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href="#In_topology"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>In topology</span> </div> </a> <ul id="toc-In_topology-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_chain_complexes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#In_chain_complexes"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>In chain complexes</span> </div> </a> <ul id="toc-In_chain_complexes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Simplicial_sets" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Simplicial_sets"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Simplicial sets</span> </div> </a> <ul id="toc-Simplicial_sets-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Properties" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Properties"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Properties</span> </div> </a> <ul id="toc-Properties-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Constructions_with_cofibrations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Constructions_with_cofibrations"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Constructions with cofibrations</span> </div> </a> <button aria-controls="toc-Constructions_with_cofibrations-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 Constructions with cofibrations subsection</span> </button> <ul id="toc-Constructions_with_cofibrations-sublist" class="vector-toc-list"> <li id="toc-Cofibrant_replacement" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Cofibrant_replacement"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Cofibrant replacement</span> </div> </a> <ul id="toc-Cofibrant_replacement-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Cofiber" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Cofiber"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Cofiber</span> </div> </a> <ul id="toc-Cofiber-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div 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class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, in particular <a href="/wiki/Homotopy_theory" title="Homotopy theory">homotopy theory</a>, a <a href="/wiki/Continuous_mapping" class="mw-redirect" title="Continuous mapping">continuous mapping</a> between <a href="/wiki/Topological_space" title="Topological space">topological spaces</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 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>,</dd></dl> <p>is a <i>cofibration</i> if it has the <a href="/wiki/Homotopy_extension_property" title="Homotopy extension property">homotopy extension property</a> with respect to all topological 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 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/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span>. 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 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> is a cofibration if for each topological 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 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/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span>, and for any continuous 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 f,f':A\to S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>,</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo>:</mo> <mi>A</mi> <mo stretchy="false">→</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f,f':A\to S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df86b692f836cc8395e6111aaad8b7207187bd04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.111ex; height:2.843ex;" alt="{\displaystyle f,f':A\to S}"></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:X\to S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g:X\to S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e47ede09815b7c6ae5642d86f9753723466d3e54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.146ex; height:2.509ex;" alt="{\displaystyle g:X\to S}"></span> with <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 i=f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>∘</mo> <mi>i</mi> <mo>=</mo> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g\circ i=f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fee40bffbaf01a4f5cd735807d4db383c1116800" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.49ex; height:2.509ex;" alt="{\displaystyle g\circ i=f}"></span>, for any <a href="/wiki/Homotopy" title="Homotopy">homotopy</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 h:A\times I\to S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>:</mo> <mi>A</mi> <mo>×</mo> <mi>I</mi> <mo stretchy="false">→</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h:A\times I\to S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c03659be75821143454e40c3e36b1d49c4e4f530" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.145ex; height:2.176ex;" alt="{\displaystyle h:A\times I\to S}"></span> from <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> 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 f'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mo>′</mo> </msup> </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/258eaada38956fb69b8cb1a2eef46bcb97d3126b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.005ex; height:2.843ex;" alt="{\displaystyle f'}"></span>, there is a continuous 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':X\to S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>g</mi> <mo>′</mo> </msup> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g':X\to S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c152fb20ed4b1adb1cdb581eeb84f2e95dc722bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.833ex; height:2.843ex;" alt="{\displaystyle g':X\to S}"></span> and 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 S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>h</mi> <mo>′</mo> </msup> <mo>:</mo> <mi>X</mi> <mo>×</mo> <mi>I</mi> <mo stretchy="false">→</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h':X\times I\to S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b933ab200582a45922001e7e18dc8eaedf675099" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.066ex; height:2.509ex;" alt="{\displaystyle h':X\times I\to S}"></span> from <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> 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 g'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>g</mi> <mo>′</mo> </msup> </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/e7a53c0df5d85b36e3fd327c74db998f679f4f55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.803ex; height:2.843ex;" 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 h'(i(a),t)=h(a,t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>h</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>i</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>h</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h'(i(a),t)=h(a,t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71c72ead50d23a61a40edade6bedda730070406d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.898ex; height:3.009ex;" alt="{\displaystyle h'(i(a),t)=h(a,t)}"></span> for all <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\in A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∈</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\in A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a97387981adb5d65f74518e20b6785a284d7abd5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.814ex; height:2.176ex;" alt="{\displaystyle a\in 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 t\in I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>∈</mo> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t\in I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5393712b57ee03dca4934d0a967374c0b4f1bf0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.852ex; height:2.176ex;" alt="{\displaystyle t\in I}"></span>. (Here, <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> denotes 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>.) </p><p>This definition is formally dual to that of a <a href="/wiki/Fibration" title="Fibration">fibration</a>, which is required to satisfy the <a href="/wiki/Homotopy_lifting_property" title="Homotopy lifting property">homotopy lifting property</a> with respect to all spaces; this is one instance of the broader <a href="/wiki/Eckmann%E2%80%93Hilton_duality" title="Eckmann–Hilton duality">Eckmann–Hilton duality</a> in topology. </p><p>Cofibrations are a fundamental concept of homotopy theory. Quillen has proposed the notion of <a href="/wiki/Model_category" title="Model category">model category</a> as a formal framework for doing homotopy theory in more general categories; a model category is endowed with three distinguished classes of morphisms called <i>fibrations</i>, <i>cofibrations</i> and <i>weak equivalences</i> satisfying certain lifting and factorization axioms. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cofibration&action=edit&section=1" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Homotopy_theory">Homotopy theory</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cofibration&action=edit&section=2" title="Edit section: Homotopy theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In what 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 I=[0,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo>=</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I=[0,1]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87ec65159c44769434523e46928bc1b82681f842" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.923ex; height:2.843ex;" alt="{\displaystyle I=[0,1]}"></span> denote the unit interval. </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 i\colon 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\colon A\to X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6a64c55f7fe66fe8e1eb4b91766d3de6a11809f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.174ex; height:2.176ex;" alt="{\displaystyle i\colon A\to X}"></span> of topological spaces is called a <b>cofibration</b><sup id="cite_ref-:0_1-0" class="reference"><a href="#cite_note-:0-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup>pg 51</sup> if for any 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 S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>A</mi> <mo stretchy="false">→</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:A\to S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50df612a8f8b9f7f964783f2e0fcaa4c8d3eb4db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.072ex; height:2.509ex;" alt="{\displaystyle f:A\to S}"></span> such that there is an extension 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> (meaning: 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 f':X\to S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f':X\to S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5ffd390b314cc0a242be5a5a521a3203ffdafca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.036ex; height:2.843ex;" alt="{\displaystyle f':X\to S}"></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 i=f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo>∘</mo> <mi>i</mi> <mo>=</mo> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f'\circ i=f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59b3630e541788e895c31d556e06eae956be0a71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.379ex; height:2.843ex;" alt="{\displaystyle f'\circ i=f}"></span>), we can extend a homotopy 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 H:A\times I\to S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo>:</mo> <mi>A</mi> <mo>×</mo> <mi>I</mi> <mo stretchy="false">→</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H:A\times I\to S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2a712add4eeacc0a84a409870530e5b53805fcc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.869ex; height:2.176ex;" alt="{\displaystyle H:A\times I\to S}"></span> to a homotopy 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 H':X\times I\to S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>H</mi> <mo>′</mo> </msup> <mo>:</mo> <mi>X</mi> <mo>×</mo> <mi>I</mi> <mo stretchy="false">→</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H':X\times I\to S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b984dfa4e68d5f8c384756a6ef9130fceb5bc631" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.831ex; height:2.509ex;" alt="{\displaystyle H':X\times I\to S}"></span>, where</p><blockquote><p><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 {\begin{aligned}H(a,0)&=f(a)\\H'(x,0)&=f'(x)\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>H</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <msup> <mi>H</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}H(a,0)&=f(a)\\H'(x,0)&=f'(x)\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b23459f5845bfc073f502ea264912339f4a61355" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:17.118ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}H(a,0)&=f(a)\\H'(x,0)&=f'(x)\end{aligned}}}"></span></p></blockquote><p>We can encode this condition in the following commutative diagram</p><blockquote><p><span class="mw-default-size" typeof="mw:File/Frameless"><a href="/wiki/File:Cofibration_diagram.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b3/Cofibration_diagram.svg/220px-Cofibration_diagram.svg.png" decoding="async" width="220" height="178" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b3/Cofibration_diagram.svg/330px-Cofibration_diagram.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b3/Cofibration_diagram.svg/440px-Cofibration_diagram.svg.png 2x" data-file-width="305" data-file-height="247" /></a></span></p></blockquote><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 S^{I}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{I}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de3a8ba7feb02944f38966dda7393e60cc871e47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.582ex; height:2.676ex;" alt="{\displaystyle S^{I}}"></span> is the <a href="/wiki/Path_space_fibration" title="Path space fibration">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 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/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span> equipped with the compact-open topology. </p><p>For the notion of a cofibration in a model category, see <a href="/wiki/Model_category" title="Model category">model category</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cofibration&action=edit&section=3" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="In_topology">In topology</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cofibration&action=edit&section=4" title="Edit section: In topology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Topologists have long studied notions of "good subspace embedding", many of which imply that the map is a cofibration, or the converse, or have similar formal properties with regards to homology. In 1937, Borsuk proved 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> is a <i>binormal space</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 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 normal, and its product with 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 X\times I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>×</mo> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\times I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8bc149f64b58d6355772426e4209ce90fe7cf55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.992ex; height:2.176ex;" alt="{\displaystyle X\times I}"></span> is normal) then every closed subspace 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> has the homotopy extension property with respect to any absolute neighborhood retract. Likewise, 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 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> is a closed subspace 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> and the subspace inclusion <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\times I\cup X\times {1}\subset X\times I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>×</mo> <mi>I</mi> <mo>∪</mo> <mi>X</mi> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mo>⊂</mo> <mi>X</mi> <mo>×</mo> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\times I\cup X\times {1}\subset X\times I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8feea57549f64ead68936beb85b4e13cbbd45e56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:23.411ex; height:2.176ex;" alt="{\displaystyle A\times I\cup X\times {1}\subset X\times I}"></span> is an absolute neighborhood retract, then the inclusion 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 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> into <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 a cofibration.<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><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> Hatcher's introductory textbook <i>Algebraic Topology</i> uses a technical notion of <i>good pair</i> which has the same long exact sequence in singular homology associated to a cofibration, but it is not equivalent. The notion of cofibration is distinguished from these because its homotopy-theoretic definition is more amenable to formal analysis and generalization. </p><p>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 continuous map between topological spaces, there is an associated topological 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 Mf}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Mf}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c953a4820cd670853f2057aab0b24393f7ad5eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.721ex; height:2.509ex;" alt="{\displaystyle Mf}"></span> called the <a href="/wiki/Mapping_cylinder" title="Mapping cylinder">mapping cylinder</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>. There is a canonical subspace embedding <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:X\to Mf}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>M</mi> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i:X\to Mf}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/949b4305d5bcdd91de8fbb876aa92ece799236b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.054ex; height:2.509ex;" alt="{\displaystyle i:X\to Mf}"></span> and a 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 r:Mf\to Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>:</mo> <mi>M</mi> <mi>f</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r:Mf\to Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a813006b900bb7d7b5593bc3b279c45487358452" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.094ex; height:2.509ex;" alt="{\displaystyle r:Mf\to Y}"></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 r\circ i=f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>∘</mo> <mi>i</mi> <mo>=</mo> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r\circ i=f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d333958be6c3ce19d1547e69ea4ea56ea4e98349" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.423ex; height:2.509ex;" alt="{\displaystyle r\circ i=f}"></span> as pictured in the commutative diagram below. Moreover, <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> is a cofibration 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 r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> is a homotopy equivalence. This result can be summarized by saying that "every map is equivalent in the homotopy category to a cofibration." </p> <dl><dd><span typeof="mw:File/Frameless"><a href="/wiki/File:Mapping_cylinder_from_X_to_Y.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Mapping_cylinder_from_X_to_Y.png/100px-Mapping_cylinder_from_X_to_Y.png" decoding="async" width="100" height="108" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Mapping_cylinder_from_X_to_Y.png/150px-Mapping_cylinder_from_X_to_Y.png 1.5x, //upload.wikimedia.org/wikipedia/commons/d/d9/Mapping_cylinder_from_X_to_Y.png 2x" data-file-width="179" data-file-height="193" /></a></span></dd></dl> <p>Arne Strøm has proved a strengthening of this result, that every 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> factors as the composition of a cofibration and a homotopy equivalence which is also a <a href="/wiki/Fibration" title="Fibration">fibration</a>.<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> </p><p>A topological 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> with distinguished basepoint <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/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> is said to be <i>well-pointed</i> if the inclusion 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 X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mo stretchy="false">→</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {x}\to X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c344e07ba0d245cbd619bd8b658d5efa8fd145d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.924ex; height:2.176ex;" alt="{\displaystyle {x}\to X}"></span> is a cofibration. </p><p>The inclusion 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 S^{n-1}\to D^{n}}"> <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>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{n-1}\to D^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dce64d76ba0c193a9282f1797f4535c39cfb5949" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.597ex; height:2.676ex;" alt="{\displaystyle S^{n-1}\to D^{n}}"></span> of the boundary sphere of a solid disk is a cofibration for every <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>. </p><p>A frequently used fact is that a cellular inclusion is a cofibration (so, for instance, 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,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> is a <a href="/wiki/CW-pair" class="mw-redirect" title="CW-pair">CW pair</a>, 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 A\to X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">→</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\to X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/029a5ef0c363b51efff80e67c6f60062bf6ed95a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.337ex; height:2.176ex;" alt="{\displaystyle A\to X}"></span> is a cofibration). This follows from the previous fact and the fact that cofibrations are stable under pushout, because pushouts are the gluing maps to the <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-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-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbd0b0f32b28f51962943ee9ede4fb34198a2521" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n-1}"></span> skeleton. </p> <div class="mw-heading mw-heading3"><h3 id="In_chain_complexes">In chain complexes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cofibration&action=edit&section=5" title="Edit section: In chain complexes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <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 {\mathcal {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/280ae03440942ab348c2ca9b8db6b56ffa9618f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.903ex; height:2.343ex;" alt="{\displaystyle {\mathcal {A}}}"></span> be an <a href="/wiki/Abelian_category" title="Abelian category">Abelian category</a> with enough projectives. </p><p> If we 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 C_{+}({\mathcal {A}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">A</mi> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{+}({\mathcal {A}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c36ab1b565cebe5d44c6f30a1a9be8395ab2cc3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.885ex; height:2.843ex;" alt="{\displaystyle C_{+}({\mathcal {A}})}"></span> be the category of chain complexes which are <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> in degrees <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 q<<0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo><<</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q<<0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e95d8f67a5643aadfc925f2f960b27ee64ae6e95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.139ex; height:2.509ex;" alt="{\displaystyle q<<0}"></span>, then there is a model category structure<sup id="cite_ref-:1_5-0" class="reference"><a href="#cite_note-:1-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup>pg 1.2</sup> where the weak equivalences are the <a href="/wiki/Quasi-isomorphism" title="Quasi-isomorphism">quasi-isomorphisms</a>, the fibrations are the epimorphisms, and the cofibrations are maps</p><blockquote><p><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:C_{\bullet }\to D_{\bullet }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>:</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> </mrow> </msub> <mo stretchy="false">→</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i:C_{\bullet }\to D_{\bullet }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13d739bb50781282aacd328d69a058d5b97e548d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.26ex; margin-bottom: -0.412ex; width:12.048ex; height:2.509ex;" alt="{\displaystyle i:C_{\bullet }\to D_{\bullet }}"></span></p></blockquote><p>which are degreewise monic and the cokernel complex <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 {\text{Coker}}(i)_{\bullet }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>Coker</mtext> </mrow> <mo stretchy="false">(</mo> <mi>i</mi> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{Coker}}(i)_{\bullet }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e24d39a7bd8509843b03cb4b2aa9c47bc329f01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.678ex; height:2.843ex;" alt="{\displaystyle {\text{Coker}}(i)_{\bullet }}"></span> is a complex of <a href="/wiki/Projective_object" title="Projective object">projective objects</a> 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 {\mathcal {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/280ae03440942ab348c2ca9b8db6b56ffa9618f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.903ex; height:2.343ex;" alt="{\displaystyle {\mathcal {A}}}"></span>. It follows that the cofibrant objects are the complexes whose objects are all projective. </p><div class="mw-heading mw-heading3"><h3 id="Simplicial_sets">Simplicial sets</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cofibration&action=edit&section=6" title="Edit section: Simplicial sets"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The 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 {\textbf {SSet}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">SSet</mtext> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\textbf {SSet}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b64de09f1f6a34c3b7614e678087994d40e83069" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.235ex; height:2.176ex;" alt="{\displaystyle {\textbf {SSet}}}"></span> of simplicial sets<sup id="cite_ref-:1_5-1" class="reference"><a href="#cite_note-:1-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup>pg 1.3</sup> there is a model category structure where the fibrations are precisely the Kan fibrations, cofibrations are all injective maps, and weak equivalences are simplicial maps which become homotopy equivalences after applying the geometric realization functor. </p> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cofibration&action=edit&section=7" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>For <a href="/wiki/Hausdorff_space" title="Hausdorff space">Hausdorff spaces</a>, every cofibration is a closed inclusion (injective with closed image); the result also generalizes to <a href="/wiki/Weak_Hausdorff_space" title="Weak Hausdorff space">weak Hausdorff spaces</a>.</li> <li>The <a href="/wiki/Pushout_(category_theory)" title="Pushout (category theory)">pushout</a> of a cofibration is a cofibration. That is, 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 g\colon A\to B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>:</mo> <mi>A</mi> <mo stretchy="false">→</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g\colon A\to B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42b48b552cb66962a64e6914b8793b47d6606532" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.271ex; height:2.509ex;" alt="{\displaystyle g\colon A\to B}"></span> is any (continuous) map (between compactly generated spaces), 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 i\colon 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\colon A\to X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6a64c55f7fe66fe8e1eb4b91766d3de6a11809f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.174ex; height:2.176ex;" alt="{\displaystyle i\colon A\to X}"></span> is a cofibration, then the induced 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 B\to B\cup _{g}X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo stretchy="false">→</mo> <mi>B</mi> <msub> <mo>∪</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> </mrow> </msub> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B\to B\cup _{g}X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3dc36e6fec1e9a1a85eb7c7ec3a7e749ebe31b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.726ex; height:2.843ex;" alt="{\displaystyle B\to B\cup _{g}X}"></span> is a cofibration.</li> <li>The <a href="/wiki/Mapping_cylinder" title="Mapping cylinder">mapping cylinder</a> can be understood as the pushout 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 i\colon 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\colon A\to X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6a64c55f7fe66fe8e1eb4b91766d3de6a11809f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.174ex; height:2.176ex;" alt="{\displaystyle i\colon A\to X}"></span> and the embedding (at one end of 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 i_{0}\colon 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}\colon A\to A\times I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1db2369d5492f9bd79ad4180e38d22e3ba0460ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.003ex; height:2.509ex;" alt="{\displaystyle i_{0}\colon A\to A\times I}"></span>. That is, the mapping cylinder can be defined 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 Mi=X\cup _{i}(A\times I)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mi>i</mi> <mo>=</mo> <mi>X</mi> <msub> <mo>∪</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>A</mi> <mo>×</mo> <mi>I</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Mi=X\cup _{i}(A\times I)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f842260fbef4a01218a221a60f9594c074c0959e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.27ex; height:2.843ex;" alt="{\displaystyle Mi=X\cup _{i}(A\times I)}"></span>. By the <a href="/wiki/Universal_property" title="Universal property">universal property</a> of the pushout, <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> is a cofibration precisely when a mapping cylinder can be constructed for every space <i>X</i>.</li> <li>There is a cofibration (<i>A</i>, <i>X</i>), if and only if there is a <a href="/wiki/Deformation_retract" class="mw-redirect" title="Deformation retract">retraction</a> from <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 I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>×</mo> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\times I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8bc149f64b58d6355772426e4209ce90fe7cf55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.992ex; height:2.176ex;" alt="{\displaystyle X\times I}"></span> 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 (A\times I)\cup (X\times \{0\})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>×</mo> <mi>I</mi> <mo stretchy="false">)</mo> <mo>∪</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>×</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A\times I)\cup (X\times \{0\})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c7b2ded7feaa47dff4b262ed4c1bccf06f1eac4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.264ex; height:2.843ex;" alt="{\displaystyle (A\times I)\cup (X\times \{0\})}"></span>, since this is the <a href="/wiki/Pushout_(category_theory)" title="Pushout (category theory)">pushout</a> and thus induces maps to every space sensible in the diagram.</li> <li>Similar equivalences can be stated for deformation-retract pairs, and for neighborhood deformation-retract pairs.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Constructions_with_cofibrations">Constructions with cofibrations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cofibration&action=edit&section=8" title="Edit section: Constructions with cofibrations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Cofibrant_replacement">Cofibrant replacement</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cofibration&action=edit&section=9" title="Edit section: Cofibrant replacement"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Note that in a model 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 {\mathcal {M}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {M}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2cc2abebd45ec020509a0ec548b67c9a2cb7cecd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.791ex; height:2.176ex;" alt="{\displaystyle {\mathcal {M}}}"></span> 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 i:*\to X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>:</mo> <mo>∗</mo> <mo stretchy="false">→</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i:*\to X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b09698140ea7314c68b585281102839c2c3fc7d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.496ex; height:2.176ex;" alt="{\displaystyle i:*\to X}"></span> is not a cofibration, then the mapping cylinder <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 Mi}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Mi}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c719f09555540f7118e20d2066617234f4ae57f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.245ex; height:2.176ex;" alt="{\displaystyle Mi}"></span> forms a <b>cofibrant replacement</b>. In fact, if we work in just the category of topological spaces, the cofibrant replacement for any map from a point to a space forms a cofibrant replacement. </p> <div class="mw-heading mw-heading3"><h3 id="Cofiber">Cofiber</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cofibration&action=edit&section=10" title="Edit section: Cofiber"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div><p> For a cofibration <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\to X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">→</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\to X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/029a5ef0c363b51efff80e67c6f60062bf6ed95a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.337ex; height:2.176ex;" alt="{\displaystyle A\to X}"></span> we define the <b>cofiber</b> to be the induced quotient 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/A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>A</mi> </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/65cb5f024326b383f3bc0a2e6afe5bcc0c67b3a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.886ex; height:2.843ex;" alt="{\displaystyle X/A}"></span>. In general, 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 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>, the <b>cofiber</b><sup id="cite_ref-:0_1-1" class="reference"><a href="#cite_note-:0-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup>pg 59</sup> is defined as the quotient space</p><blockquote><p><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 C_{f}=M_{f}/(X\times \{0\})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>×</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{f}=M_{f}/(X\times \{0\})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a437763372602d56e28f30751ad0e94cb1d2b903" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.567ex; height:3.009ex;" alt="{\displaystyle C_{f}=M_{f}/(X\times \{0\})}"></span></p></blockquote><p>which is the mapping cone 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>. Homotopically, the cofiber acts as a homotopy cokernel of 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 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>. In fact, for pointed topological spaces, the <a href="/wiki/Homotopy_colimit" class="mw-redirect" title="Homotopy colimit">homotopy colimit</a> of</p><blockquote><p><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 {\underset {\to }{\text{hocolim}}}\left({\begin{matrix}X&\xrightarrow {f} &Y\\\downarrow &&\\\end{matrix}}\right)=C_{f}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <munder> <mtext>hocolim</mtext> <mo stretchy="false">→</mo> </munder> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>X</mi> </mtd> <mtd> <mover> <mo>→</mo> <mpadded width="+0.611em" lspace="0.278em" voffset=".15em"> <mi>f</mi> </mpadded> </mover> </mtd> <mtd> <mi>Y</mi> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">↓</mo> </mtd> <mtd /> <mtd /> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\underset {\to }{\text{hocolim}}}\left({\begin{matrix}X&\xrightarrow {f} &Y\\\downarrow &&\\\end{matrix}}\right)=C_{f}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5900e396282e09da64bbbeb1a3bd2c673235129e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; margin-top: -0.271ex; width:29.318ex; height:7.509ex;" alt="{\displaystyle {\underset {\to }{\text{hocolim}}}\left({\begin{matrix}X&\xrightarrow {f} &Y\\\downarrow &&\\\end{matrix}}\right)=C_{f}}"></span></p></blockquote><p>In fact, the sequence 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 Y\to C_{f}}"> <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> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\to Y\to C_{f}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f22adfd0a979cc9418614a3dbe603068e43fff72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.78ex; height:2.843ex;" alt="{\displaystyle X\to Y\to C_{f}}"></span> comes equipped with the <a href="/wiki/Cofiber_sequence" class="mw-redirect" title="Cofiber sequence">cofiber sequence</a> which acts like a <a href="/wiki/Distinguished_triangle" class="mw-redirect" title="Distinguished triangle">distinguished triangle</a> in triangulated categories. </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=Cofibration&action=edit&section=11" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Fibration" title="Fibration">Fibration</a></li> <li><a href="/wiki/Homotopy_colimit" class="mw-redirect" title="Homotopy colimit">Homotopy colimit</a></li> <li><a href="/wiki/Homotopy_fiber" title="Homotopy fiber">Homotopy fiber</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=Cofibration&action=edit&section=12" 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"><ol class="references"> <li id="cite_note-:0-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_1-1"><sup><i><b>b</b></i></sup></a></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 id="CITEREFMay,_J._Peter.1999" class="citation book cs1">May, J. Peter. (1999). <a rel="nofollow" class="external text" href="https://www.worldcat.org/oclc/41266205"><i>A concise course in algebraic topology</i></a>. Chicago: University of Chicago Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-226-51182-0" title="Special:BookSources/0-226-51182-0"><bdi>0-226-51182-0</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/41266205">41266205</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+concise+course+in+algebraic+topology&rft.place=Chicago&rft.pub=University+of+Chicago+Press&rft.date=1999&rft_id=info%3Aoclcnum%2F41266205&rft.isbn=0-226-51182-0&rft.au=May%2C+J.+Peter.&rft_id=https%3A%2F%2Fwww.worldcat.org%2Foclc%2F41266205&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACofibration" class="Z3988"></span></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">Edwin Spanier, <i>Algebraic Topology</i>, 1966, p. 57.</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">Garth Warner, <i>Topics in Topology and Homotopy Theory</i>, section 6.</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">Arne Strøm, The homotopy category is a homotopy category</span> </li> <li id="cite_note-:1-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-:1_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:1_5-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFQuillen,_Daniel_G.1967" class="citation book cs1">Quillen, Daniel G. (1967). <a rel="nofollow" class="external text" href="https://www.worldcat.org/oclc/294862881"><i>Homotopical algebra</i></a>. Berlin: 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>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/294862881">294862881</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.place=Berlin&rft.pub=Springer-Verlag&rft.date=1967&rft_id=info%3Aoclcnum%2F294862881&rft.isbn=978-3-540-03914-3&rft.au=Quillen%2C+Daniel+G.&rft_id=https%3A%2F%2Fwww.worldcat.org%2Foclc%2F294862881&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACofibration" class="Z3988"></span></span> </li> </ol></div></div> <ul><li><a rel="nofollow" class="external text" href="http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf">Peter May, "A Concise Course in Algebraic Topology"</a> : chapter 6 defines and discusses cofibrations, and they are used throughout</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBrown" class="citation book cs1"><a href="/wiki/Ronald_Brown_(mathematician)" title="Ronald Brown (mathematician)">Brown, Ronald</a>. "7. Cofibrations". <a rel="nofollow" class="external text" href="http://groupoids.org.uk/topgpds.html"><i>Topology and Groupoids</i></a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4196-2722-4" title="Special:BookSources/978-1-4196-2722-4"><bdi>978-1-4196-2722-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=7.+Cofibrations&rft.btitle=Topology+and+Groupoids&rft.isbn=978-1-4196-2722-4&rft.aulast=Brown&rft.aufirst=Ronald&rft_id=http%3A%2F%2Fgroupoids.org.uk%2Ftopgpds.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACofibration" class="Z3988"></span> Chapter 7 has many results not found elsewhere.</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 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href="/wiki/Template:Manifolds" title="Template:Manifolds"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Manifolds" title="Template talk:Manifolds"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Manifolds" title="Special:EditPage/Template:Manifolds"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Manifolds_(Glossary)" style="font-size:114%;margin:0 4em"><a href="/wiki/Manifold" title="Manifold">Manifolds</a> (<a href="/wiki/Glossary_of_differential_geometry_and_topology" title="Glossary of differential geometry and topology">Glossary</a>)</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Basic concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Topological_manifold" title="Topological manifold">Topological manifold</a> <ul><li><a href="/wiki/Atlas_(topology)" title="Atlas (topology)">Atlas</a></li></ul></li> <li><a href="/wiki/Differentiable_manifold" title="Differentiable manifold">Differentiable/Smooth manifold</a> <ul><li><a href="/wiki/Differential_structure" title="Differential structure">Differential structure</a></li> <li><a href="/wiki/Smooth_structure" title="Smooth structure">Smooth atlas</a></li></ul></li> <li><a href="/wiki/Submanifold" title="Submanifold">Submanifold</a></li> <li><a href="/wiki/Riemannian_manifold" title="Riemannian manifold">Riemannian manifold</a></li> <li><a href="/wiki/Smoothness" title="Smoothness">Smooth map</a></li> <li><a href="/wiki/Submersion_(mathematics)" title="Submersion (mathematics)">Submersion</a></li> <li><a href="/wiki/Pushforward_(differential)" title="Pushforward (differential)">Pushforward</a></li> <li><a href="/wiki/Tangent_space" title="Tangent space">Tangent space</a></li> <li><a href="/wiki/Differential_form" title="Differential form">Differential form</a></li> <li><a href="/wiki/Vector_field" title="Vector field">Vector field</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Main results <span style="font-size:85%;"><a href="/wiki/Category:Theorems_in_differential_geometry" title="Category:Theorems in differential geometry">(list)</a></span></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Atiyah%E2%80%93Singer_index_theorem" title="Atiyah–Singer index theorem">Atiyah–Singer index</a></li> <li><a href="/wiki/Darboux%27s_theorem" title="Darboux's theorem">Darboux's</a></li> <li><a href="/wiki/De_Rham_cohomology#De_Rham's_theorem" title="De Rham cohomology">De Rham's</a></li> <li><a href="/wiki/Frobenius_theorem_(differential_topology)" title="Frobenius theorem (differential topology)">Frobenius</a></li> <li><a href="/wiki/Generalized_Stokes_theorem" title="Generalized Stokes theorem">Generalized Stokes</a></li> <li><a href="/wiki/Hopf%E2%80%93Rinow_theorem" title="Hopf–Rinow theorem">Hopf–Rinow</a></li> <li><a href="/wiki/Noether%27s_theorem" title="Noether's theorem">Noether's</a></li> <li><a href="/wiki/Sard%27s_theorem" title="Sard's theorem">Sard's</a></li> <li><a href="/wiki/Whitney_embedding_theorem" title="Whitney embedding theorem">Whitney embedding</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Smoothness" title="Smoothness">Maps</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Differentiable_curve" title="Differentiable curve">Curve</a></li> <li><a href="/wiki/Diffeomorphism" title="Diffeomorphism">Diffeomorphism</a> <ul><li><a href="/wiki/Local_diffeomorphism" title="Local diffeomorphism">Local</a></li></ul></li> <li><a href="/wiki/Geodesic" title="Geodesic">Geodesic</a></li> <li><a href="/wiki/Exponential_map_(Riemannian_geometry)" title="Exponential map (Riemannian geometry)">Exponential map</a> <ul><li><a href="/wiki/Exponential_map_(Lie_theory)" title="Exponential map (Lie theory)">in Lie theory</a></li></ul></li> <li><a href="/wiki/Foliation" title="Foliation">Foliation</a></li> <li><a href="/wiki/Immersion_(mathematics)" title="Immersion (mathematics)">Immersion</a></li> <li><a href="/wiki/Integral_curve" title="Integral curve">Integral curve</a></li> <li><a href="/wiki/Lie_derivative" title="Lie derivative">Lie derivative</a></li> <li><a href="/wiki/Section_(fiber_bundle)" title="Section (fiber bundle)">Section</a></li> <li><a href="/wiki/Submersion_(mathematics)" title="Submersion (mathematics)">Submersion</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types of<br />manifolds</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Closed_manifold" title="Closed manifold">Closed</a></li> <li>(<a href="/wiki/Almost_complex_manifold" title="Almost complex manifold">Almost</a>) <a href="/wiki/Complex_manifold" title="Complex manifold">Complex</a></li> <li>(<a href="/wiki/Almost-contact_manifold" title="Almost-contact manifold">Almost</a>) <a href="/wiki/Contact_manifold" class="mw-redirect" title="Contact manifold">Contact</a></li> <li><a href="/wiki/Fibered_manifold" title="Fibered manifold">Fibered</a></li> <li><a href="/wiki/Finsler_manifold" title="Finsler manifold">Finsler</a></li> <li><a href="/wiki/Flat_manifold" title="Flat manifold">Flat</a></li> <li><a href="/wiki/G-structure_on_a_manifold" title="G-structure on a manifold">G-structure</a></li> <li><a href="/wiki/Hadamard_manifold" title="Hadamard manifold">Hadamard</a></li> <li><a href="/wiki/Hermitian_manifold" title="Hermitian manifold">Hermitian</a></li> <li><a href="/wiki/Hyperbolic_manifold" title="Hyperbolic manifold">Hyperbolic</a></li> <li><a href="/wiki/K%C3%A4hler_manifold" title="Kähler manifold">Kähler</a></li> <li><a href="/wiki/Kenmotsu_manifold" title="Kenmotsu manifold">Kenmotsu</a></li> <li><a href="/wiki/Lie_group" title="Lie group">Lie group</a> <ul><li><a href="/wiki/Lie_group%E2%80%93Lie_algebra_correspondence" title="Lie group–Lie algebra correspondence">Lie algebra</a></li></ul></li> <li><a href="/wiki/Manifold_with_boundary" class="mw-redirect" title="Manifold with boundary">Manifold with boundary</a></li> <li><a href="/wiki/Orientability" title="Orientability">Oriented</a></li> <li><a href="/wiki/Parallelizable_manifold" title="Parallelizable manifold">Parallelizable</a></li> <li><a href="/wiki/Poisson_manifold" title="Poisson manifold">Poisson</a></li> <li><a href="/wiki/Prime_manifold" title="Prime manifold">Prime</a></li> <li><a href="/wiki/Quaternionic_manifold" title="Quaternionic manifold">Quaternionic</a></li> <li><a href="/wiki/Hypercomplex_manifold" title="Hypercomplex manifold">Hypercomplex</a></li> <li>(<a href="/wiki/Pseudo-Riemannian_manifold" title="Pseudo-Riemannian manifold">Pseudo−</a>, <a href="/wiki/Sub-Riemannian_manifold" title="Sub-Riemannian manifold">Sub−</a>) <a href="/wiki/Riemannian_manifold" title="Riemannian manifold">Riemannian</a></li> <li><a href="/wiki/Rizza_manifold" title="Rizza manifold">Rizza</a></li> <li>(<a href="/wiki/Almost_symplectic_manifold" title="Almost symplectic manifold">Almost</a>) <a href="/wiki/Symplectic_manifold" title="Symplectic manifold">Symplectic</a></li> <li><a href="/wiki/Tame_manifold" title="Tame manifold">Tame</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Tensor" title="Tensor">Tensors</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Vectors</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Distribution_(differential_geometry)" title="Distribution (differential geometry)">Distribution</a></li> <li><a href="/wiki/Lie_bracket_of_vector_fields" title="Lie bracket of vector fields">Lie bracket</a></li> <li><a href="/wiki/Pushforward_(differential)" title="Pushforward (differential)">Pushforward</a></li> <li><a href="/wiki/Tangent_space" title="Tangent space">Tangent space</a> <ul><li><a href="/wiki/Tangent_bundle" title="Tangent bundle">bundle</a></li></ul></li> <li><a href="/wiki/Torsion_tensor" title="Torsion tensor">Torsion</a></li> <li><a href="/wiki/Vector_field" title="Vector field">Vector field</a></li> <li><a href="/wiki/Vector_flow" title="Vector flow">Vector flow</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Covectors</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Closed_and_exact_differential_forms" title="Closed and exact differential forms">Closed/Exact</a></li> <li><a href="/wiki/Covariant_derivative" title="Covariant derivative">Covariant derivative</a></li> <li><a href="/wiki/Cotangent_space" title="Cotangent space">Cotangent space</a> <ul><li><a href="/wiki/Cotangent_bundle" title="Cotangent bundle">bundle</a></li></ul></li> <li><a href="/wiki/De_Rham_cohomology" title="De Rham cohomology">De Rham cohomology</a></li> <li><a href="/wiki/Differential_form" title="Differential form">Differential form</a> <ul><li><a href="/wiki/Vector-valued_differential_form" title="Vector-valued differential form">Vector-valued</a></li></ul></li> <li><a href="/wiki/Exterior_derivative" title="Exterior derivative">Exterior derivative</a></li> <li><a href="/wiki/Interior_product" title="Interior product">Interior product</a></li> <li><a href="/wiki/Pullback_(differential_geometry)" title="Pullback (differential geometry)">Pullback</a></li> <li><a href="/wiki/Ricci_curvature" title="Ricci curvature">Ricci curvature</a> <ul><li><a href="/wiki/Ricci_flow" title="Ricci flow">flow</a></li></ul></li> <li><a href="/wiki/Riemann_curvature_tensor" title="Riemann curvature tensor">Riemann curvature tensor</a></li> <li><a href="/wiki/Tensor_field" title="Tensor field">Tensor field</a> <ul><li><a href="/wiki/Tensor_density" title="Tensor density">density</a></li></ul></li> <li><a href="/wiki/Volume_form" title="Volume form">Volume form</a></li> <li><a href="/wiki/Wedge_product" class="mw-redirect" title="Wedge product">Wedge product</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Fiber_bundle" title="Fiber bundle">Bundles</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Adjoint_bundle" title="Adjoint bundle">Adjoint</a></li> <li><a href="/wiki/Affine_bundle" title="Affine bundle">Affine</a></li> <li><a href="/wiki/Associated_bundle" title="Associated bundle">Associated</a></li> <li><a href="/wiki/Cotangent_bundle" title="Cotangent bundle">Cotangent</a></li> <li><a href="/wiki/Dual_bundle" title="Dual bundle">Dual</a></li> <li><a href="/wiki/Fiber_bundle" title="Fiber bundle">Fiber</a></li> <li>(<a class="mw-selflink selflink">Co</a>) <a href="/wiki/Fibration" title="Fibration">Fibration</a></li> <li><a href="/wiki/Jet_bundle" title="Jet bundle">Jet</a></li> <li><a href="/wiki/Lie_algebra_bundle" title="Lie algebra bundle">Lie algebra</a></li> <li>(<a href="/wiki/Stable_normal_bundle" title="Stable normal bundle">Stable</a>) <a href="/wiki/Normal_bundle" title="Normal bundle">Normal</a></li> <li><a href="/wiki/Principal_bundle" title="Principal bundle">Principal</a></li> <li><a href="/wiki/Spinor_bundle" title="Spinor bundle">Spinor</a></li> <li><a href="/wiki/Subbundle" title="Subbundle">Subbundle</a></li> <li><a href="/wiki/Tangent_bundle" title="Tangent bundle">Tangent</a></li> <li><a href="/wiki/Tensor_bundle" title="Tensor bundle">Tensor</a></li> <li><a href="/wiki/Vector_bundle" title="Vector bundle">Vector</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Connection_(mathematics)" title="Connection (mathematics)">Connections</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Affine_connection" title="Affine connection">Affine</a></li> <li><a href="/wiki/Cartan_connection" title="Cartan connection">Cartan</a></li> <li><a href="/wiki/Ehresmann_connection" title="Ehresmann connection">Ehresmann</a></li> <li><a href="/wiki/Connection_form" title="Connection form">Form</a></li> <li><a href="/wiki/Connection_(fibred_manifold)" title="Connection (fibred manifold)">Generalized</a></li> <li><a href="/wiki/Koszul_connection" class="mw-redirect" title="Koszul connection">Koszul</a></li> <li><a href="/wiki/Levi-Civita_connection" title="Levi-Civita connection">Levi-Civita</a></li> <li><a href="/wiki/Connection_(principal_bundle)" title="Connection (principal bundle)">Principal</a></li> <li><a href="/wiki/Connection_(vector_bundle)" title="Connection (vector bundle)">Vector</a></li> <li><a href="/wiki/Parallel_transport" title="Parallel transport">Parallel transport</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Classification_of_manifolds" title="Classification of manifolds">Classification of manifolds</a></li> <li><a href="/wiki/Gauge_theory_(mathematics)" title="Gauge theory (mathematics)">Gauge theory</a></li> <li><a href="/wiki/History_of_manifolds_and_varieties" title="History of manifolds and varieties">History</a></li> <li><a href="/wiki/Morse_theory" title="Morse theory">Morse theory</a></li> <li><a href="/wiki/Moving_frame" title="Moving frame">Moving frame</a></li> <li><a href="/wiki/Singularity_theory" title="Singularity theory">Singularity theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Generalizations</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Banach_manifold" title="Banach manifold">Banach manifold</a></li> <li><a href="/wiki/Diffeology" title="Diffeology">Diffeology</a></li> <li><a href="/wiki/Diffiety" title="Diffiety">Diffiety</a></li> <li><a href="/wiki/Fr%C3%A9chet_manifold" title="Fréchet manifold">Fréchet manifold</a></li> <li><a href="/wiki/K-theory" title="K-theory">K-theory</a></li> <li><a href="/wiki/Orbifold" title="Orbifold">Orbifold</a></li> <li><a href="/wiki/Secondary_calculus_and_cohomological_physics" title="Secondary calculus and cohomological physics">Secondary calculus</a> <ul><li><a href="/wiki/Differential_calculus_over_commutative_algebras" title="Differential calculus over commutative algebras">over commutative algebras</a></li></ul></li> <li><a href="/wiki/Sheaf_(mathematics)" title="Sheaf (mathematics)">Sheaf</a></li> <li><a href="/wiki/Stratifold" title="Stratifold">Stratifold</a></li> <li><a href="/wiki/Supermanifold" title="Supermanifold">Supermanifold</a></li> <li><a href="/wiki/Stratified_space" title="Stratified space">Stratified space</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved 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