff4b6fbfb6c80a8da83e90ad28f0ab3483" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.594ex; height:2.176ex;" alt="{\displaystyle X\times Y}"></span> by the <a href="/wiki/Wedge_sum" title="Wedge sum">wedge sum</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\vee Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>∨</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\vee Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/091f7bb09b74960d59d46cc57a297ae37aece6e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.336ex; height:2.176ex;" alt="{\displaystyle X\vee Y}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Homotopy">Homotopy</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Homotopy_theory&action=edit&section=4" title="Edit section: Homotopy"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Homotopy" title="Homotopy">Homotopy</a></div> <p>Let <i>I</i> denote the unit interval <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,1]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/738f7d23bb2d9642bab520020873cccbef49768d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.653ex; height:2.843ex;" alt="{\displaystyle [0,1]}"></span>. A map </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h:X\times I\to Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>:</mo> <mi>X</mi> <mo>×</mo> <mi>I</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h:X\times I\to Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f240a8efc3ead4c041585116d590cdfb0e4df75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.656ex; height:2.176ex;" alt="{\displaystyle h:X\times I\to Y}"></span></dd></dl> <p>is called a <a href="/wiki/Homotopy" title="Homotopy">homotopy</a> from the map <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08c1c908b03c3f63383c7199465c7fd0b105030f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.393ex; height:2.509ex;" alt="{\displaystyle h_{0}}"></span> to the map <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14e8880a2e4243a2fe5157e574a0547ef3d5d373" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.393ex; height:2.509ex;" alt="{\displaystyle h_{1}}"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{t}(x)=h(x,t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>h</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h_{t}(x)=h(x,t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04b80f80ea8df70ed6fda9cd793469d578025501" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.754ex; height:2.843ex;" alt="{\displaystyle h_{t}(x)=h(x,t)}"></span>. Intuitively, we may think of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}"></span> as a path from the map <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08c1c908b03c3f63383c7199465c7fd0b105030f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.393ex; height:2.509ex;" alt="{\displaystyle h_{0}}"></span> to the map <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14e8880a2e4243a2fe5157e574a0547ef3d5d373" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.393ex; height:2.509ex;" alt="{\displaystyle h_{1}}"></span>. Indeed, a homotopy can be shown to be an <a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence relation</a>. When <i>X</i>, <i>Y</i> are pointed spaces, the maps <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h_{t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8dbf3d8bfe322f68ff6400385578f8d78e1ba7c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.165ex; height:2.509ex;" alt="{\displaystyle h_{t}}"></span> are required to preserve the basepoint and the homotopy <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}"></span> is called a <a href="/w/index.php?title=Based_homotopy&action=edit&redlink=1" class="new" title="Based homotopy (page does not exist)">based homotopy</a>. A based homotopy is the same as a (based) map <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\wedge I_{+}\to Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>∧</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\wedge I_{+}\to Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a9ae72755c47f18e95a6484b51dc4fd34ee7475" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.484ex; height:2.509ex;" alt="{\displaystyle X\wedge I_{+}\to Y}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03b29b97c4ce7b7a7551cf9f537a5fa42c14c145" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.534ex; height:2.509ex;" alt="{\displaystyle I_{+}}"></span> is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"></span> together with a disjoint basepoint.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>Given a pointed space <i>X</i> and an <a href="/wiki/Integer" title="Integer">integer</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce8a1b7b3bc3c790054d93629fc3b08cd1da1fd0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 0}"></span>, let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{n}X=[S^{n},X]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>X</mi> <mo>=</mo> <mo stretchy="false">[</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>,</mo> <mi>X</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi _{n}X=[S^{n},X]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01933c254562233f07b4a64a076e6aa6337e066a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.67ex; height:2.843ex;" alt="{\displaystyle \pi _{n}X=[S^{n},X]}"></span> be the homotopy classes of based maps <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{n}\to X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">→</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{n}\to X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21881e91b66e1c3c1fd8803e144986cb52bec9ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.334ex; height:2.343ex;" alt="{\displaystyle S^{n}\to X}"></span> from a (pointed) <i>n</i>-sphere <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee006452a59bf1eb29983b4412348b66517a2d23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.74ex; height:2.343ex;" alt="{\displaystyle S^{n}}"></span> to <i>X</i>. As it turns out, </p> <ul><li>for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≥</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\geq 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8ce9ce38d06f6bf5a3fe063118c09c2b6202bfe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 1}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{n}X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi _{n}X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/309cf040236632f263afb0b16d69c0f0fa4f2140" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.524ex; height:2.509ex;" alt="{\displaystyle \pi _{n}X}"></span> are <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">groups</a> called <a href="/wiki/Homotopy_group" title="Homotopy group">homotopy groups</a>; in particular, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{1}X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi _{1}X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b73939ad53b202c43c810c0d625ca2d2c10946b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.359ex; height:2.509ex;" alt="{\displaystyle \pi _{1}X}"></span> is called the <a href="/wiki/Fundamental_group" title="Fundamental group">fundamental group</a> of <i>X</i>,</li> <li>for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≥</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\geq 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6bf67f9d06ca3af619657f8d20ee1322da77174" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 2}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{n}X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi _{n}X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/309cf040236632f263afb0b16d69c0f0fa4f2140" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.524ex; height:2.509ex;" alt="{\displaystyle \pi _{n}X}"></span> are <a href="/wiki/Abelian_group" title="Abelian group">abelian groups</a> by the <a href="/wiki/Eckmann%E2%80%93Hilton_argument" title="Eckmann–Hilton argument">Eckmann–Hilton argument</a>,</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{0}X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi _{0}X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20676791fe5303f461e0c23f2eb5c8acd8740635" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.359ex; height:2.509ex;" alt="{\displaystyle \pi _{0}X}"></span> can be identified with the set of path-connected components in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>.</li></ul> <p>Every group is the fundamental group of some space.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>A map <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> is called a <a href="/wiki/Homotopy_equivalence" class="mw-redirect" title="Homotopy equivalence">homotopy equivalence</a> if there is another map <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"></span> such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\circ g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∘</mo> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\circ g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2f61ca7838709fbae07dce9c0d513770f10cfae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.589ex; height:2.509ex;" alt="{\displaystyle f\circ g}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\circ f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>∘</mo> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g\circ f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10b5ad4985af48d0fb7efa3c8afa5ad7d42bfc92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.589ex; height:2.509ex;" alt="{\displaystyle g\circ f}"></span> are both homotopic to the identities. Two spaces are said to be homotopy equivalent if there is a homotopy equivalence between them. A homotopy equivalence class of spaces is then called a <a href="/wiki/Homotopy_type" class="mw-redirect" title="Homotopy type">homotopy type</a>. There is a weaker notion: a map <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:X\to Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abd1e080abef4bbdab67b43819c6431e7561361c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.583ex; height:2.509ex;" alt="{\displaystyle f:X\to Y}"></span> is said to be a <a href="/wiki/Weak_homotopy_equivalence" class="mw-redirect" title="Weak homotopy equivalence">weak homotopy equivalence</a> if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{*}:\pi _{n}(X)\to \pi _{n}(Y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msub> <mo>:</mo> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{*}:\pi _{n}(X)\to \pi _{n}(Y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79fa1c64342226715f9327abdcb794322fb82a7f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.204ex; height:2.843ex;" alt="{\displaystyle f_{*}:\pi _{n}(X)\to \pi _{n}(Y)}"></span> is an isomorphism for each <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce8a1b7b3bc3c790054d93629fc3b08cd1da1fd0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 0}"></span> and each choice of a base point. A homotopy equivalence is a weak homotopy equivalence but the converse need not be true. </p><p>Through the adjunction </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Map} (X\times I,Y)=\operatorname {Map} (X,\operatorname {Map} (I,Y)),\,\,h\mapsto (x\mapsto h(x,\cdot ))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Map</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>×</mo> <mi>I</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>Map</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Map</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>I</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mi>h</mi> <mo stretchy="false">↦</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">↦</mo> <mi>h</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mo>⋅</mo> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Map} (X\times I,Y)=\operatorname {Map} (X,\operatorname {Map} (I,Y)),\,\,h\mapsto (x\mapsto h(x,\cdot ))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/527d6a194bf5615639a453ef75143bcba8cb2967" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:57.749ex; height:2.843ex;" alt="{\displaystyle \operatorname {Map} (X\times I,Y)=\operatorname {Map} (X,\operatorname {Map} (I,Y)),\,\,h\mapsto (x\mapsto h(x,\cdot ))}"></span>,</dd></dl> <p>a homotopy <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h:X\times I\to Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>:</mo> <mi>X</mi> <mo>×</mo> <mi>I</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h:X\times I\to Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f240a8efc3ead4c041585116d590cdfb0e4df75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.656ex; height:2.176ex;" alt="{\displaystyle h:X\times I\to Y}"></span> is sometimes viewed as a map <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\to Y^{I}=\operatorname {Map} (I,Y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo stretchy="false">→</mo> <msup> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msup> <mo>=</mo> <mi>Map</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>I</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\to Y^{I}=\operatorname {Map} (I,Y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e2fa58803125782416ad8d133cb2c740c4c2a7c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.028ex; height:3.176ex;" alt="{\displaystyle X\to Y^{I}=\operatorname {Map} (I,Y)}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="CW_complex">CW complex</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Homotopy_theory&action=edit&section=5" title="Edit section: CW complex"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/CW_complex" title="CW complex">CW complex</a></div> <p>A <a href="/wiki/CW_complex" title="CW complex">CW complex</a> is a space that has a filtration <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\supset \cdots \supset X^{n}\supset X^{n-1}\supset \cdots \supset X^{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>⊃</mo> <mo>⋯</mo> <mo>⊃</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>⊃</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⊃</mo> <mo>⋯</mo> <mo>⊃</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\supset \cdots \supset X^{n}\supset X^{n-1}\supset \cdots \supset X^{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a171084199cd91bc40152db48e752b9f3545297" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:34.501ex; height:2.676ex;" alt="{\displaystyle X\supset \cdots \supset X^{n}\supset X^{n-1}\supset \cdots \supset X^{0}}"></span> whose union is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> and such that </p> <ol><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X^{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e7ed80088727b5ba6be077bad40afbd304e84de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.051ex; height:2.676ex;" alt="{\displaystyle X^{0}}"></span> is a discrete space, called the set of 0-cells (vertices) in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>.</li> <li>Each <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/268db8293666fefd75cfb00513706171948edf09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.215ex; height:2.343ex;" alt="{\displaystyle X^{n}}"></span> is obtained by attaching several <i>n</i>-disks, <i>n</i>-cells, to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{n-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X^{n-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98ccc911504085b636c3b947ad7534a657f18dee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.316ex; height:2.676ex;" alt="{\displaystyle X^{n-1}}"></span> via maps <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{n-1}\to X^{n-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">→</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{n-1}\to X^{n-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed7855b6b6419f3438d4a945794d0caf166fb7ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.771ex; height:2.676ex;" alt="{\displaystyle S^{n-1}\to X^{n-1}}"></span>; i.e., the boundary of an n-disk is identified with the image of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{n-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{n-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/742bebb03fe630674b18823a59d2c75efd0066e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.841ex; height:2.676ex;" alt="{\displaystyle S^{n-1}}"></span> in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{n-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X^{n-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98ccc911504085b636c3b947ad7534a657f18dee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.316ex; height:2.676ex;" alt="{\displaystyle X^{n-1}}"></span>.</li> <li>A subset <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></span> is open if and only if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\cap X^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>∩</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U\cap X^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60a7c5137e3642ff6d22d6cbc8d6720411cdec86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.58ex; height:2.343ex;" alt="{\displaystyle U\cap X^{n}}"></span> is open for each <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>.</li></ol> <p>For example, a sphere <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee006452a59bf1eb29983b4412348b66517a2d23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.74ex; height:2.343ex;" alt="{\displaystyle S^{n}}"></span> has two cells: one 0-cell and one <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>-cell, since <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee006452a59bf1eb29983b4412348b66517a2d23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.74ex; height:2.343ex;" alt="{\displaystyle S^{n}}"></span> can be obtained by collapsing the boundary <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{n-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{n-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/742bebb03fe630674b18823a59d2c75efd0066e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.841ex; height:2.676ex;" alt="{\displaystyle S^{n-1}}"></span> of the <i>n</i>-disk to a point. In general, every manifold has the homotopy type of a CW complex;<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> in fact, <a href="/wiki/Morse_theory" title="Morse theory">Morse theory</a> implies that a compact manifold has the homotopy type of a finite CW complex.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (September 2024)">citation needed</span></a></i>]</sup> </p><p>Remarkably, <a href="/wiki/Whitehead%27s_theorem" class="mw-redirect" title="Whitehead's theorem">Whitehead's theorem</a> says that for CW complexes, a weak homotopy equivalence and a homotopy equivalence are the same thing. </p><p>Another important result is the approximation theorem. First, the <a href="/wiki/Homotopy_category" title="Homotopy category">homotopy category</a> of spaces is the category where an object is a space but a morphism is the homotopy class of a map. Then </p> <style data-mw-deduplicate="TemplateStyles:r1110004140">.mw-parser-output .math_theorem{margin:1em 2em;padding:0.5em 1em 0.4em;border:1px solid #aaa;overflow:hidden}@media(max-width:500px){.mw-parser-output .math_theorem{margin:1em 0em;padding:0.5em 0.5em 0.4em}}</style><div class="math_theorem" style=""> <p><strong class="theorem-name"><a href="/wiki/CW_approximation" class="mw-redirect" title="CW approximation">CW approximation</a></strong><span class="theoreme-tiret"> — </span><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> There exist a functor (called the CW approximation functor) </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Theta :\operatorname {Ho} ({\textrm {spaces}})\to \operatorname {Ho} ({\textrm {CW}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Θ</mi> <mo>:</mo> <mi>Ho</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>spaces</mtext> </mrow> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>Ho</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>CW</mtext> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Theta :\operatorname {Ho} ({\textrm {spaces}})\to \operatorname {Ho} ({\textrm {CW}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e4e5293ea913ffcabab12a1bbde2bca66dea67c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.208ex; height:2.843ex;" alt="{\displaystyle \Theta :\operatorname {Ho} ({\textrm {spaces}})\to \operatorname {Ho} ({\textrm {CW}})}"></span></dd></dl> <p>from the homotopy category of spaces to the homotopy category of CW complexes as well as a natural transformation </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta :i\circ \Theta \to \operatorname {Id} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>:</mo> <mi>i</mi> <mo>∘</mo> <mi mathvariant="normal">Θ</mi> <mo stretchy="false">→</mo> <mi>Id</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta :i\circ \Theta \to \operatorname {Id} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24f2f46c7e2d87fea944783b58f2b0a5a4afe449" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.226ex; height:2.509ex;" alt="{\displaystyle \theta :i\circ \Theta \to \operatorname {Id} ,}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i:\operatorname {Ho} ({\textrm {CW}})\hookrightarrow \operatorname {Ho} ({\textrm {spaces}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>:</mo> <mi>Ho</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>CW</mtext> </mrow> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">↪</mo> <mi>Ho</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>spaces</mtext> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i:\operatorname {Ho} ({\textrm {CW}})\hookrightarrow \operatorname {Ho} ({\textrm {spaces}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95604963b664deaa4cb7850f63d39917b82dbd67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.495ex; height:2.843ex;" alt="{\displaystyle i:\operatorname {Ho} ({\textrm {CW}})\hookrightarrow \operatorname {Ho} ({\textrm {spaces}})}"></span>, such that each <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta _{X}:i(\Theta (X))\to X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>:</mo> <mi>i</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">Θ</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta _{X}:i(\Theta (X))\to X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f9cd1839d29889d07bd58e5fa7898ad4d21862c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.463ex; height:2.843ex;" alt="{\displaystyle \theta _{X}:i(\Theta (X))\to X}"></span> is a weak homotopy equivalence. </p><p>Similar statements also hold for pairs and excisive triads.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> </div> <p>Explicitly, the above approximation functor can be defined as the composition of the <a href="/wiki/Singular_chain" class="mw-redirect" title="Singular chain">singular chain</a> functor <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d3325410f0949bfb47c30694d58411ab83c7137" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.293ex; margin-bottom: -0.379ex; width:2.479ex; height:2.509ex;" alt="{\displaystyle S_{*}}"></span> followed by the geometric realization functor; see <a href="#Simplicial_set">§ Simplicial set</a>. </p><p>The above theorem justifies a common habit of working only with CW complexes. For example, given a space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>, one can just define the homology of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> to the homology of the CW approximation of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> (the cell structure of a CW complex determines the natural homology, the <a href="/wiki/Cellular_homology" title="Cellular homology">cellular homology</a> and that can be taken to be the homology of the complex.) </p> <div class="mw-heading mw-heading3"><h3 id="Cofibration_and_fibration">Cofibration and fibration</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Homotopy_theory&action=edit&section=6" title="Edit section: Cofibration and fibration"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A map <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:A\to X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>A</mi> <mo stretchy="false">→</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:A\to X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23200a6d204a3980f2ba2bb829f254094c7d7e7d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.553ex; height:2.509ex;" alt="{\displaystyle f:A\to X}"></span> is called a <a href="/wiki/Cofibration" title="Cofibration">cofibration</a> if given: </p> <ol><li>A map <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{0}:X\to Z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h_{0}:X\to Z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e835a1751594e3ce73108c0101e090e9a275c493" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.605ex; height:2.509ex;" alt="{\displaystyle h_{0}:X\to Z}"></span>, and</li> <li>A homotopy <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{t}:A\to Z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>:</mo> <mi>A</mi> <mo stretchy="false">→</mo> <mi>Z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{t}:A\to Z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba4b435ba3383dd590571704d8b7066abb868355" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.91ex; height:2.509ex;" alt="{\displaystyle g_{t}:A\to Z}"></span></li></ol> <p>such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{0}\circ f=g_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>∘</mo> <mi>f</mi> <mo>=</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h_{0}\circ f=g_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddb9202cdc1ab7ee1c855d64845bca53540dedfc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.128ex; height:2.509ex;" alt="{\displaystyle h_{0}\circ f=g_{0}}"></span>, there exists a homotopy <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{t}:X\to Z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h_{t}:X\to Z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2a3b006de07338252900e446478f6a6f4955006" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.377ex; height:2.509ex;" alt="{\displaystyle h_{t}:X\to Z}"></span> that extends <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08c1c908b03c3f63383c7199465c7fd0b105030f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.393ex; height:2.509ex;" alt="{\displaystyle h_{0}}"></span> and such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{t}\circ f=g_{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>∘</mo> <mi>f</mi> <mo>=</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h_{t}\circ f=g_{t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1beb124f689b567d787fbc2f98a38dfc5304f44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.672ex; height:2.509ex;" alt="{\displaystyle h_{t}\circ f=g_{t}}"></span>. An example is a <a href="/wiki/Neighborhood_deformation_retract" class="mw-redirect" title="Neighborhood deformation retract">neighborhood deformation retract</a>; that is, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> contains a <a href="/wiki/Mapping_cylinder" title="Mapping cylinder">mapping cylinder</a> neighborhood of a closed subspace <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> the inclusion (e.g., a <a href="/wiki/Tubular_neighborhood" title="Tubular neighborhood">tubular neighborhood</a> of a closed submanifold).<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> In fact, a cofibration can be characterized as a neighborhood deformation retract pair.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> Another basic example is a <a href="/wiki/CW_pair" class="mw-redirect" title="CW pair">CW pair</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,A)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d53eff80e8e569a9ce3e2f20adf4e9bb17feca0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.566ex; height:2.843ex;" alt="{\displaystyle (X,A)}"></span>; many often work only with CW complexes and the notion of a cofibration there is then often implicit. </p><p>A <a href="/wiki/Fibration" title="Fibration">fibration</a> in the sense of Hurewicz is the dual notion of a cofibration: that is, a map <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p:X\to B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p:X\to B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d178a180f28c19c746685eaa14fd7071ee60eece" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:10.554ex; height:2.509ex;" alt="{\displaystyle p:X\to B}"></span> is a fibration if given (1) a map <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{0}:Z\to X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>:</mo> <mi>Z</mi> <mo stretchy="false">→</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h_{0}:Z\to X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47ebad73de729f0ec29c72279e68a6e8bb2031e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.605ex; height:2.509ex;" alt="{\displaystyle h_{0}:Z\to X}"></span> and (2) a homotopy <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{t}:Z\to B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>:</mo> <mi>Z</mi> <mo stretchy="false">→</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{t}:Z\to B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb740774f751b4f47394a47a45ebfb297ca39771" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.931ex; height:2.509ex;" alt="{\displaystyle g_{t}:Z\to B}"></span> such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\circ h_{0}=g_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>∘</mo> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\circ h_{0}=g_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/977353bc68c4b8102b70459de9d30c98f9ce88ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:11.109ex; height:2.509ex;" alt="{\displaystyle p\circ h_{0}=g_{0}}"></span>, there exists a homotopy <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{t}:Z\to X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>:</mo> <mi>Z</mi> <mo stretchy="false">→</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h_{t}:Z\to X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/729bc141fec1ea676628149aac00591bde0d3c80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.377ex; height:2.509ex;" alt="{\displaystyle h_{t}:Z\to X}"></span> that extends <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08c1c908b03c3f63383c7199465c7fd0b105030f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.393ex; height:2.509ex;" alt="{\displaystyle h_{0}}"></span> and such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\circ h_{t}=g_{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>∘</mo> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\circ h_{t}=g_{t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af02e5c5bc01896a68b31a30a1e4ebbde6701cf2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:10.652ex; height:2.509ex;" alt="{\displaystyle p\circ h_{t}=g_{t}}"></span>. </p><p>While a cofibration is characterized by the existence of a retract, a fibration is characterized by the existence of a section called the <a href="/w/index.php?title=Path_lifting&action=edit&redlink=1" class="new" title="Path lifting (page does not exist)">path lifting</a> as follows. Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p':Np\to B^{I}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>p</mi> <mo>′</mo> </msup> <mo>:</mo> <mi>N</mi> <mi>p</mi> <mo stretchy="false">→</mo> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p':Np\to B^{I}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46fe86e289c43d3f3611a70f9e7f09d0bac36f98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:13.553ex; height:3.009ex;" alt="{\displaystyle p':Np\to B^{I}}"></span> be the pull-back of a map <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p:E\to B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>:</mo> <mi>E</mi> <mo stretchy="false">→</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p:E\to B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c8fc956bbe1fa571b4fdd9dafb69dcd8642a0a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:10.35ex; height:2.509ex;" alt="{\displaystyle p:E\to B}"></span> along <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi \mapsto \chi (1):B^{I}\to B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>χ</mi> <mo stretchy="false">↦</mo> <mi>χ</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>:</mo> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msup> <mo stretchy="false">→</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi \mapsto \chi (1):B^{I}\to B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4180aae06ae17fcbcb2f0615e40aa0f16aef90" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.636ex; height:3.176ex;" alt="{\displaystyle \chi \mapsto \chi (1):B^{I}\to B}"></span>, called the <a href="/wiki/Mapping_path_space" class="mw-redirect" title="Mapping path space">mapping path space</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span>.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> Viewing <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>p</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40e623e3163571a220ed60ecb31aa78c24104b85" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.944ex; height:2.843ex;" alt="{\displaystyle p'}"></span> as a homotopy <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Np\times I\to B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mi>p</mi> <mo>×</mo> <mi>I</mi> <mo stretchy="false">→</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Np\times I\to B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9479387e4fdb5345c4344f559f602f5635ef771d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.623ex; height:2.509ex;" alt="{\displaystyle Np\times I\to B}"></span> (see <a href="#Homotopy">§ Homotopy</a>), if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> is a fibration, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>p</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40e623e3163571a220ed60ecb31aa78c24104b85" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.944ex; height:2.843ex;" alt="{\displaystyle p'}"></span> gives a homotopy <sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s:Np\to E^{I}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>:</mo> <mi>N</mi> <mi>p</mi> <mo stretchy="false">→</mo> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s:Np\to E^{I}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbd4738475487043baa416cd14189c6285cd91b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.729ex; height:3.009ex;" alt="{\displaystyle s:Np\to E^{I}}"></span></dd></dl> <p>such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s(e,\chi )(0)=e,\,(p^{I}\circ s)(e,\chi )=\chi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo stretchy="false">(</mo> <mi>e</mi> <mo>,</mo> <mi>χ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>e</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msup> <mo>∘</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>e</mi> <mo>,</mo> <mi>χ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>χ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s(e,\chi )(0)=e,\,(p^{I}\circ s)(e,\chi )=\chi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1d2d0facb2cbf75cc05121cca64ab25e6642ee2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.307ex; height:3.176ex;" alt="{\displaystyle s(e,\chi )(0)=e,\,(p^{I}\circ s)(e,\chi )=\chi }"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p^{I}:E^{I}\to B^{I}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msup> <mo>:</mo> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msup> <mo stretchy="false">→</mo> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p^{I}:E^{I}\to B^{I}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db1e5edc7958251f837450d16d05f07885c87cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:13.55ex; height:3.009ex;" alt="{\displaystyle p^{I}:E^{I}\to B^{I}}"></span> is given by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span>.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> This <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span> is called the path lifting associated to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span>. Conversely, if there is a path lifting <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span>, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> is a fibration as a required homotopy is obtained via <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span>. </p><p>A basic example of a fibration is a <a href="/wiki/Covering_map" class="mw-redirect" title="Covering map">covering map</a> as it comes with a unique path lifting. If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></span> is a <a href="/wiki/Principal_bundle" title="Principal bundle">principal <i>G</i>-bundle</a> over a paracompact space, that is, a space with a <a href="/wiki/Group_action#Remarkable_properties_of_actions" title="Group action">free and transitive</a> (topological) <a href="/wiki/Group_action" title="Group action">group action</a> of a (<a href="/wiki/Topological_group" title="Topological group">topological</a>) group, then the projection map <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p:E\to X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>:</mo> <mi>E</mi> <mo stretchy="false">→</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p:E\to X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd5c4236298a8edc5123246d49d3a8c21107f0cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:10.566ex; height:2.509ex;" alt="{\displaystyle p:E\to X}"></span> is a fibration, because a Hurewicz fibration can be checked locally on a paracompact space.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p><p>While a cofibration is injective with closed image,<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> a fibration need not be surjective. </p><p>There are also based versions of a cofibration and a fibration (namely, the maps are required to be based).<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Lifting_property">Lifting property</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Homotopy_theory&action=edit&section=7" title="Edit section: Lifting property"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A pair of maps <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i:A\to X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>:</mo> <mi>A</mi> <mo stretchy="false">→</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i:A\to X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97059160d1002162c22e1f2f5e4c2aee2afaf629" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.077ex; height:2.176ex;" alt="{\displaystyle i:A\to X}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p:E\to B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>:</mo> <mi>E</mi> <mo stretchy="false">→</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p:E\to B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c8fc956bbe1fa571b4fdd9dafb69dcd8642a0a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:10.35ex; height:2.509ex;" alt="{\displaystyle p:E\to B}"></span> is said to satisfy the <a href="/wiki/Lifting_property" title="Lifting property">lifting property</a><sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> if for each commutative square diagram </p> <dl><dd><span typeof="mw:File/Frameless"><a href="/wiki/File:Lifting_property_diagram.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Lifting_property_diagram.png/150px-Lifting_property_diagram.png" decoding="async" width="150" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Lifting_property_diagram.png/225px-Lifting_property_diagram.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Lifting_property_diagram.png/300px-Lifting_property_diagram.png 2x" data-file-width="2560" data-file-height="2560" /></a></span></dd></dl> <p>there is a map <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span> that makes the above diagram still commute. (The notion originates in the theory of <a href="/wiki/Model_category" title="Model category">model categories</a>.) </p><p>Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">c</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {c}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b21924b960341255be18e538e51404718f29cbc0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.905ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {c}}}"></span> be a class of maps. Then a map <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p:E\to B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>:</mo> <mi>E</mi> <mo stretchy="false">→</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p:E\to B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c8fc956bbe1fa571b4fdd9dafb69dcd8642a0a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:10.35ex; height:2.509ex;" alt="{\displaystyle p:E\to B}"></span> is said to satisfy the <a href="/wiki/Right_lifting_property" class="mw-redirect" title="Right lifting property">right lifting property</a> or the RLP if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> satisfies the above lifting property for each <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span> in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">c</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {c}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b21924b960341255be18e538e51404718f29cbc0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.905ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {c}}}"></span>. Similarly, a map <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i:A\to X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>:</mo> <mi>A</mi> <mo stretchy="false">→</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i:A\to X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97059160d1002162c22e1f2f5e4c2aee2afaf629" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.077ex; height:2.176ex;" alt="{\displaystyle i:A\to X}"></span> is said to satisfy the <a href="/wiki/Left_lifting_property" class="mw-redirect" title="Left lifting property">left lifting property</a> or the LLP if it satisfies the lifting property for each <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">c</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {c}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b21924b960341255be18e538e51404718f29cbc0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.905ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {c}}}"></span>. </p><p>For example, a Hurewicz fibration is exactly a map <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p:E\to B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>:</mo> <mi>E</mi> <mo stretchy="false">→</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p:E\to B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c8fc956bbe1fa571b4fdd9dafb69dcd8642a0a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:10.35ex; height:2.509ex;" alt="{\displaystyle p:E\to B}"></span> that satisfies the RLP for the inclusions <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i_{0}:A\to A\times I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>:</mo> <mi>A</mi> <mo stretchy="false">→</mo> <mi>A</mi> <mo>×</mo> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i_{0}:A\to A\times I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c231234a8d2dc7a24716e86de39aba7534b7146" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.906ex; height:2.509ex;" alt="{\displaystyle i_{0}:A\to A\times I}"></span>. A <a href="/wiki/Serre_fibration" class="mw-redirect" title="Serre fibration">Serre fibration</a> is a map satisfying the RLP for the inclusions <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i:S^{n-1}\to D^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>:</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">→</mo> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i:S^{n-1}\to D^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/796199b301dc6c0315beaa3695eecb9db6a522c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.337ex; height:2.676ex;" alt="{\displaystyle i:S^{n-1}\to D^{n}}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b27b44abb494176cfeb76818591f178f034f28e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.854ex; height:2.676ex;" alt="{\displaystyle S^{-1}}"></span> is the empty set. A Hurewicz fibration is a Serre fibration and the converse holds for CW complexes.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p><p>On the other hand, a cofibration is exactly a map satisfying the LLP for evaluation maps <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p:B^{I}\to B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>:</mo> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msup> <mo stretchy="false">→</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p:B^{I}\to B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f39eab96c3f2c05a3a3ec7446bad679ab9db10a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:11.399ex; height:3.009ex;" alt="{\displaystyle p:B^{I}\to B}"></span> at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Loop_and_suspension">Loop and suspension</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Homotopy_theory&action=edit&section=8" title="Edit section: Loop and suspension"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>On the category of pointed spaces, there are two important functors: the <a href="/wiki/Loop_functor" class="mw-redirect" title="Loop functor">loop functor</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }"></span> and the (reduced) <a href="/wiki/Suspension_functor" class="mw-redirect" title="Suspension functor">suspension functor</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e1f558f53cda207614abdf90162266c70bc5c1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Sigma }"></span>, which are in the <a href="/wiki/Adjoint_functor" class="mw-redirect" title="Adjoint functor">adjoint relation</a>. Precisely, they are defined as<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega X=\operatorname {Map} (S^{1},X)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> <mi>X</mi> <mo>=</mo> <mi>Map</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>,</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega X=\operatorname {Map} (S^{1},X)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be19b146eaa63d0d4f30affcbfc4c5232afb1a65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.742ex; height:3.176ex;" alt="{\displaystyle \Omega X=\operatorname {Map} (S^{1},X)}"></span>, and</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma X=X\wedge S^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Σ</mi> <mi>X</mi> <mo>=</mo> <mi>X</mi> <mo>∧</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma X=X\wedge S^{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a4d0c76d6820007190b14819cdab1dd7f873e1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.895ex; height:2.676ex;" alt="{\displaystyle \Sigma X=X\wedge S^{1}}"></span>.</li></ul> <p>Because of the adjoint relation between a smash product and a mapping space, we have: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Map} (\Sigma X,Y)=\operatorname {Map} (X,\Omega Y).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Map</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">Σ</mi> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>Map</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi mathvariant="normal">Ω</mi> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Map} (\Sigma X,Y)=\operatorname {Map} (X,\Omega Y).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8aa20d7e37fe5ff4b62ddd519e3533955eee634" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.466ex; height:2.843ex;" alt="{\displaystyle \operatorname {Map} (\Sigma X,Y)=\operatorname {Map} (X,\Omega Y).}"></span></dd></dl> <p>These functors are used to construct <a href="/wiki/Fiber_sequence" class="mw-redirect" title="Fiber sequence">fiber sequences</a> and <a href="/wiki/Cofiber_sequence" class="mw-redirect" title="Cofiber sequence">cofiber sequences</a>. Namely, if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:X\to Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abd1e080abef4bbdab67b43819c6431e7561361c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.583ex; height:2.509ex;" alt="{\displaystyle f:X\to Y}"></span> is a map, the fiber sequence generated by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> is the exact sequence<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cdots \to \Omega ^{2}Ff\to \Omega ^{2}X\to \Omega ^{2}Y\to \Omega Ff\to \Omega X\to \Omega Y\to Ff\to X\to Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⋯</mo> <mo stretchy="false">→</mo> <msup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>F</mi> <mi>f</mi> <mo stretchy="false">→</mo> <msup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>X</mi> <mo stretchy="false">→</mo> <msup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>Y</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">Ω</mi> <mi>F</mi> <mi>f</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">Ω</mi> <mi>X</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">Ω</mi> <mi>Y</mi> <mo stretchy="false">→</mo> <mi>F</mi> <mi>f</mi> <mo stretchy="false">→</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cdots \to \Omega ^{2}Ff\to \Omega ^{2}X\to \Omega ^{2}Y\to \Omega Ff\to \Omega X\to \Omega Y\to Ff\to X\to Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d78dc87443b5998f559fbc4300a9375493b2b3ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:68.799ex; height:3.009ex;" alt="{\displaystyle \cdots \to \Omega ^{2}Ff\to \Omega ^{2}X\to \Omega ^{2}Y\to \Omega Ff\to \Omega X\to \Omega Y\to Ff\to X\to Y}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Ff}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Ff}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3b8098a4dce98b91535c1705a918f091d8e16ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.019ex; height:2.509ex;" alt="{\displaystyle Ff}"></span> is the <a href="/wiki/Homotopy_fiber" title="Homotopy fiber">homotopy fiber</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>; i.e., a fiber obtained after replacing <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> by a (based) fibration. The cofibration sequence generated by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\to Y\to Cf\to \Sigma X\to \cdots ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Y</mi> <mo stretchy="false">→</mo> <mi>C</mi> <mi>f</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">Σ</mi> <mi>X</mi> <mo stretchy="false">→</mo> <mo>⋯</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\to Y\to Cf\to \Sigma X\to \cdots ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/865ccf092466a6defe1fb8bb73a560c4310a5c1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:28.67ex; height:2.509ex;" alt="{\displaystyle X\to Y\to Cf\to \Sigma X\to \cdots ,}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Cf}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Cf}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/333df6a3322fb29fd7198e3244dd70ab570f2d37" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.045ex; height:2.509ex;" alt="{\displaystyle Cf}"></span> is the homotooy cofiber of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> constructed like a homotopy fiber (use a quotient instead of a fiber.) </p><p>The functors <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega ,\Sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mi mathvariant="normal">Σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega ,\Sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5dd1af4cb0f984e5ecd9ff1b5ba4de669f4798e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.39ex; height:2.509ex;" alt="{\displaystyle \Omega ,\Sigma }"></span> restrict to the category of CW complexes in the following weak sense: a theorem of Milnor says that if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> has the homotopy type of a CW complex, then so does its loop space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b372331db039fbac1df6fa94dbc87af06a95cfa7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.658ex; height:2.176ex;" alt="{\displaystyle \Omega X}"></span>.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Classifying_spaces_and_homotopy_operations">Classifying spaces and homotopy operations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Homotopy_theory&action=edit&section=9" title="Edit section: Classifying spaces and homotopy operations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Given a topological group <i>G</i>, the <a href="/wiki/Classifying_space" title="Classifying space">classifying space</a> for <a href="/wiki/Principal_bundle" title="Principal bundle">principal <i>G</i>-bundles</a> ("the" up to equivalence) is a space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle BG}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle BG}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/773ca20b2080cb3766062a5451a01d2220e9b067" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.591ex; height:2.176ex;" alt="{\displaystyle BG}"></span> such that, for each space <i>X</i>, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [X,BG]=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>X</mi> <mo>,</mo> <mi>B</mi> <mi>G</mi> <mo stretchy="false">]</mo> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [X,BG]=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70b556f7b12eeaf4340043c849e9e196eeaf19d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.352ex; height:2.843ex;" alt="{\displaystyle [X,BG]=}"></span> {principal <i>G</i>-bundle on <i>X</i>} / ~ <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ,\,\,[f]\mapsto [f^{*}EG]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>,</mo> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">]</mo> <mo stretchy="false">↦</mo> <mo stretchy="false">[</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>E</mi> <mi>G</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ,\,\,[f]\mapsto [f^{*}EG]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/281b96260477a392b2ccad083a9146d743556f90" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.265ex; height:2.843ex;" alt="{\displaystyle ,\,\,[f]\mapsto [f^{*}EG]}"></span></dd></dl> <p>where </p> <ul><li>the left-hand side is the set of homotopy classes of maps <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\to BG}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo stretchy="false">→</mo> <mi>B</mi> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\to BG}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ab00a9aeca38e80f4567f05918c4418858784a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.185ex; height:2.176ex;" alt="{\displaystyle X\to BG}"></span>,</li> <li>~ refers isomorphism of bundles, and</li> <li>= is given by pulling-back the distinguished bundle <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle EG}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle EG}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a7ebfb4d29c3c955d9a1cabc6f2305b94c8bbce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.602ex; height:2.176ex;" alt="{\displaystyle EG}"></span> on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle BG}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle BG}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/773ca20b2080cb3766062a5451a01d2220e9b067" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.591ex; height:2.176ex;" alt="{\displaystyle BG}"></span> (called universal bundle) along a map <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\to BG}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo stretchy="false">→</mo> <mi>B</mi> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\to BG}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ab00a9aeca38e80f4567f05918c4418858784a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.185ex; height:2.176ex;" alt="{\displaystyle X\to BG}"></span>.</li></ul> <p><a href="/wiki/Brown%27s_representability_theorem" title="Brown's representability theorem">Brown's representability theorem</a> guarantees the existence of classifying spaces. </p> <div class="mw-heading mw-heading3"><h3 id="Spectrum_and_generalized_cohomology">Spectrum and generalized cohomology</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Homotopy_theory&action=edit&section=10" title="Edit section: Spectrum and generalized cohomology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Spectrum_(algebraic_topology)" class="mw-redirect" title="Spectrum (algebraic topology)">Spectrum (algebraic topology)</a> and <a href="/wiki/Generalized_cohomology" class="mw-redirect" title="Generalized cohomology">Generalized cohomology</a></div> <p>The idea that a classifying space classifies principal bundles can be pushed further. For example, one might try to classify cohomology classes: given an <a href="/wiki/Abelian_group" title="Abelian group">abelian group</a> <i>A</i> (such as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></span>), </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [X,K(A,n)]=\operatorname {H} ^{n}(X;A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>X</mi> <mo>,</mo> <mi>K</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>=</mo> <msup> <mi mathvariant="normal">H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>;</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [X,K(A,n)]=\operatorname {H} ^{n}(X;A)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4751d41be66b271a292e8bf33341bd2829febc6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.981ex; height:2.843ex;" alt="{\displaystyle [X,K(A,n)]=\operatorname {H} ^{n}(X;A)}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(A,n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K(A,n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aa5cc0602bbce5d12b4b2b6e1444efe75ae51e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.047ex; height:2.843ex;" alt="{\displaystyle K(A,n)}"></span> is the <a href="/wiki/Eilenberg%E2%80%93MacLane_space" title="Eilenberg–MacLane space">Eilenberg–MacLane space</a>. The above equation leads to the notion of a generalized cohomology theory; i.e., a <a href="/wiki/Contravariant_functor" class="mw-redirect" title="Contravariant functor">contravariant functor</a> from the category of spaces to the <a href="/wiki/Category_of_abelian_groups" title="Category of abelian groups">category of abelian groups</a> that satisfies the axioms generalizing ordinary cohomology theory. As it turns out, such a functor may not be <a href="/wiki/Representable_functor" title="Representable functor">representable</a> by a space but it can always be represented by a sequence of (pointed) spaces with structure maps called a spectrum. In other words, to give a generalized cohomology theory is to give a spectrum. A <a href="/wiki/K-theory" title="K-theory">K-theory</a> is an example of a generalized cohomology theory. </p><p>A basic example of a spectrum is a <a href="/wiki/Sphere_spectrum" title="Sphere spectrum">sphere spectrum</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{0}\to S^{1}\to S^{2}\to \cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo stretchy="false">→</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo stretchy="false">→</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">→</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{0}\to S^{1}\to S^{2}\to \cdots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/674deaec2e7c73fd6545abf9d04ec2d8222eab05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:21.293ex; height:2.676ex;" alt="{\displaystyle S^{0}\to S^{1}\to S^{2}\to \cdots }"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Ring_spectrum_and_module_spectrum">Ring spectrum and module spectrum</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Homotopy_theory&action=edit&section=11" title="Edit section: Ring spectrum and module spectrum"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-Expand_section plainlinks metadata ambox mbox-small-left ambox-content" role="presentation"><tbody><tr><td class="mbox-image"><span typeof="mw:File"><a href="/wiki/File:Wiki_letter_w_cropped.svg" class="mw-file-description"><img alt="[icon]" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/20px-Wiki_letter_w_cropped.svg.png" decoding="async" width="20" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/30px-Wiki_letter_w_cropped.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/40px-Wiki_letter_w_cropped.svg.png 2x" data-file-width="44" data-file-height="31" /></a></span></td><td class="mbox-text"><div class="mbox-text-span">This section <b>needs expansion</b>. You can help by <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Homotopy_theory&action=edit&section=">adding to it</a>. <span class="date-container"><i>(<span class="date">September 2024</span>)</i></span></div></td></tr></tbody></table> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Ring_spectrum" title="Ring spectrum">Ring spectrum</a> and <a href="/wiki/Module_spectrum" title="Module spectrum">Module spectrum</a></div> <div class="mw-heading mw-heading2"><h2 id="Key_theorems">Key theorems</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Homotopy_theory&action=edit&section=12" title="Edit section: Key theorems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Seifert%E2%80%93van_Kampen_theorem" class="mw-redirect" title="Seifert–van Kampen theorem">Seifert–van Kampen theorem</a></li> <li><a href="/wiki/Homotopy_excision_theorem" title="Homotopy excision theorem">Homotopy excision theorem</a></li> <li><a href="/wiki/Freudenthal_suspension_theorem" title="Freudenthal suspension theorem">Freudenthal suspension theorem</a> (a corollary of the excision theorem)</li> <li><a href="/wiki/Landweber_exact_functor_theorem" title="Landweber exact functor theorem">Landweber exact functor theorem</a></li> <li><a href="/wiki/Dold%E2%80%93Kan_correspondence" title="Dold–Kan correspondence">Dold–Kan correspondence</a></li> <li><a href="/wiki/Eckmann%E2%80%93Hilton_argument" title="Eckmann–Hilton argument">Eckmann–Hilton argument</a> - this shows for instance higher homotopy groups are <a href="/wiki/Abelian_group" title="Abelian group">abelian</a>.</li> <li><a href="/wiki/Universal_coefficient_theorem" title="Universal coefficient theorem">Universal coefficient theorem</a></li> <li><a href="/wiki/Dold%E2%80%93Thom_theorem" title="Dold–Thom theorem">Dold–Thom theorem</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Obstruction_theory_and_characteristic_class">Obstruction theory and characteristic class</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Homotopy_theory&action=edit&section=13" title="Edit section: Obstruction theory and characteristic class"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-Expand_section plainlinks metadata ambox mbox-small-left ambox-content" role="presentation"><tbody><tr><td class="mbox-image"><span typeof="mw:File"><a href="/wiki/File:Wiki_letter_w_cropped.svg" class="mw-file-description"><img alt="[icon]" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/20px-Wiki_letter_w_cropped.svg.png" decoding="async" width="20" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/30px-Wiki_letter_w_cropped.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/40px-Wiki_letter_w_cropped.svg.png 2x" data-file-width="44" data-file-height="31" /></a></span></td><td class="mbox-text"><div class="mbox-text-span">This section <b>needs expansion</b>. You can help by <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Homotopy_theory&action=edit&section=">adding to it</a>. <span class="date-container"><i>(<span class="date">May 2020</span>)</i></span></div></td></tr></tbody></table> <p>See also: <a href="/wiki/Characteristic_class" title="Characteristic class">Characteristic class</a>, <a href="/wiki/Postnikov_tower" class="mw-redirect" title="Postnikov tower">Postnikov tower</a>, <a href="/wiki/Whitehead_torsion" title="Whitehead torsion">Whitehead torsion</a> </p> <div class="mw-heading mw-heading2"><h2 id="Localization_and_completion_of_a_space">Localization and completion of a space</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Homotopy_theory&action=edit&section=14" title="Edit section: Localization and completion of a space"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-Expand_section plainlinks metadata ambox mbox-small-left ambox-content" role="presentation"><tbody><tr><td class="mbox-image"><span typeof="mw:File"><a href="/wiki/File:Wiki_letter_w_cropped.svg" class="mw-file-description"><img alt="[icon]" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/20px-Wiki_letter_w_cropped.svg.png" decoding="async" width="20" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/30px-Wiki_letter_w_cropped.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/40px-Wiki_letter_w_cropped.svg.png 2x" data-file-width="44" data-file-height="31" /></a></span></td><td class="mbox-text"><div class="mbox-text-span">This section <b>needs expansion</b>. You can help by <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Homotopy_theory&action=edit&section=">adding to it</a>. <span class="date-container"><i>(<span class="date">May 2020</span>)</i></span></div></td></tr></tbody></table> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Localization_of_a_topological_space" title="Localization of a topological space">Localization of a topological space</a></div> <div class="mw-heading mw-heading2"><h2 id="Specific_theories">Specific theories</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Homotopy_theory&action=edit&section=15" title="Edit section: Specific theories"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There are several specific theories </p> <ul><li><a href="/wiki/Simple_homotopy_theory" title="Simple homotopy theory">simple homotopy theory</a></li> <li><a href="/wiki/Stable_homotopy_theory" title="Stable homotopy theory">stable homotopy theory</a></li> <li><a href="/wiki/Chromatic_homotopy_theory" title="Chromatic homotopy theory">chromatic homotopy theory</a></li> <li><a href="/wiki/Rational_homotopy_theory" title="Rational homotopy theory">rational homotopy theory</a></li> <li><a href="/w/index.php?title=P-adic_homotopy_theory&action=edit&redlink=1" class="new" title="P-adic homotopy theory (page does not exist)">p-adic homotopy theory</a></li> <li><a href="/w/index.php?title=Equivariant_homotopy_theory&action=edit&redlink=1" class="new" title="Equivariant homotopy theory (page does not exist)">equivariant homotopy theory</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Homotopy_hypothesis">Homotopy hypothesis</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Homotopy_theory&action=edit&section=16" title="Edit section: Homotopy hypothesis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Homotopy_hypothesis" title="Homotopy hypothesis">Homotopy hypothesis</a></div> <p>One of the basic questions in the foundations of homotopy theory is the nature of a space. The <a href="/wiki/Homotopy_hypothesis" title="Homotopy hypothesis">homotopy hypothesis</a> asks whether a space is something fundamentally algebraic. </p><p>If one prefers to work with a space instead of a pointed space, there is the notion of a <a href="/wiki/Fundamental_groupoid" title="Fundamental groupoid">fundamental groupoid</a> (and higher variants): by definition, the fundamental groupoid of a space <i>X</i> is the <a href="/wiki/Category_(mathematics)" title="Category (mathematics)">category</a> where the <a href="/wiki/Object_(category_theory)" class="mw-redirect" title="Object (category theory)">objects</a> are the points of <i>X</i> and the <a href="/wiki/Morphism" title="Morphism">morphisms</a> are paths. </p> <div class="mw-heading mw-heading2"><h2 id="Abstract_homotopy_theory">Abstract homotopy theory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Homotopy_theory&action=edit&section=17" title="Edit section: Abstract homotopy theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Abstract homotopy theory is an axiomatic approach to homotopy theory. Such axiomatization is useful for non-traditional applications of homotopy theory. One approach to axiomatization is by Quillen's <a href="/wiki/Model_category" title="Model category">model categories</a>. A model category is a category with a choice of three classes of maps called weak equivalences, cofibrations and fibrations, subject to the axioms that are reminiscent of facts in algebraic topology. For example, the category of (reasonable) topological spaces has a structure of a model category where a weak equivalence is a weak homotopy equivalence, a cofibration a certain retract and a fibration a Serre fibration.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> Another example is the category of non-negatively graded chain complexes over a fixed base ring.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Algebraic_homotopy" title="Algebraic homotopy">Algebraic homotopy</a></div> <div class="mw-heading mw-heading3"><h3 id="Simplicial_set">Simplicial set</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Homotopy_theory&action=edit&section=18" title="Edit section: Simplicial set"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Simplicial_set" title="Simplicial set">Simplicial set</a> and <a href="/wiki/Simplicial_homotopy_theory" class="mw-redirect" title="Simplicial homotopy theory">simplicial homotopy theory</a></div> <p>A <a href="/wiki/Simplicial_set" title="Simplicial set">simplicial set</a> is an abstract generalization of a <a href="/wiki/Simplicial_complex" title="Simplicial complex">simplicial complex</a> and can play a role of a "space" in some sense. Despite the name, it is not a set but is a sequence of sets together with the certain maps (face and degeneracy) between those sets. </p><p>For example, given a space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>, for each integer <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce8a1b7b3bc3c790054d93629fc3b08cd1da1fd0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 0}"></span>, let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{n}X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{n}X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52fba2d5367b62ae540877131ffde8925f0f5532" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.623ex; height:2.509ex;" alt="{\displaystyle S_{n}X}"></span> be the set of all maps from the <i>n</i>-simplex to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>. Then the sequence <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{n}X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{n}X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52fba2d5367b62ae540877131ffde8925f0f5532" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.623ex; height:2.509ex;" alt="{\displaystyle S_{n}X}"></span> of sets is a simplicial set.<sup id="cite_ref-May_simplicial_22-0" class="reference"><a href="#cite_note-May_simplicial-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> Each simplicial set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K=\{K_{n}\}_{n\geq 0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>≥</mo> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K=\{K_{n}\}_{n\geq 0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b86aba796b67931dbedb9089b20934097a8d9eca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14ex; height:2.843ex;" alt="{\displaystyle K=\{K_{n}\}_{n\geq 0}}"></span> has a naturally associated chain complex and the homology of that chain complex is the homology of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span>. The <a href="/wiki/Singular_homology" title="Singular homology">singular homology</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> is precisely the homology of the simplicial set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{*}X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msub> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{*}X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6184c0414df27e52aaf2c0412771b6d6ea60fd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.293ex; margin-bottom: -0.379ex; width:4.459ex; height:2.509ex;" alt="{\displaystyle S_{*}X}"></span>. Also, the <a href="/wiki/Simplicial_set#Geometric_realization" title="Simplicial set">geometric realization</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\cdot |}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\cdot |}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4570d0a1c9fb8f2f413f0b73ce846dd1eb1dca3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.973ex; height:2.843ex;" alt="{\displaystyle |\cdot |}"></span> of a simplicial set is a CW complex and the composition <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\mapsto |S_{*}X|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo stretchy="false">↦</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\mapsto |S_{*}X|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6c0e7be2b4c1e2722d4d30ca14e4f84d6be3e3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.347ex; height:2.843ex;" alt="{\displaystyle X\mapsto |S_{*}X|}"></span> is precisely the CW approximation functor. </p><p>Another important example is a category or more precisely the <a href="/wiki/Nerve_of_a_category" class="mw-redirect" title="Nerve of a category">nerve of a category</a>, which is a simplicial set. In fact, a simplicial set is the nerve of some category if and only if it satisfies the <a href="/wiki/Segal_condition" class="mw-redirect" title="Segal condition">Segal conditions</a> (a theorem of Grothendieck). Each category is completely determined by its nerve. In this way, a category can be viewed as a special kind of a simplicial set, and this observation is used to generalize a category. Namely, an <a href="/wiki/Infinity_category" class="mw-redirect" title="Infinity category"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c26c105004f30c27aa7c2a9c601550a4183b1f21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.676ex;" alt="{\displaystyle \infty }"></span>-category</a> or an <a href="/wiki/Infinity_groupoid" class="mw-redirect" title="Infinity groupoid"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c26c105004f30c27aa7c2a9c601550a4183b1f21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.676ex;" alt="{\displaystyle \infty }"></span>-groupoid</a> is defined as particular kinds of simplicial sets. </p><p>Since simplicial sets are sort of abstract spaces (if not topological spaces), it is possible to develop the homotopy theory on them, which is called the <a href="/wiki/Simplicial_homotopy_theory" class="mw-redirect" title="Simplicial homotopy theory">simplicial homotopy theory</a>.<sup id="cite_ref-May_simplicial_22-1" class="reference"><a href="#cite_note-May_simplicial-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Homotopy_theory&action=edit&section=19" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Highly_structured_ring_spectrum" title="Highly structured ring spectrum">Highly structured ring spectrum</a></li> <li><a href="/wiki/Homotopy_type_theory" title="Homotopy type theory">Homotopy type theory</a></li> <li><a href="/wiki/Pursuing_Stacks" title="Pursuing Stacks">Pursuing Stacks</a></li> <li><a href="/wiki/Shape_theory_(mathematics)" title="Shape theory (mathematics)">Shape theory</a></li> <li><a href="/wiki/Moduli_stack_of_formal_group_laws" title="Moduli stack of formal group laws">Moduli stack of formal group laws</a></li> <li><a href="/wiki/Crossed_module" title="Crossed module">Crossed module</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Homotopy_theory&action=edit&section=20" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><a href="#CITEREFMay">May</a>, Ch. 8. § 3.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><a href="#CITEREFMay">May</a>, Ch 4. § 5.</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><a href="#CITEREFMilnor1959">Milnor 1959</a>, Corollary 1. NB: "second countable" implies "separable".</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><a href="#CITEREFMay">May</a>, Ch. 10., § 5</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><a href="#CITEREFMay">May</a>, Ch. 10., § 6</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><a href="#CITEREFMay">May</a>, Ch. 10., § 7</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><a href="#CITEREFHatcher">Hatcher</a>, Example 0.15.</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><a href="#CITEREFMay">May</a>, Ch 6. § 4.</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text">Some authors use <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi \mapsto \chi (0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>χ</mi> <mo stretchy="false">↦</mo> <mi>χ</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi \mapsto \chi (0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/175d4ac82c0ccf05ffd0030a792a1677006670de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.496ex; height:2.843ex;" alt="{\displaystyle \chi \mapsto \chi (0)}"></span>. The definition here is from <a href="#CITEREFMay">May</a>, Ch. 8., § 5.</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><a href="#CITEREFMay">May</a>, Ch. 7., § 2.</span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> in the reference should be <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p^{I}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p^{I}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95c2f3bd90ec450c694e7221db900baebfc3c03c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.32ex; height:3.009ex;" alt="{\displaystyle p^{I}}"></span>.</span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><a href="#CITEREFMay">May</a>, Ch. 7., § 4.</span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><a href="#CITEREFMay">May</a>, Ch. 6., Problem (1)</span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><a href="#CITEREFMay">May</a>, Ch 8. § 3. and § 5.</span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><a href="#CITEREFMayPonto">May & Ponto</a>, Definition 14.1.5.</span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://ncatlab.org/nlab/show/a+Serre+fibration+between+CW-complexes+is+a+Hurewicz+fibration">"A Serre fibration between CW-complexes is a Hurewicz fibration in nLab"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=A+Serre+fibration+between+CW-complexes+is+a+Hurewicz+fibration+in+nLab&rft_id=https%3A%2F%2Fncatlab.org%2Fnlab%2Fshow%2Fa%2BSerre%2Bfibration%2Bbetween%2BCW-complexes%2Bis%2Ba%2BHurewicz%2Bfibration&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHomotopy+theory" class="Z3988"></span></span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><a href="#CITEREFMay">May</a>, Ch. 8, § 2.</span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><a href="#CITEREFMay">May</a>, Ch. 8, § 6.</span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><a href="#CITEREFMilnor1959">Milnor 1959</a>, Theorem 3.</span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><a href="#CITEREFDwyerSpalinski">Dwyer & Spalinski</a>, Example 3.5.</span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"><a href="#CITEREFDwyerSpalinski">Dwyer & Spalinski</a>, Example 3.7.</span> </li> <li id="cite_note-May_simplicial-22"><span class="mw-cite-backlink">^ <a href="#cite_ref-May_simplicial_22-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-May_simplicial_22-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFMay">May</a>, Ch. 16, § 4.</span> </li> </ol></div></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBottTu1995" class="citation book cs1">Bott, Raoul; Tu, Loring W. (1995). <i>Differential Forms in Algebraic Topology</i>. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-038790613-3" title="Special:BookSources/978-038790613-3"><bdi>978-038790613-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Differential+Forms+in+Algebraic+Topology&rft.pub=Springer&rft.date=1995&rft.isbn=978-038790613-3&rft.aulast=Bott&rft.aufirst=Raoul&rft.au=Tu%2C+Loring+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHomotopy+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMay" class="citation web cs1">May, J. Peter. <a rel="nofollow" class="external text" href="http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf">"A Concise Course in Algebraic Topology"</a> <span class="cs1-format">(PDF)</span>. <i><a href="/wiki/University_of_Chicago" title="University of Chicago">University of Chicago</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=University+of+Chicago&rft.atitle=A+Concise+Course+in+Algebraic+Topology&rft.aulast=May&rft.aufirst=J.+Peter&rft_id=http%3A%2F%2Fwww.math.uchicago.edu%2F~may%2FCONCISE%2FConciseRevised.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHomotopy+theory" class="Z3988"></span></li></ul> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMayPonto" class="citation book cs1">May, J. Peter; Ponto, Kate. <a rel="nofollow" class="external text" href="https://www.maths.ed.ac.uk/~v1ranick/papers/mayponto.pdf"><i>More Concise Algebraic Topology: Localization, completion, and model categories</i></a> <span class="cs1-format">(PDF)</span>. <a href="/wiki/University_of_Chicago_Press" title="University of Chicago Press">University of Chicago Press</a>. p. 215. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-022651178-8" title="Special:BookSources/978-022651178-8"><bdi>978-022651178-8</bdi></a> – via <a href="/wiki/University_of_Edinburgh" title="University of Edinburgh">University of Edinburgh</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=More+Concise+Algebraic+Topology%3A+Localization%2C+completion%2C+and+model+categories&rft.pages=215&rft.pub=University+of+Chicago+Press&rft.isbn=978-022651178-8&rft.aulast=May&rft.aufirst=J.+Peter&rft.au=Ponto%2C+Kate&rft_id=https%3A%2F%2Fwww.maths.ed.ac.uk%2F~v1ranick%2Fpapers%2Fmayponto.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHomotopy+theory" class="Z3988"></span></li></ul> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWhitehead1978" class="citation book cs1"><a href="/wiki/George_W._Whitehead" title="George W. Whitehead">Whitehead, George William</a> (1978). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=wlrvAAAAMAAJ"><i>Elements of homotopy theory</i></a>. Graduate Texts in Mathematics. Vol. 61 (3rd ed.). New York-Berlin: Springer-Verlag. pp. xxi+744. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-90336-1" title="Special:BookSources/978-0-387-90336-1"><bdi>978-0-387-90336-1</bdi></a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0516508">0516508</a><span class="reference-accessdate">. Retrieved <span class="nowrap">September 6,</span> 2011</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elements+of+homotopy+theory&rft.place=New+York-Berlin&rft.series=Graduate+Texts+in+Mathematics&rft.pages=xxi%2B744&rft.edition=3rd&rft.pub=Springer-Verlag&rft.date=1978&rft.isbn=978-0-387-90336-1&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0516508%23id-name%3DMR&rft.aulast=Whitehead&rft.aufirst=George+William&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DwlrvAAAAMAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHomotopy+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBrown2006" class="citation web cs1">Brown, Ronald (2006). <a rel="nofollow" class="external text" href="http://arquivo.pt/wayback/20160514115224/http://www.bangor.ac.uk/r.brown/topgpds.html">"Topology and groupoids"</a>. Booksurge LLC. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1-4196-2722-8" title="Special:BookSources/1-4196-2722-8"><bdi>1-4196-2722-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Topology+and+groupoids&rft.pub=Booksurge+LLC&rft.date=2006&rft.isbn=1-4196-2722-8&rft.aulast=Brown&rft.aufirst=Ronald&rft_id=http%3A%2F%2Farquivo.pt%2Fwayback%2F20160514115224%2Fhttp%3A%2F%2Fwww.bangor.ac.uk%2Fr.brown%2Ftopgpds.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHomotopy+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://ncatlab.org/nlab/show/homotopical+algebra">"Homotopical algebra"</a>. <i>nLab</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=nLab&rft.atitle=Homotopical+algebra&rft_id=https%3A%2F%2Fncatlab.org%2Fnlab%2Fshow%2Fhomotopical%2Balgebra&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHomotopy+theory" class="Z3988"></span></li></ul> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDwyerSpalinski" class="citation book cs1">Dwyer, W.G.; Spalinski, J. "Homotopy Theories and Model Categories". In James, I.M. (ed.). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=xoM5DxQZihQC&printsec=copyright#v=onepage&q&f=false"><i>Handbook of Algebraic Topology</i></a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-444-81779-4" title="Special:BookSources/0-444-81779-4"><bdi>0-444-81779-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Homotopy+Theories+and+Model+Categories&rft.btitle=Handbook+of+Algebraic+Topology&rft.isbn=0-444-81779-4&rft.aulast=Dwyer&rft.aufirst=W.G.&rft.au=Spalinski%2C+J.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DxoM5DxQZihQC%26printsec%3Dcopyright%23v%3Donepage%26q%26f%3Dfalse&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHomotopy+theory" class="Z3988"></span></li></ul> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHatcher" class="citation web cs1">Hatcher, Allen. <a rel="nofollow" class="external text" href="http://pi.math.cornell.edu/~hatcher/AT/ATpage.html">"Algebraic topology"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Algebraic+topology&rft.aulast=Hatcher&rft.aufirst=Allen&rft_id=http%3A%2F%2Fpi.math.cornell.edu%2F~hatcher%2FAT%2FATpage.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHomotopy+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMilnor1959" class="citation journal cs1">Milnor, John (1959). "On spaces having the homotopy type of 𝐶𝑊-complex". <i><a href="/wiki/Transactions_of_the_American_Mathematical_Society" title="Transactions of the American Mathematical Society">Transactions of the American Mathematical Society</a></i>. <b>90</b> (2): 272–280. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0002-9947-1959-0100267-4">10.1090/S0002-9947-1959-0100267-4</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0002-9947">0002-9947</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:123048606">123048606</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Transactions+of+the+American+Mathematical+Society&rft.atitle=On+spaces+having+the+homotopy+type+of+%F0%9D%90%B6%F0%9D%91%8A-complex&rft.volume=90&rft.issue=2&rft.pages=272-280&rft.date=1959&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A123048606%23id-name%3DS2CID&rft.issn=0002-9947&rft_id=info%3Adoi%2F10.1090%2FS0002-9947-1959-0100267-4&rft.aulast=Milnor&rft.aufirst=John&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHomotopy+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSpanier" class="citation book cs1">Spanier, Edwin. <i>Algebraic topology</i>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-94426-5" title="Special:BookSources/978-0-387-94426-5"><bdi>978-0-387-94426-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebraic+topology&rft.isbn=978-0-387-94426-5&rft.aulast=Spanier&rft.aufirst=Edwin&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHomotopy+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSullivan1974" class="citation journal cs1">Sullivan, Dennis (July 1974). <a rel="nofollow" class="external text" href="https://math.univ-cotedazur.fr/~cazanave/Gdt/ImJ/Sullivan.pdf">"Genetics of homotopy theory and the Adams conjecture"</a> <span class="cs1-format">(PDF)</span>. <i><a href="/wiki/Annals_of_Mathematics" title="Annals of Mathematics">Annals of Mathematics</a></i>. 2. <b>100</b> (1): 1–79 – via Math - <a href="/wiki/C%C3%B4te_d%27Azur_University" title="Côte d'Azur University">Côte d'Azur University</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+Mathematics&rft.atitle=Genetics+of+homotopy+theory+and+the+Adams+conjecture&rft.volume=100&rft.issue=1&rft.pages=1-79&rft.date=1974-07&rft.aulast=Sullivan&rft.aufirst=Dennis&rft_id=https%3A%2F%2Fmath.univ-cotedazur.fr%2F~cazanave%2FGdt%2FImJ%2FSullivan.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHomotopy+theory" class="Z3988"></span></li></ul> <p><br /> </p> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Homotopy_theory&action=edit&section=21" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCisinski2015" class="citation web cs1">Cisinski, Denis-Charles (March 2015). <a rel="nofollow" class="external text" href="http://www.math.univ-toulouse.fr/~dcisinsk/1097.pdf">"Higher Categories And Topos Theory(in french)"</a> <span class="cs1-format">(PDF)</span>. <i>Math - <a href="/wiki/University_of_Toulouse" title="University of Toulouse">University of Toulouse</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Math+-+University+of+Toulouse&rft.atitle=Higher+Categories+And+Topos+Theory%28in+french%29&rft.date=2015-03&rft.aulast=Cisinski&rft.aufirst=Denis-Charles&rft_id=http%3A%2F%2Fwww.math.univ-toulouse.fr%2F~dcisinsk%2F1097.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHomotopy+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPorter2010" class="citation web cs1">Porter, Timothy (February 12, 2010). <a rel="nofollow" class="external text" href="http://ncatlab.org/nlab/files/Abstract-Homotopy.pdf">"Abstract Homotopy Theory: The Interaction Of Category Theory And Homotopy Theory: A Revised Version Of The 2001 Article"</a> <span class="cs1-format">(PDF)</span>. <i>nLab</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=nLab&rft.atitle=Abstract+Homotopy+Theory%3A+The+Interaction+Of+Category+Theory+And+Homotopy+Theory%3A+A+Revised+Version+Of+The+2001+Article&rft.date=2010-02-12&rft.aulast=Porter&rft.aufirst=Timothy&rft_id=http%3A%2F%2Fncatlab.org%2Fnlab%2Ffiles%2FAbstract-Homotopy.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHomotopy+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://uregina.ca/~franklam/Math527/Math527.html">"Math 527 - Homotopy Theory Spring 2013, Section F1"</a>. <a href="/wiki/University_of_Illinois_Urbana-Champaign" title="University of Illinois Urbana-Champaign">University of Illinois Urbana-Champaign</a> – via <a href="/wiki/University_of_Regina" title="University of Regina">University of Regina</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Math+527+-+Homotopy+Theory+Spring+2013%2C+Section+F1&rft.pub=University+of+Illinois+Urbana-Champaign&rft_id=https%3A%2F%2Furegina.ca%2F~franklam%2FMath527%2FMath527.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHomotopy+theory" class="Z3988"></span>, lectures by Martin Frankland</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFQuillen1967" class="citation book cs1">Quillen, D. (1967). <i>Homotopical algebra</i>. Lectures Notes in Math. Vol. 43. Springer Verlag. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-03914-3" title="Special:BookSources/978-3-540-03914-3"><bdi>978-3-540-03914-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Homotopical+algebra&rft.series=Lectures+Notes+in+Math&rft.pub=Springer+Verlag&rft.date=1967&rft.isbn=978-3-540-03914-3&rft.aulast=Quillen&rft.aufirst=D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHomotopy+theory" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Homotopy_theory&action=edit&section=22" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://ncatlab.org/nlab/show/homotopy+theory">"Homotopy theory"</a>. <i>ncatlab.org</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=ncatlab.org&rft.atitle=Homotopy+theory&rft_id=https%3A%2F%2Fncatlab.org%2Fnlab%2Fshow%2Fhomotopy%2Btheory&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHomotopy+theory" class="Z3988"></span></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl 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