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class="vector-toc-numb">2</span> <span>Properties</span> </div> </a> <ul id="toc-Properties-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bundles_and_sheaves" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bundles_and_sheaves"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Bundles and sheaves</span> </div> </a> <ul id="toc-Bundles_and_sheaves-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 class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sources" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Sources"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Sources</span> </div> </a> <ul id="toc-Sources-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " 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id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" 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>, a <b>pullback bundle</b> or <b>induced bundle</b><sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><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> is the <a href="/wiki/Fiber_bundle" title="Fiber bundle">fiber bundle</a> that is induced by a map of its base-space. Given a fiber bundle <span class="texhtml"><i>π</i> : <i>E</i> → <i>B</i></span> and a <a href="/wiki/Continuous_(topology)" class="mw-redirect" title="Continuous (topology)">continuous map</a> <span class="texhtml"><i>f</i> : <i>B</i>′ → <i>B</i></span> one can define a "pullback" of <span class="texhtml"><i>E</i></span> by <span class="texhtml"><i>f</i></span> as a bundle <span class="texhtml"><i>f</i><sup>*</sup><i>E</i></span> over <span class="texhtml"><i>B</i>′</span>. The fiber of <span class="texhtml"><i>f</i><sup>*</sup><i>E</i></span> over a point <span class="texhtml"><i>b</i>′</span> in <span class="texhtml"><i>B</i>′</span> is just the fiber of <span class="texhtml"><i>E</i></span> over <span class="texhtml"><i>f</i>(<i>b</i>′)</span>. Thus <span class="texhtml"><i>f</i><sup>*</sup><i>E</i></span> is the <a href="/wiki/Disjoint_union" title="Disjoint union">disjoint union</a> of all these fibers equipped with a suitable <a href="/wiki/Topological_space" title="Topological space">topology</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Formal_definition">Formal definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pullback_bundle&action=edit&section=1" title="Edit section: Formal definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <span class="texhtml"><i>π</i> : <i>E</i> → <i>B</i></span> be a fiber bundle with abstract fiber <span class="texhtml"><i>F</i></span> and let <span class="texhtml"><i>f</i> : <i>B</i>′ → <i>B</i></span> be a <a href="/wiki/Continuous_(topology)" class="mw-redirect" title="Continuous (topology)">continuous map</a>. Define the <b>pullback bundle</b> by </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 f^{*}E=\{(b',e)\in B'\times E\mid f(b')=\pi (e)\}\subseteq B'\times E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>E</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <msup> <mi>b</mi> <mo>′</mo> </msup> <mo>,</mo> <mi>e</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <msup> <mi>B</mi> <mo>′</mo> </msup> <mo>×</mo> <mi>E</mi> <mo>∣</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msup> <mi>b</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mi>π</mi> <mo stretchy="false">(</mo> <mi>e</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mo>⊆</mo> <msup> <mi>B</mi> <mo>′</mo> </msup> <mo>×</mo> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{*}E=\{(b',e)\in B'\times E\mid f(b')=\pi (e)\}\subseteq B'\times E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2360262490f052361892bede06c5735331734448" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:49.282ex; height:3.009ex;" alt="{\displaystyle f^{*}E=\{(b',e)\in B'\times E\mid f(b')=\pi (e)\}\subseteq B'\times E}"></span></dd></dl> <p>and equip it with the <a href="/wiki/Subspace_topology" title="Subspace topology">subspace topology</a> and the <a href="/wiki/Projection_map" class="mw-redirect" title="Projection map">projection map</a> <span class="texhtml"><i>π</i>′ : <i>f</i><sup>*</sup><i>E</i> → <i>B</i>′</span> given by the projection onto the first factor, i.e., </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 \pi '(b',e)=b'.\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>π</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <msup> <mi>b</mi> <mo>′</mo> </msup> <mo>,</mo> <mi>e</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>b</mi> <mo>′</mo> </msup> <mo>.</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi '(b',e)=b'.\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f23159639522cae9cfbd39ab6d2c8f253712b944" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.443ex; height:3.009ex;" alt="{\displaystyle \pi '(b',e)=b'.\,}"></span></dd></dl> <p>The projection onto the second factor gives a map </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h\colon f^{*}E\to E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>:</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>E</mi> <mo stretchy="false">→</mo> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h\colon f^{*}E\to E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41d464f9423a5146b76c5f72ac345dc0ae05d147" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.913ex; height:2.676ex;" alt="{\displaystyle h\colon f^{*}E\to E}"></span></dd></dl> <p>such that the following diagram <a href="/wiki/Commutative_diagram" title="Commutative diagram">commutes</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 {\begin{array}{ccc}f^{\ast }E&{\stackrel {h}{\longrightarrow }}&E\\{\pi }'\downarrow &&\downarrow \pi \\B'&{\stackrel {f}{\longrightarrow }}&B\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="center center center" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>E</mi> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo stretchy="false">⟶</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </mover> </mrow> </mrow> </mtd> <mtd> <mi>E</mi> </mtd> </mtr> <mtr> <mtd> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">↓</mo> </mtd> <mtd /> <mtd> <mo stretchy="false">↓</mo> <mi>π</mi> </mtd> </mtr> <mtr> <mtd> <msup> <mi>B</mi> <mo>′</mo> </msup> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo stretchy="false">⟶</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </mover> </mrow> </mrow> </mtd> <mtd> <mi>B</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{ccc}f^{\ast }E&{\stackrel {h}{\longrightarrow }}&E\\{\pi }'\downarrow &&\downarrow \pi \\B'&{\stackrel {f}{\longrightarrow }}&B\end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ceacba424ed97cafdc7ac8385514413c1c93f6ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.338ex; width:16.492ex; height:11.843ex;" alt="{\displaystyle {\begin{array}{ccc}f^{\ast }E&{\stackrel {h}{\longrightarrow }}&E\\{\pi }'\downarrow &&\downarrow \pi \\B'&{\stackrel {f}{\longrightarrow }}&B\end{array}}}"></span></dd></dl> <p>If <span class="texhtml">(<i>U</i>, <i>φ</i>)</span> is a <a href="/wiki/Local_trivialization" class="mw-redirect" title="Local trivialization">local trivialization</a> of <span class="texhtml"><i>E</i></span> then <span class="texhtml">(<i>f</i><sup>−1</sup><i>U</i>, <i>ψ</i>)</span> is a local trivialization of <span class="texhtml"><i>f</i><sup>*</sup><i>E</i></span> where </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 \psi (b',e)=(b',{\mbox{proj}}_{2}(\varphi (e))).\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">(</mo> <msup> <mi>b</mi> <mo>′</mo> </msup> <mo>,</mo> <mi>e</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mi>b</mi> <mo>′</mo> </msup> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>proj</mtext> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>e</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>.</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (b',e)=(b',{\mbox{proj}}_{2}(\varphi (e))).\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/780647f83586dd0fcfc980eafdeb8349ffb1c557" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.135ex; height:3.009ex;" alt="{\displaystyle \psi (b',e)=(b',{\mbox{proj}}_{2}(\varphi (e))).\,}"></span></dd></dl> <p>It then follows that <span class="texhtml"><i>f</i><sup>*</sup><i>E</i></span> is a fiber bundle over <span class="texhtml"><i>B</i>′</span> with fiber <span class="texhtml"><i>F</i></span>. The bundle <span class="texhtml"><i>f</i><sup>*</sup><i>E</i></span> is called the <b>pullback of <i>E</i> by <span class="texhtml"><i>f</i></span> </b> or the <b>bundle induced by <span class="texhtml"><i>f</i></span></b>. The map <span class="texhtml"><i>h</i></span> is then a <a href="/wiki/Bundle_morphism" class="mw-redirect" title="Bundle morphism">bundle morphism</a> covering <span class="texhtml"><i>f</i></span>. </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=Pullback_bundle&action=edit&section=2" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Any <a href="/wiki/Section_(fiber_bundle)" title="Section (fiber bundle)">section</a> <span class="texhtml"><i>s</i></span> of <span class="texhtml"><i>E</i></span> over <span class="texhtml"><i>B</i></span> induces a section of <span class="texhtml"><i>f</i><sup>*</sup><i>E</i></span>, called the <b>pullback section</b> <span class="texhtml"><i>f</i><sup>*</sup><i>s</i></span>, simply by defining </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 f^{*}s(b'):=(b',s(f(b'))\ )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>s</mi> <mo stretchy="false">(</mo> <msup> <mi>b</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo>:=</mo> <mo stretchy="false">(</mo> <msup> <mi>b</mi> <mo>′</mo> </msup> <mo>,</mo> <mi>s</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msup> <mi>b</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mtext> </mtext> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{*}s(b'):=(b',s(f(b'))\ )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aed641c183ea0954d49bb8743582b12f45c89ae9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.478ex; height:3.009ex;" alt="{\displaystyle f^{*}s(b'):=(b',s(f(b'))\ )}"></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 b'\in B'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>b</mi> <mo>′</mo> </msup> <mo>∈</mo> <msup> <mi>B</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b'\in B'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b32e20be129456ee21174ec0265b40b0f0851b51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.972ex; height:2.509ex;" alt="{\displaystyle b'\in B'}"></span>.</dd></dl> <p>If the bundle <span class="texhtml"><i>E</i> → <i>B</i></span> has <a href="/wiki/Structure_group" class="mw-redirect" title="Structure group">structure group</a> <span class="texhtml"><i>G</i></span> with transition functions <span class="texhtml"><i>t</i><sub><i>ij</i></sub></span> (with respect to a family of local trivializations <span class="texhtml">{(<i>U</i><sub><i>i</i></sub>, <i>φ</i><sub><i>i</i></sub>)}</span>) then the pullback bundle <span class="texhtml"><i>f</i><sup>*</sup><i>E</i></span> also has structure group <span class="texhtml"><i>G</i></span>. The transition functions in <span class="texhtml"><i>f</i><sup>*</sup><i>E</i></span> are given by </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 f^{*}t_{ij}=t_{ij}\circ f.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>∘</mo> <mi>f</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{*}t_{ij}=t_{ij}\circ f.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ddf4342e5cc1300152058bbf9270ce14878fba1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.227ex; height:3.009ex;" alt="{\displaystyle f^{*}t_{ij}=t_{ij}\circ f.}"></span></dd></dl> <p>If <span class="texhtml"><i>E</i> → <i>B</i></span> is a <a href="/wiki/Vector_bundle" title="Vector bundle">vector bundle</a> or <a href="/wiki/Principal_bundle" title="Principal bundle">principal bundle</a> then so is the pullback <span class="texhtml"><i>f</i><sup>*</sup><i>E</i></span>. In the case of a principal bundle the right <a href="/wiki/Group_action_(mathematics)" class="mw-redirect" title="Group action (mathematics)">action</a> of <span class="texhtml"><i>G</i></span> on <span class="texhtml"><i>f</i><sup>*</sup><i>E</i></span> is given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,e)\cdot g=(x,e\cdot g)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>e</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>g</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>e</mi> <mo>⋅</mo> <mi>g</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,e)\cdot g=(x,e\cdot g)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0c2173b89d0e86f776ed0dafc145d0c55ea70d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.201ex; height:2.843ex;" alt="{\displaystyle (x,e)\cdot g=(x,e\cdot g)}"></span></dd></dl> <p>It then follows that the map <span class="texhtml"><i>h</i></span> covering <span class="texhtml"><i>f</i></span> is <a href="/wiki/Equivariant" class="mw-redirect" title="Equivariant">equivariant</a> and so defines a morphism of principal bundles. </p><p>In the language of <a href="/wiki/Category_theory" title="Category theory">category theory</a>, the pullback bundle construction is an example of the more general <a href="/wiki/Categorical_pullback" class="mw-redirect" title="Categorical pullback">categorical pullback</a>. As such it satisfies the corresponding <a href="/wiki/Universal_property" title="Universal property">universal property</a>. </p><p>The construction of the pullback bundle can be carried out in subcategories of the category of <a href="/wiki/Topological_spaces" class="mw-redirect" title="Topological spaces">topological spaces</a>, such as the category of <a href="/wiki/Smooth_manifold" class="mw-redirect" title="Smooth manifold">smooth manifolds</a>. The latter construction is useful in <a href="/wiki/Differential_geometry_and_topology" class="mw-redirect" title="Differential geometry and topology">differential geometry and topology</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Bundles_and_sheaves">Bundles and sheaves</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pullback_bundle&action=edit&section=3" title="Edit section: Bundles and sheaves"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Bundles may also be described by their <a href="/wiki/Sheaf_(mathematics)" title="Sheaf (mathematics)">sheaves of sections</a>. The pullback of bundles then corresponds to the <a href="/wiki/Inverse_image_functor" title="Inverse image functor">inverse image of sheaves</a>, which is a <a href="/wiki/Covariance_and_contravariance_of_functors" class="mw-redirect" title="Covariance and contravariance of functors">contravariant</a> functor. A sheaf, however, is more naturally a <a href="/wiki/Covariance_and_contravariance_of_functors" class="mw-redirect" title="Covariance and contravariance of functors">covariant</a> object, since it has a <a href="/wiki/Direct_image_sheaf" class="mw-redirect" title="Direct image sheaf">pushforward</a>, called the <a href="/wiki/Direct_image_of_a_sheaf" class="mw-redirect" title="Direct image of a sheaf">direct image of a sheaf</a>. The tension and interplay between bundles and sheaves, or inverse and direct image, can be advantageous in many areas of geometry. However, the direct image of a sheaf of sections of a bundle is <i>not</i> in general the sheaf of sections of some direct image bundle, so that although the notion of a 'pushforward of a bundle' is defined in some contexts (for example, the pushforward by a diffeomorphism), in general it is better understood in the category of sheaves, because the objects it creates cannot in general be bundles. </p> <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=Pullback_bundle&action=edit&section=4" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><a href="#CITEREFSteenrod1951">Steenrod 1951</a>, p. 47</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><a href="#CITEREFHusemoller1994">Husemoller 1994</a>, p. 18</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><a href="#CITEREFLawsonMichelsohn1989">Lawson & Michelsohn 1989</a>, p. 374</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Sources">Sources</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pullback_bundle&action=edit&section=5" title="Edit section: Sources"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><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="CITEREFSteenrod1951" class="citation book cs1"><a href="/wiki/Norman_Steenrod" title="Norman Steenrod">Steenrod, Norman</a> (1951). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/topologyoffibreb0000stee"><i>The Topology of Fibre Bundles</i></a></span>. Princeton: <a href="/wiki/Princeton_University_Press" title="Princeton University Press">Princeton University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-691-00548-6" title="Special:BookSources/0-691-00548-6"><bdi>0-691-00548-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Topology+of+Fibre+Bundles&rft.place=Princeton&rft.pub=Princeton+University+Press&rft.date=1951&rft.isbn=0-691-00548-6&rft.aulast=Steenrod&rft.aufirst=Norman&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Ftopologyoffibreb0000stee&rfr_id=info%3Asid%2Fen.wikipedia.org%3APullback+bundle" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHusemoller1994" class="citation book cs1"><a href="/wiki/Dale_Husemoller" title="Dale Husemoller">Husemoller, Dale</a> (1994). <i>Fibre Bundles</i>. Graduate Texts in Mathematics. Vol. 20 (Third ed.). New York: <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-94087-8" title="Special:BookSources/978-0-387-94087-8"><bdi>978-0-387-94087-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fibre+Bundles&rft.place=New+York&rft.series=Graduate+Texts+in+Mathematics&rft.edition=Third&rft.pub=Springer-Verlag&rft.date=1994&rft.isbn=978-0-387-94087-8&rft.aulast=Husemoller&rft.aufirst=Dale&rfr_id=info%3Asid%2Fen.wikipedia.org%3APullback+bundle" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLawsonMichelsohn1989" class="citation book cs1"><a href="/wiki/H._Blaine_Lawson" title="H. Blaine Lawson">Lawson, H. Blaine</a>; <a href="/wiki/Marie-Louise_Michelsohn" title="Marie-Louise Michelsohn">Michelsohn, Marie-Louise</a> (1989). <i>Spin Geometry</i>. <a href="/wiki/Princeton_University_Press" title="Princeton University Press">Princeton University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-08542-5" title="Special:BookSources/978-0-691-08542-5"><bdi>978-0-691-08542-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Spin+Geometry&rft.pub=Princeton+University+Press&rft.date=1989&rft.isbn=978-0-691-08542-5&rft.aulast=Lawson&rft.aufirst=H.+Blaine&rft.au=Michelsohn%2C+Marie-Louise&rfr_id=info%3Asid%2Fen.wikipedia.org%3APullback+bundle" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pullback_bundle&action=edit&section=6" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSharpe1997" class="citation book cs1">Sharpe, R. W. (1997). <i>Differential Geometry: Cartan's Generalization of Klein's Erlangen Program</i>. Graduate Texts in Mathematics. Vol. 166. New York: Springer-Verlag. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-94732-9" title="Special:BookSources/0-387-94732-9"><bdi>0-387-94732-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Differential+Geometry%3A+Cartan%27s+Generalization+of+Klein%27s+Erlangen+Program&rft.place=New+York&rft.series=Graduate+Texts+in+Mathematics&rft.pub=Springer-Verlag&rft.date=1997&rft.isbn=0-387-94732-9&rft.aulast=Sharpe&rft.aufirst=R.+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APullback+bundle" class="Z3988"></span></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Pullback_bundle&oldid=1165206921">https://en.wikipedia.org/w/index.php?title=Pullback_bundle&oldid=1165206921</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Category</a>: <ul><li><a href="/wiki/Category:Fiber_bundles" title="Category:Fiber bundles">Fiber bundles</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 13 July 2023, at 17:57<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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