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width: 0.3em;"></span> <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/10606/#Item_1" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <blockquote> <p>This entry is about base change of <a class="existingWikiWord" href="/nlab/show/slice+categories">slice categories</a>. For base change in <a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a> see at <a class="existingWikiWord" href="/nlab/show/change+of+enriching+category">change of enriching category</a>.</p> </blockquote> <hr /> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="limits_and_colimits">Limits and colimits</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/limit">limits and colimits</a></strong></p> <h2 id="1categorical">1-Categorical</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/limit">limit and colimit</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/limits+and+colimits+by+example">limits and colimits by example</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/commutativity+of+limits+and+colimits">commutativity of limits and colimits</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/small+limit">small limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/filtered+colimit">filtered colimit</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/directed+colimit">directed colimit</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/sequential+colimit">sequential colimit</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sifted+colimit">sifted colimit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connected+limit">connected limit</a>, <a class="existingWikiWord" href="/nlab/show/wide+pullback">wide pullback</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/preserved+limit">preserved limit</a>, <a class="existingWikiWord" href="/nlab/show/reflected+limit">reflected limit</a>, <a class="existingWikiWord" href="/nlab/show/created+limit">created limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/product">product</a>, <a class="existingWikiWord" href="/nlab/show/fiber+product">fiber product</a>, <a class="existingWikiWord" href="/nlab/show/base+change">base change</a>, <a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a>, <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a>, <a class="existingWikiWord" href="/nlab/show/pushout">pushout</a>, <a class="existingWikiWord" href="/nlab/show/cobase+change">cobase change</a>, <a class="existingWikiWord" href="/nlab/show/equalizer">equalizer</a>, <a class="existingWikiWord" href="/nlab/show/coequalizer">coequalizer</a>, <a class="existingWikiWord" href="/nlab/show/join">join</a>, <a class="existingWikiWord" href="/nlab/show/meet">meet</a>, <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a>, <a class="existingWikiWord" href="/nlab/show/initial+object">initial object</a>, <a class="existingWikiWord" href="/nlab/show/direct+product">direct product</a>, <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/finite+limit">finite limit</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/exact+functor">exact functor</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kan+extension">Kan extension</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Yoneda+extension">Yoneda extension</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/weighted+limit">weighted limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/end">end and coend</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fibered+limit">fibered limit</a></p> </li> </ul> <h2 id="2categorical">2-Categorical</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-limit">2-limit</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/inserter">inserter</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/isoinserter">isoinserter</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equifier">equifier</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/inverter">inverter</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/PIE-limit">PIE-limit</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-pullback">2-pullback</a>, <a class="existingWikiWord" href="/nlab/show/comma+object">comma object</a></p> </li> </ul> <h2 id="1categorical_2">(∞,1)-Categorical</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-limit">(∞,1)-limit</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-pullback">(∞,1)-pullback</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fiber+sequence">fiber sequence</a></li> </ul> </li> </ul> </li> </ul> <h3 id="modelcategorical">Model-categorical</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+Kan+extension">homotopy Kan extension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+limit">homotopy limit</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+product">homotopy product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+equalizer">homotopy equalizer</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+fiber">homotopy fiber</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+totalization">homotopy totalization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+end">homotopy end</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+colimit">homotopy colimit</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coproduct">homotopy coproduct</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coequalizer">homotopy coequalizer</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+cofiber">homotopy cofiber</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cocone">mapping cocone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+pushout">homotopy pushout</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+realization">homotopy realization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coend">homotopy coend</a></p> </li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/infinity-limits+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="topos_theory">Topos theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/topos+theory">topos theory</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Toposes">Toposes</a></li> </ul> <h2 id="background">Background</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category">category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> </li> </ul> </li> </ul> <h2 id="toposes">Toposes</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%280%2C1%29-topos">(0,1)-topos</a>, <a class="existingWikiWord" href="/nlab/show/Heyting+algebra">Heyting algebra</a>, <a class="existingWikiWord" href="/nlab/show/locale">locale</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pretopos">pretopos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topos">topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+topos">Grothendieck topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+presheaves">category of presheaves</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/presheaf">presheaf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/representable+functor">representable presheaf</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+sheaves">category of sheaves</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/site">site</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/sieve">sieve</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/coverage">coverage</a>, <a class="existingWikiWord" href="/nlab/show/Grothendieck+pretopology">pretopology</a>, <a class="existingWikiWord" href="/nlab/show/Grothendieck+topology">topology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sheafification">sheafification</a></p> </li> </ul> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quasitopos">quasitopos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/base+topos">base topos</a>, <a class="existingWikiWord" href="/nlab/show/indexed+topos">indexed topos</a></p> </li> </ul> <h2 id="toc_internal_logic">Internal Logic</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/categorical+semantics">categorical semantics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/internal+logic">internal logic</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/subobject+classifier">subobject classifier</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/natural+numbers+object">natural numbers object</a></p> </li> </ul> </li> </ul> <h2 id="topos_morphisms">Topos morphisms</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/logical+morphism">logical morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+morphism">geometric morphism</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/direct+image">direct image</a>/<a class="existingWikiWord" href="/nlab/show/inverse+image">inverse image</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/global+section">global sections</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+embedding">geometric embedding</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/surjective+geometric+morphism">surjective geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/essential+geometric+morphism">essential geometric morphism</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+connected+geometric+morphism">locally connected geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connected+geometric+morphism">connected geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/totally+connected+geometric+morphism">totally connected geometric morphism</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%C3%A9tale+geometric+morphism">étale geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/open+geometric+morphism">open geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/proper+geometric+morphism">proper geometric morphism</a>, <a class="existingWikiWord" href="/nlab/show/compact+topos">compact topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/separated+geometric+morphism">separated geometric morphism</a>, <a class="existingWikiWord" href="/nlab/show/Hausdorff+topos">Hausdorff topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+geometric+morphism">local geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bounded+geometric+morphism">bounded geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/base+change">base change</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/localic+geometric+morphism">localic geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hyperconnected+geometric+morphism">hyperconnected geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/atomic+geometric+morphism">atomic geometric morphism</a></p> </li> </ul> </li> </ul> <h2 id="extra_stuff_structure_properties">Extra stuff, structure, properties</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+locale">topological locale</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/localic+topos">localic topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/petit+topos">petit topos/gros topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+connected+topos">locally connected topos</a>, <a class="existingWikiWord" href="/nlab/show/connected+topos">connected topos</a>, <a class="existingWikiWord" href="/nlab/show/totally+connected+topos">totally connected topos</a>, <a class="existingWikiWord" href="/nlab/show/strongly+connected+topos">strongly connected topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+topos">local topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohesive+topos">cohesive topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/classifying+topos">classifying topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+topos">smooth topos</a></p> </li> </ul> <h2 id="cohomology_and_homotopy">Cohomology and homotopy</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+groups+in+an+%28infinity%2C1%29-topos">homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">model structure on simplicial presheaves</a></p> </li> </ul> <h2 id="in_higher_category_theory">In higher category theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+topos+theory">higher topos theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%280%2C1%29-topos">(0,1)-topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/%280%2C1%29-site">(0,1)-site</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-topos">2-topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-site">2-site</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-sheaf">2-sheaf</a>, <a class="existingWikiWord" href="/nlab/show/stack">stack</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-site">(∞,1)-site</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-sheaf">(∞,1)-sheaf</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stack">∞-stack</a>, <a class="existingWikiWord" href="/nlab/show/derived+stack">derived stack</a></p> </li> </ul> </li> </ul> <h2 id="theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Diaconescu%27s+theorem">Diaconescu's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Barr%27s+theorem">Barr's theorem</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/topos+theory+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <ul> <li><a href='#pullback'>Pullback</a></li> <li><a href='#in_a_fibered_category'>In a fibered category</a></li> <li><a href='#GeometricMorphism'>Base change geometric morphisms</a></li> </ul> <li><a href='#properties'>Properties</a></li> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#AlongDeloopingsOfGroupHomomorphisms'>Along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>H</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}H \to \mathbf{B}G</annotation></semantics></math></a></li> <li><a href='#AlongPointInclusionIntoBG'>Along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\ast \to \mathbf{B}G</annotation></semantics></math></a></li> <li><a href='#along__3'>Along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo stretchy="false">/</mo><mi>G</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">V/G \to \mathbf{B}G</annotation></semantics></math></a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f : X \to Y</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a> in a <a class="existingWikiWord" href="/nlab/show/category">category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> with <a class="existingWikiWord" href="/nlab/show/pullbacks">pullbacks</a>, there is an induced <a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup><mo>:</mo><mi>C</mi><mo stretchy="false">/</mo><mi>Y</mi><mo>→</mo><mi>C</mi><mo stretchy="false">/</mo><mi>X</mi></mrow><annotation encoding="application/x-tex"> f^* : C/Y \to C/X </annotation></semantics></math></div> <p>of <a class="existingWikiWord" href="/nlab/show/over-categories">over-categories</a>. This is the <em>base change</em> morphism. If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/topos">topos</a>, then this refines to an <a class="existingWikiWord" href="/nlab/show/essential+geometric+morphism">essential geometric morphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>f</mi> <mo>!</mo></msub><mo>⊣</mo><msup><mi>f</mi> <mo>*</mo></msup><mo>⊣</mo><msub><mi>f</mi> <mo>*</mo></msub><mo stretchy="false">)</mo><mo>:</mo><mi>C</mi><mo stretchy="false">/</mo><mi>X</mi><mo>→</mo><mi>C</mi><mo stretchy="false">/</mo><mi>Y</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (f_! \dashv f^* \dashv f_*) : C/X \to C/Y \,. </annotation></semantics></math></div> <p>More generally, such a triple adjunction holds whenever <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/locally+cartesian+closed+category">locally cartesian closed</a>, and indeed this <a class="existingWikiWord" href="/nlab/show/locally+cartesian+closed+category#DependentProductImpliesLocalCartesinClosure">characterises</a> locally cartesian closed categories. The <a class="existingWikiWord" href="/nlab/show/duality">dual</a> concept is <a class="existingWikiWord" href="/nlab/show/cobase+change">cobase change</a>.</p> <h2 id="definition">Definition</h2> <h3 id="pullback">Pullback</h3> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f : X \to Y</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a> in a <a class="existingWikiWord" href="/nlab/show/category">category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> with <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a>s, there is an induced <a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup><mo>:</mo><mi>C</mi><mo stretchy="false">/</mo><mi>Y</mi><mo>→</mo><mi>C</mi><mo stretchy="false">/</mo><mi>X</mi></mrow><annotation encoding="application/x-tex"> f^* : C/Y \to C/X </annotation></semantics></math></div> <p>of <a class="existingWikiWord" href="/nlab/show/over-categories">over-categories</a>. It is on objects given by <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a>/<a class="existingWikiWord" href="/nlab/show/fiber+product">fiber product</a> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>p</mi><mo>:</mo><mi>K</mi><mo>→</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>↦</mo><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mi>X</mi><msub><mo>×</mo> <mi>Y</mi></msub><mi>K</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>K</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msup><mi>p</mi> <mo>*</mo></msup></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mi>f</mi></mover></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (p : K \to Y) \mapsto \left( \array{ X \times_Y K &\to & K \\ {}^{\mathllap{p^*}}\downarrow && \downarrow \\ X &\stackrel{f}{\to}& Y } \right) \,. </annotation></semantics></math></div> <p>On morphisms, it follows from the universal property of pullback</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mi>K</mi></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mi>g</mi></mover></mtd> <mtd></mtd> <mtd><mi>K</mi><mo>′</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mi>p</mi></msub><mo>↘</mo></mtd> <mtd></mtd> <mtd><msub><mo>↙</mo> <mrow><mi>p</mi><mo>′</mo></mrow></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mo>}</mo></mrow><mo>↦</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mi>X</mi><msub><mo>×</mo> <mi>Y</mi></msub><mi>K</mi></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mrow><msup><mi>g</mi> <mo>*</mo></msup></mrow></mover></mtd> <mtd></mtd> <mtd><mi>X</mi><msub><mo>×</mo> <mi>Y</mi></msub><mi>K</mi><mo>′</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mrow><msup><mi>p</mi> <mo>*</mo></msup></mrow></msub><mo>↘</mo></mtd> <mtd></mtd> <mtd><msub><mo>↙</mo> <mrow><mi>p</mi><msup><mo>′</mo> <mo>*</mo></msup></mrow></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>X</mi></mtd></mtr></mtable></mrow><mo>}</mo></mrow></mrow><annotation encoding="application/x-tex"> \left\lbrace \array{ K &&\stackrel{g}{\to}&& K' \\ & {}_p \searrow && \swarrow_{p'} \\ && Y } \right\rbrace \mapsto \left\lbrace \array{ X \times_Y K &&\stackrel{g^*}{\to}&& X \times_Y K' \\ & {}_{p^*} \searrow && \swarrow_{p'^*} \\ && X } \right\rbrace </annotation></semantics></math></div> <p>by observing that this square commutes</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>X</mi><msub><mo>×</mo> <mi>Y</mi></msub><mi>K</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mrow></mrow> <mrow><msup><mi>p</mi> <mo>*</mo></msup></mrow></msup><mo>↙</mo></mtd> <mtd></mtd> <mtd><msup><mo>↘</mo> <mrow><mi>g</mi><mo>∘</mo><msub><mi>p</mi> <mi>K</mi></msub></mrow></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>X</mi></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>K</mi><mo>′</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msub><mrow></mrow> <mi>f</mi></msub><mo>↘</mo></mtd> <mtd></mtd> <mtd><msub><mo>↙</mo> <mrow><mi>p</mi><mo>′</mo></mrow></msub></mtd> <mtd></mtd> <mtd></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>Y</mi></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ &&&& X \times_Y K \\& && {}^{p^*}\swarrow && \searrow^{g \circ p_K} \\ && X &&&& K' \\ & && {}_f\searrow & & \swarrow_{p'} && \\ &&&& Y &&&& } \,. </annotation></semantics></math></div> <h3 id="in_a_fibered_category">In a fibered category</h3> <p>The concept of base change generalises from this case to other <a class="existingWikiWord" href="/nlab/show/fibered+category">fibred categories</a>.</p> <h3 id="GeometricMorphism">Base change geometric morphisms</h3> <div class="num_prop" id="BaseChangeIsEssentialGeometricMorphism"> <h6 id="proposition">Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/topos">topos</a> (or <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a>, etc.) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f : X \to Y</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>, then base change induces an <a class="existingWikiWord" href="/nlab/show/essential+geometric+morphism">essential geometric morphism</a> between over-toposes/<a class="existingWikiWord" href="/nlab/show/over-%28%E2%88%9E%2C1%29-topos">over-(∞,1)-topos</a>es</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mi>f</mi></munder><mo>⊣</mo><msup><mi>f</mi> <mo>*</mo></msup><mo>⊣</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mi>f</mi></munder><mo stretchy="false">)</mo><mo>:</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">/</mo><mi>X</mi><mover><mover><munder><mo>→</mo><mrow><msub><mi>f</mi> <mo>*</mo></msub></mrow></munder><mover><mo>←</mo><mrow><msup><mi>f</mi> <mo>*</mo></msup></mrow></mover></mover><mover><mo>→</mo><mrow><msub><mi>f</mi> <mo>!</mo></msub></mrow></mover></mover><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">/</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex"> (\sum_f \dashv f^* \dashv \prod_f) : \mathbf{H}/X \stackrel{\overset{f_!}{\to}}{\stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}}} \mathbf{H}/Y </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mo>!</mo></msub></mrow><annotation encoding="application/x-tex">f_!</annotation></semantics></math> is given by postcomposition with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">f^*</annotation></semantics></math> by <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math>.</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>That we have <a class="existingWikiWord" href="/nlab/show/adjoint+functor">adjoint functor</a>s/<a class="existingWikiWord" href="/nlab/show/adjoint+%28%E2%88%9E%2C1%29-functor">adjoint (∞,1)-functor</a>s <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>f</mi> <mo>!</mo></msub><mo>⊣</mo><msup><mi>f</mi> <mo>*</mo></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(f_! \dashv f^*)</annotation></semantics></math> follows directly from the universal property of the pullback. The fact that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">f^*</annotation></semantics></math> has a further <a class="existingWikiWord" href="/nlab/show/right+adjoint">right adjoint</a> is due to the fact that it preserves all small <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a>s/<a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-colimit">(∞,1)-colimit</a>s by the fact that in a topos we have <a class="existingWikiWord" href="/nlab/show/universal+colimits">universal colimits</a> and then by the <a class="existingWikiWord" href="/nlab/show/adjoint+functor+theorem">adjoint functor theorem</a>/<a class="existingWikiWord" href="/nlab/show/adjoint+%28%E2%88%9E%2C1%29-functor+theorem">adjoint (∞,1)-functor theorem</a>.</p> </div> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>The (<a class="existingWikiWord" href="/nlab/show/comonad">co-</a>)<a class="existingWikiWord" href="/nlab/show/monads">monads</a> induced by the <a class="existingWikiWord" href="/nlab/show/adjoint+triple">adjoint triple</a> in prop. <a class="maruku-ref" href="#BaseChangeIsEssentialGeometricMorphism"></a> have special names in some contexts:</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mo>*</mo></msub><msup><mi>f</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">f_\ast f^\ast</annotation></semantics></math> is also called the <a class="existingWikiWord" href="/nlab/show/function+monad">function monad</a> (or “<a class="existingWikiWord" href="/nlab/show/reader+monad">reader monad</a>”, see at <em><a class="existingWikiWord" href="/nlab/show/monad+%28in+computer+science%29">monad (in computer science)</a></em>).</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mo>!</mo></msub><msup><mi>f</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">f_! f^\ast</annotation></semantics></math> is also called the “<a class="existingWikiWord" href="/nlab/show/writer+comonad">writer comonad</a>” (in computer science)</p> </li> <li> <p>in <a class="existingWikiWord" href="/nlab/show/modal+type+theory">modal type theory</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup><msub><mi>f</mi> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">f^\ast f_\ast</annotation></semantics></math> is <em><a class="existingWikiWord" href="/nlab/show/necessity">necessity</a></em> while <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup><msub><mi>f</mi> <mo>!</mo></msub></mrow><annotation encoding="application/x-tex">f^\ast f_!</annotation></semantics></math> is <em><a class="existingWikiWord" href="/nlab/show/possibility">possibility</a></em>.</p> </li> </ul> </div> <div class="num_prop"> <h6 id="proposition_2">Proposition</h6> <p>Here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">f^\ast</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/cartesian+closed+functor">cartesian closed functor</a>, hence base change of toposes constitutes a cartesian <a class="existingWikiWord" href="/nlab/show/Wirthm%C3%BCller+context">Wirthmüller context</a>.</p> </div> <p>See at <em><a class="existingWikiWord" href="/nlab/show/cartesian+closed+functor">cartesian closed functor</a></em> for the proof.</p> <div class="num_prop"> <h6 id="proposition_3">Proposition</h6> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">f^*</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/logical+functor">logical functor</a>. Hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>f</mi> <mo>*</mo></msup><mo>⊣</mo><msub><mi>f</mi> <mo>*</mo></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(f^* \dashv f_*)</annotation></semantics></math> is also an <a class="existingWikiWord" href="/nlab/show/atomic+geometric+morphism">atomic geometric morphism</a>.</p> </div> <p>This appears for instance as (<a href="#MacLaneMoerdijk">MacLaneMoerdijk, theorem IV.7.2</a>).</p> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>By prop. <a class="maruku-ref" href="#BaseChangeIsEssentialGeometricMorphism"></a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">f^*</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/right+adjoint">right adjoint</a> and hence preserves all <a class="existingWikiWord" href="/nlab/show/limit">limit</a>s, in particular <a class="existingWikiWord" href="/nlab/show/finite+limit">finite limit</a>s.</p> <p>Notice that the <a class="existingWikiWord" href="/nlab/show/subobject+classifier">subobject classifier</a> of an <a class="existingWikiWord" href="/nlab/show/over+topos">over topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">/</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\mathbf{H}/X</annotation></semantics></math> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>p</mi> <mn>2</mn></msub><mo>:</mo><msub><mi>Ω</mi> <mstyle mathvariant="bold"><mi>H</mi></mstyle></msub><mo>×</mo><mi>X</mi><mo>→</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(p_2 : \Omega_{\mathbf{H}} \times X \to X)</annotation></semantics></math>. This <a class="existingWikiWord" href="/nlab/show/product">product</a> is preserved by the <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> by which <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">f^*</annotation></semantics></math> acts, hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">f^*</annotation></semantics></math> preserves the subobject classifier.</p> <p>To show that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">f^*</annotation></semantics></math> is logical it therefore remains to show that it also preserves <a class="existingWikiWord" href="/nlab/show/exponential+object">exponential object</a>s. (…)</p> </div> <div class="num_defn"> <h6 id="definition_2">Definition</h6> <p>A (necessarily essential and atomic) geometric morphism of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>f</mi> <mo>*</mo></msup><mo>⊣</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mi>f</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(f^* \dashv \prod_f)</annotation></semantics></math> is called the <strong>base change geometric morphism</strong> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/right+adjoint">right adjoint</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mo>*</mo></msub><mo>=</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mi>f</mi></msub></mrow><annotation encoding="application/x-tex">f_* = \prod_f</annotation></semantics></math> is also called the <a class="existingWikiWord" href="/nlab/show/dependent+product">dependent product</a> relative to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mo>!</mo></msub><mo>=</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mi>f</mi></msub></mrow><annotation encoding="application/x-tex">f_! = \sum_f</annotation></semantics></math> is also called the <a class="existingWikiWord" href="/nlab/show/dependent+sum">dependent sum</a> relative to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math>.</p> <p>In the case <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>=</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">Y = *</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a>, the base change geometric morphism is also called an <strong><a class="existingWikiWord" href="/nlab/show/etale+geometric+morphism">etale geometric morphism</a></strong>. See there for more details</p> </div> <h2 id="properties">Properties</h2> <div class="num_prop"> <h6 id="proposition_4">Proposition</h6> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/locally+cartesian+closed+category">locally cartesian closed category</a> then for every morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f \colon X \to Y</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/inverse+image">inverse image</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup><mo lspace="verythinmathspace">:</mo><msub><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mi>Y</mi></mrow></msub><mo>→</mo><msub><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow></msub></mrow><annotation encoding="application/x-tex">f^* \colon \mathcal{C}_{/Y} \to \mathcal{C}_{/X}</annotation></semantics></math> of the base change is a <a class="existingWikiWord" href="/nlab/show/cartesian+closed+functor">cartesian closed functor</a>.</p> </div> <p>See at <em><a href="cartesian%20closed%20functor#Examples">cartesian closed functor – Examples</a></em> for a proof.</p> <h2 id="examples">Examples</h2> <h3 id="AlongDeloopingsOfGroupHomomorphisms">Along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>H</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}H \to \mathbf{B}G</annotation></semantics></math></h3> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> an group object in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> (an <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a>), then the <a class="existingWikiWord" href="/nlab/show/slice+%28%E2%88%9E%2C1%29-topos">slice (∞,1)-topos</a> over its <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> may be identified with the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/%E2%88%9E-actions">∞-actions</a> (see there for more):</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Act</mi> <mi>G</mi></msub><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo><mo>≃</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Act_G(\mathbf{H}) \simeq \mathbf{H}_{/\mathbf{B}G} \,. </annotation></semantics></math></div> <p>Under this identification, then left and right base change long a morphism of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>H</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}H \to \mathbf{B}G</annotation></semantics></math> (corresponding to an <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a> homomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo>→</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">H \to G</annotation></semantics></math>) corresponds to forming <a class="existingWikiWord" href="/nlab/show/induced+representations">induced representations</a> and <a class="existingWikiWord" href="/nlab/show/coinduced+representations">coinduced representations</a>, respectively.</p> <h3 id="AlongPointInclusionIntoBG">Along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\ast \to \mathbf{B}G</annotation></semantics></math></h3> <p>As the special case of the <a href="#AlongDeloopingsOfGroupHomomorphisms">above</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">H = 1</annotation></semantics></math> the trivial group we obtain the following:</p> <div class="num_prop" id="CyclicLoopSpace"> <h6 id="proposition_5">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> be any <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> be a group object in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> (an <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a>). Then the base change along the canonical point inclusion</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>i</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mo>*</mo><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex"> i \;\colon\; \ast \to \mathbf{B}G </annotation></semantics></math></div> <p>into the <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> takes the following form:</p> <p>There is a pair of <a class="existingWikiWord" href="/nlab/show/adjoint+%E2%88%9E-functors">adjoint ∞-functors</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><munderover><mo>⊥</mo><munder><mo>⟶</mo><mrow><msub><mi>i</mi> <mo>*</mo></msub><mo>≃</mo><mo stretchy="false">[</mo><mi>G</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo><mo stretchy="false">/</mo><mi>G</mi></mrow></munder><mover><mo>⟵</mo><mrow><msup><mi>i</mi> <mo>*</mo></msup><mo>≃</mo><mi>hofib</mi></mrow></mover></munderover><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow></msub><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \mathbf{H} \underoverset { \underset{i_\ast \simeq [G,-]/G}{\longrightarrow}} { \overset{i^\ast \simeq hofib}{\longleftarrow}} {\bot} \mathbf{H}_{/\mathbf{B}G} \,, </annotation></semantics></math></div> <p>where</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>hofib</mi></mrow><annotation encoding="application/x-tex">hofib</annotation></semantics></math> denotes the operation of taking the <a class="existingWikiWord" href="/nlab/show/homotopy+fiber">homotopy fiber</a> of a map to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math> over the canonical basepoint;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>G</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[G,-]</annotation></semantics></math> denotes the <a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>G</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo><mo stretchy="false">/</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">[G,-]/G</annotation></semantics></math> denotes the <a class="existingWikiWord" href="/nlab/show/homotopy+quotient">homotopy quotient</a> by the <a class="existingWikiWord" href="/nlab/show/conjugation+action">conjugation</a> <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> equipped with its canonical <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a> by left multiplication and the argument regarded as equipped with its trivial <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-action</p> <p>(for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>=</mo><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">G = S^1</annotation></semantics></math> then this is the <a class="existingWikiWord" href="/nlab/show/cyclic+loop+space">cyclic loop space</a> construction).</p> </li> </ul> <p>Hence for</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>X</mi><mo stretchy="false">^</mo></mover><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\hat X \to X</annotation></semantics></math> a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/coefficient">coefficient</a> object, such as for some <a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential</a> <a class="existingWikiWord" href="/nlab/show/generalized+cohomology+theory">generalized cohomology theory</a></p> </li> </ul> <p>then there is a <a class="existingWikiWord" href="/nlab/show/natural+equivalence">natural equivalence</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><munder><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mover><mi>X</mi><mo stretchy="false">^</mo></mover><mspace width="thickmathspace"></mspace><mo>,</mo><mspace width="thickmathspace"></mspace><mi>A</mi><mo stretchy="false">)</mo></mrow><mo>⏟</mo></munder><mfrac linethickness="0"><mrow><mtext>original</mtext></mrow><mrow><mtext>fluxes</mtext></mrow></mfrac></munder><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><munderover><mo>≃</mo><munder><mo>⟵</mo><mi>oxidation</mi></munder><mover><mo>⟶</mo><mi>reduction</mi></mover></munderover><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><munder><munder><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mspace width="thickmathspace"></mspace><mo>,</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">[</mo><mi>G</mi><mo>,</mo><mi>A</mi><mo stretchy="false">]</mo><mo stretchy="false">/</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><mo>⏟</mo></munder><mfrac linethickness="0"><mrow><mtext>doubly</mtext></mrow><mrow><mfrac linethickness="0"><mrow><mtext>dimensionally reduced</mtext></mrow><mrow><mtext>fluxes</mtext></mrow></mfrac></mrow></mfrac></munder></mrow><annotation encoding="application/x-tex"> \underset{ \text{original} \atop \text{fluxes} }{ \underbrace{ \mathbf{H}(\hat X\;,\; A) } } \;\; \underoverset {\underset{oxidation}{\longleftarrow}} {\overset{reduction}{\longrightarrow}} {\simeq} \;\; \underset{ \text{doubly} \atop { \text{dimensionally reduced} \atop \text{fluxes} } }{ \underbrace{ \mathbf{H}(X \;,\; [G,A]/G) } } </annotation></semantics></math></div> <p>given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><mover><mi>X</mi><mo stretchy="false">^</mo></mover><mo>⟶</mo><mi>A</mi><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>↔</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd></mtd> <mtd><mo>⟶</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">[</mo><mi>G</mi><mo>,</mo><mi>A</mi><mo stretchy="false">]</mo><mo stretchy="false">/</mo><mi>G</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd><mo>↙</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd></mtr></mtable></mrow><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> \left( \hat X \longrightarrow A \right) \;\;\; \leftrightarrow \;\;\; \left( \array{ X && \longrightarrow && [G,A]/G \\ & \searrow && \swarrow \\ && \mathbf{B}G } \right) </annotation></semantics></math></div></div> <div class="proof" id="DimensionalReductionAbstractly"> <h6 id="proof_3">Proof</h6> <p>The statement that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>i</mi> <mo>*</mo></msup><mo>≃</mo><mi>hofib</mi></mrow><annotation encoding="application/x-tex">i^\ast \simeq hofib</annotation></semantics></math> follows immediately by the definitions. What we need to see is that the <a class="existingWikiWord" href="/nlab/show/dependent+product">dependent product</a> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math> is given as claimed.</p> <p>To that end, first observe that the <a class="existingWikiWord" href="/nlab/show/conjugation+action">conjugation action</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>G</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[G,X]</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom</a> in the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/%E2%88%9E-actions">∞-actions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Act</mi> <mi>G</mi></msub><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Act_G(\mathbf{H})</annotation></semantics></math>. Under the <a class="existingWikiWord" href="/nlab/show/equivalence+of+%28%E2%88%9E%2C1%29-categories">equivalence of (∞,1)-categories</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Act</mi> <mi>G</mi></msub><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo><mo>≃</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow></msub></mrow><annotation encoding="application/x-tex"> Act_G(\mathbf{H}) \simeq \mathbf{H}_{/\mathbf{B}G} </annotation></semantics></math></div> <p>(from <a href="https://ncatlab.org/schreiber/show/Principal+%E2%88%9E-bundles+--+theory%2C+presentations+and+applications">NSS 12</a>) then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> with its canonical <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo>*</mo><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\ast \to \mathbf{B}G)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> with the trivial action is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X \times \mathbf{B}G \to \mathbf{B}G)</annotation></semantics></math>.</p> <p>Hence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>G</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo><mo stretchy="false">/</mo><mi>G</mi><mo>≃</mo><mo stretchy="false">[</mo><mo>*</mo><mo>,</mo><mi>X</mi><mo>×</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><msub><mo stretchy="false">]</mo> <mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow></msub><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>∈</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [G,X]/G \simeq [\ast, X \times \mathbf{B}G]_{\mathbf{B}G} \;\;\;\;\; \in \mathbf{H}_{/\mathbf{B}G} \,. </annotation></semantics></math></div> <p>So far this is the very definition of what <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>G</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo><mo stretchy="false">/</mo><mi>G</mi><mo>∈</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow></msub></mrow><annotation encoding="application/x-tex">[G,X]/G \in \mathbf{H}_{/\mathbf{B}G}</annotation></semantics></math> is to mean in the first place.</p> <p>But now since the <a class="existingWikiWord" href="/nlab/show/slice+%28%E2%88%9E%2C1%29-topos">slice (∞,1)-topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbf{H}_{/\mathbf{B}G}</annotation></semantics></math> is itself <a class="existingWikiWord" href="/nlab/show/cartesian+closed+%28infinity%2C1%29-category">cartesian closed</a>, via</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>E</mi><msub><mo>×</mo> <mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>⊣</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo stretchy="false">[</mo><mi>E</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mo stretchy="false">]</mo> <mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow></msub></mrow><annotation encoding="application/x-tex"> E \times_{\mathbf{B}G}(-) \;\;\; \dashv \;\;\; [E,-]_{\mathbf{B}G} </annotation></semantics></math></div> <p>it is immediate that there is the following sequence of <a class="existingWikiWord" href="/nlab/show/natural+equivalences">natural equivalences</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow></msub><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mo stretchy="false">[</mo><mi>G</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo><mo stretchy="false">/</mo><mi>G</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>≃</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow></msub><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mo stretchy="false">[</mo><mo>*</mo><mo>,</mo><mi>X</mi><mo>×</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><msub><mo stretchy="false">]</mo> <mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow></msub><mo stretchy="false">(</mo><mi>Y</mi><msub><mo>×</mo> <mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow></msub><mo>*</mo><mo>,</mo><munder><munder><mrow><mi>X</mi><mo>×</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><mo>⏟</mo></munder><mrow><msup><mi>p</mi> <mo>*</mo></msup><mi>X</mi></mrow></munder><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><munder><munder><mrow><msub><mi>p</mi> <mo>!</mo></msub><mo stretchy="false">(</mo><mi>Y</mi><msub><mo>×</mo> <mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow></msub><mo>*</mo><mo stretchy="false">)</mo></mrow><mo>⏟</mo></munder><mrow><mi>hofib</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow></munder><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>hofib</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \mathbf{H}_{/\mathbf{B}G}(Y, [G,X]/G) & \simeq \mathbf{H}_{/\mathbf{B}G}(Y, [\ast, X \times \mathbf{B}G]_{\mathbf{B}G}) \\ & \simeq \mathbf{H}_{/\mathbf{B}G}( Y \times_{\mathbf{B}G} \ast, \underset{p^\ast X}{\underbrace{X \times \mathbf{B}G }} ) \\ & \simeq \mathbf{H}( \underset{hofib(Y)}{\underbrace{p_!(Y \times_{\mathbf{B}G} \ast)}}, X ) \\ & \simeq \mathbf{H}(hofib(Y),X) \end{aligned} </annotation></semantics></math></div> <p>Here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo lspace="verythinmathspace">:</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>→</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">p \colon \mathbf{B}G \to \ast</annotation></semantics></math> denotes the terminal morphism and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mo>!</mo></msub><mo>⊣</mo><msup><mi>p</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">p_! \dashv p^\ast</annotation></semantics></math> denotes the <a class="existingWikiWord" href="/nlab/show/base+change">base change</a> along it.</p> </div> <p>See also at <em><a class="existingWikiWord" href="/nlab/show/double+dimensional+reduction">double dimensional reduction</a></em> for more on this.</p> <h3 id="along__3">Along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo stretchy="false">/</mo><mi>G</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">V/G \to \mathbf{B}G</annotation></semantics></math></h3> <p>More generally:</p> <div class="num_prop" id="RightBaseChangeAlongUniversalFiberBundleProjection"> <h6 id="proposition_6">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> be an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>∈</mo><mi>Grp</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G \in Grp(\mathbf{H})</annotation></semantics></math> an <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a>.</p> <p>Let moreover <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>∈</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">V \in \mathbf{H}</annotation></semantics></math> be an object equipped with a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math>, equivalently (by the discussion there) a <a class="existingWikiWord" href="/nlab/show/homotopy+fiber+sequence">homotopy fiber sequence</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>V</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>V</mi><mo stretchy="false">/</mo><mi>G</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>p</mi> <mi>ρ</mi></msub></mrow></mover></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ V \\ \downarrow \\ V/G & \overset{p_\rho}{\longrightarrow}& \mathbf{B}G } </annotation></semantics></math></div> <p>Then</p> <ol> <li> <p>pullback along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mi>ρ</mi></msub></mrow><annotation encoding="application/x-tex">p_\rho</annotation></semantics></math> is the operation that assigns to a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">c \colon X \to \mathbf{B}G</annotation></semantics></math> the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/fiber+%E2%88%9E-bundle">fiber ∞-bundle</a> which is <a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundle">associated</a> via <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math> to the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>P</mi> <mi>c</mi></msub></mrow><annotation encoding="application/x-tex">P_c</annotation></semantics></math> classified by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>p</mi> <mi>ρ</mi></msub><msup><mo stretchy="false">)</mo> <mo>*</mo></msup><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>c</mi><mo>↦</mo><msub><mi>P</mi> <mi>c</mi></msub><msub><mo>×</mo> <mi>G</mi></msub><mi>V</mi></mrow><annotation encoding="application/x-tex"> (p_\rho)^\ast \;\colon\; c \mapsto P_c \times_G V </annotation></semantics></math></div></li> <li> <p>the right base change along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mi>ρ</mi></msub></mrow><annotation encoding="application/x-tex">p_\rho</annotation></semantics></math> is given on objects of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">/</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \times (V/G)</annotation></semantics></math> by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>p</mi> <mi>ρ</mi></msub><msub><mo stretchy="false">)</mo> <mo>*</mo></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>×</mo><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">/</mo><mi>G</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>↦</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">[</mo><mi>V</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo><mo stretchy="false">/</mo><mi>G</mi></mrow><annotation encoding="application/x-tex"> (p_\rho)_\ast \;\colon\; X \times (V/G) \;\mapsto\; [V,X]/G </annotation></semantics></math></div></li> </ol> </div> <div class="proof"> <h6 id="proof_4">Proof</h6> <p>The first statement is <a href="https://ncatlab.org/schreiber/show/Principal+%E2%88%9E-bundles+--+theory%2C+presentations+and+applications">NSS 12, prop. 4.6</a>.</p> <p>The second statement follows as in the proof of prop. <a class="maruku-ref" href="#CyclicLoopSpace"></a>: Let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mi>Y</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>c</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mo>∈</mo><mspace width="thickmathspace"></mspace><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow></msub></mrow><annotation encoding="application/x-tex"> \left( \array{ Y \\ \downarrow^{\mathrlap{c}} \\ \mathbf{B}G } \right) \;\in\; \mathbf{H}_{/\mathbf{B}G} </annotation></semantics></math></div> <p>be any object, then there is the following sequence of <a class="existingWikiWord" href="/nlab/show/natural+equivalences">natural equivalences</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow></msub><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mo stretchy="false">[</mo><mi>V</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo><mo stretchy="false">/</mo><mi>G</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>≃</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow></msub><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mo stretchy="false">[</mo><mi>V</mi><mo stretchy="false">/</mo><mi>G</mi><mo>,</mo><mi>X</mi><mo>×</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><msub><mo stretchy="false">]</mo> <mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow></msub><mo stretchy="false">(</mo><mi>Y</mi><msub><mo>×</mo> <mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow></msub><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">/</mo><mi>G</mi><mo stretchy="false">)</mo><mo>,</mo><munder><munder><mrow><mi>X</mi><mo>×</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><mo>⏟</mo></munder><mrow><msup><mi>p</mi> <mo>*</mo></msup><mi>X</mi></mrow></munder><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow></msub><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msub><mi>p</mi> <mi>ρ</mi></msub><msub><mo stretchy="false">)</mo> <mo>!</mo></msub><mo stretchy="false">(</mo><msub><mi>P</mi> <mi>c</mi></msub><msub><mo>×</mo> <mi>G</mi></msub><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">/</mo><mi>G</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msup><mi>p</mi> <mo>*</mo></msup><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">/</mo><mi>G</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>P</mi> <mi>c</mi></msub><msub><mo>×</mo> <mi>G</mi></msub><mi>V</mi><mo>,</mo><mo stretchy="false">(</mo><msub><mi>p</mi> <mi>ρ</mi></msub><msup><mo stretchy="false">)</mo> <mo>*</mo></msup><msup><mi>p</mi> <mo>*</mo></msup><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">/</mo><mi>G</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>P</mi> <mi>c</mi></msub><msub><mo>×</mo> <mi>G</mi></msub><mi>V</mi><mo>,</mo><mi>X</mi><mo>×</mo><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">/</mo><mi>G</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \mathbf{H}_{/\mathbf{B}G}(Y, [V,X]/G) & \simeq \mathbf{H}_{/\mathbf{B}G}(Y, [V/G, X \times \mathbf{B}G]_{\mathbf{B}G}) \\ & \simeq \mathbf{H}_{/\mathbf{B}G}( Y \times_{\mathbf{B}G} (V/G), \underset{p^\ast X}{\underbrace{X \times \mathbf{B}G }} ) \\ & \simeq \mathbf{H}_{/\mathbf{B}G} ( (p_\rho)_!( P_c \times_G (V/G) ), p^\ast X ) \\ & \simeq \mathbf{H}_{/(V/G)} ( P_c \times_G V, (p_\rho)^\ast p^\ast X ) \\ & \simeq \mathbf{H}_{/(V/G)}(P_c \times_G V, X \times (V/G)) \end{aligned} </annotation></semantics></math></div> <p>where again</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo lspace="verythinmathspace">:</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>→</mo><mo>*</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> p \colon \mathbf{B}G \to \ast \,. </annotation></semantics></math></div></div> <div class="num_example" id="SymmetricPowers"> <h6 id="example">Example</h6> <p><strong>(symmetric powers)</strong></p> <p>Let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>=</mo><mi>Σ</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>Grp</mi><mo stretchy="false">(</mo><mi>Set</mi><mo stretchy="false">)</mo><mo>↪</mo><mi>Grp</mi><mo stretchy="false">(</mo><mn>∞</mn><mi>Grpd</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mi>LConst</mi></mover><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex"> G = \Sigma(n) \in Grp(Set) \hookrightarrow Grp(\infty Grpd) \overset{LConst}{\longrightarrow} \mathbf{H} </annotation></semantics></math></div> <p>be the <a class="existingWikiWord" href="/nlab/show/symmetric+group">symmetric group</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> elements, and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>=</mo><mo stretchy="false">{</mo><mn>1</mn><mo>,</mo><mi>⋯</mi><mo>,</mo><mi>n</mi><mo stretchy="false">}</mo><mo>∈</mo><mi>Set</mi><mo>↪</mo><mn>∞</mn><mi>Grpd</mi><mover><mo>⟶</mo><mi>LConst</mi></mover><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex"> V = \{1, \cdots, n\} \in Set \hookrightarrow \infty Grpd \overset{LConst}{\longrightarrow} \mathbf{H} </annotation></semantics></math></div> <p>the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-element <a class="existingWikiWord" href="/nlab/show/set">set</a> (<a class="existingWikiWord" href="/nlab/show/h-set">h-set</a>) equipped with the canonical <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Sigma(n)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/action">action</a>. Then prop. <a class="maruku-ref" href="#RightBaseChangeAlongUniversalFiberBundleProjection"></a> says that right base change of any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>p</mi> <mi>ρ</mi> <mo>*</mo></msubsup><msup><mi>p</mi> <mo>*</mo></msup><mi>X</mi></mrow><annotation encoding="application/x-tex">p_\rho^\ast p^\ast X</annotation></semantics></math> along</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mn>1</mn><mo>,</mo><mi>⋯</mi><mo>,</mo><mi>n</mi><mo stretchy="false">}</mo><mo stretchy="false">/</mo><mi>Σ</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>⟶</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Σ</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \{1, \cdots, n\}/\Sigma(n) \longrightarrow \mathbf{B}\Sigma(n) </annotation></semantics></math></div> <p>is equivalently the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>th <a class="existingWikiWord" href="/nlab/show/symmetric+power">symmetric power</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo stretchy="false">{</mo><mn>1</mn><mo>,</mo><mi>⋯</mi><mo>,</mo><mi>n</mi><mo stretchy="false">}</mo><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo><mo stretchy="false">/</mo><mi>Σ</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>≃</mo><mo stretchy="false">(</mo><msup><mi>X</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>Σ</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [\{1,\cdots, n\},X]/\Sigma(n) \simeq (X^n)/\Sigma(n) \,. </annotation></semantics></math></div></div> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><strong>base change</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/dependent+sum">dependent sum</a>, <a class="existingWikiWord" href="/nlab/show/dependent+product">dependent product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dependent+sum+type">dependent sum type</a>, <a class="existingWikiWord" href="/nlab/show/dependent+product+type">dependent product type</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/necessity">necessity</a>, <a class="existingWikiWord" href="/nlab/show/possibility">possibility</a>, <a class="existingWikiWord" href="/nlab/show/reader+monad">reader monad</a>, <a class="existingWikiWord" href="/nlab/show/writer+comonad">writer comonad</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/proper+base+change+theorem">proper base change theorem</a></p> </li> <li> <p>Base change geometric morphisms may be interpreted in terms of <a class="existingWikiWord" href="/nlab/show/fiber+integration">fiber integration</a>. See <a class="existingWikiWord" href="/nlab/show/integral+transforms+on+sheaves">integral transforms on sheaves</a> for more on this.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/change+of+enriching+category">change of enriching category</a></p> </li> </ul> <div> <p><strong>Notions of pullback:</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/pullback">pullback</a>, <a class="existingWikiWord" href="/nlab/show/fiber+product">fiber product</a> (<a class="existingWikiWord" href="/nlab/show/limit">limit</a> over a <a class="existingWikiWord" href="/nlab/show/cospan">cospan</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/wide+pullback">wide pullback</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/lax+pullback">lax pullback</a>, <a class="existingWikiWord" href="/nlab/show/comma+object">comma object</a> (<a class="existingWikiWord" href="/nlab/show/lax+limit">lax limit</a> over a cospan)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-pullback">(∞,1)-pullback</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a>, (<a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-limit">(∞,1)-limit</a> over a cospan)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/base+change">base change</a>, <a class="existingWikiWord" href="/nlab/show/context+extension">context extension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pullback+bundle">pullback bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/contravariant+functor">contravariant functor</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/pullback+in+cohomology">pullback in cohomology</a>, <a class="existingWikiWord" href="/nlab/show/d-invariant">d-invariant</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pullback+of+differential+forms">pullback of differential forms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pullback+of+a+distribution">pullback of a distribution</a></p> </li> </ul> </li> </ul> </div> <h2 id="references">References</h2> <p>A general discussion that applies (also) to <a class="existingWikiWord" href="/nlab/show/enriched+categories">enriched categories</a> and <a class="existingWikiWord" href="/nlab/show/internal+categories">internal categories</a> is in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Dominic+Verity">Dominic Verity</a>, <em>Enriched categories, internal categories and change of base</em> Ph.D. thesis, Cambridge University (1992), reprinted as Reprints in Theory and Applications of Categories, No. 20 (2011) pp 1-266 (<a href="http://www.tac.mta.ca/tac/reprints/articles/20/tr20abs.html">TAC</a>)</li> </ul> <p>Discussion in the context of <a class="existingWikiWord" href="/nlab/show/topos+theory">topos theory</a> is around example A.4.1.2 of</p> <ul> <li id="Johnstone"><a class="existingWikiWord" href="/nlab/show/Peter+Johnstone">Peter Johnstone</a>, <em><a class="existingWikiWord" href="/nlab/show/Sketches+of+an+Elephant">Sketches of an Elephant</a></em></li> </ul> <p>and around theorem IV.7.2 in</p> <ul> <li id="MacLaneMoerdijk"><a class="existingWikiWord" href="/nlab/show/Saunders+MacLane">Saunders MacLane</a>, <a class="existingWikiWord" href="/nlab/show/Ieke+Moerdijk">Ieke Moerdijk</a>, <em><a class="existingWikiWord" href="/nlab/show/Sheaves+in+Geometry+and+Logic">Sheaves in Geometry and Logic</a></em></li> </ul> <p>Discussion in the context of <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-topos+theory">(infinity,1)-topos theory</a> is in section 6.3.5 of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, <em><a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">Higher Topos Theory</a></em></li> </ul> <p>See also</p> <ul> <li>A. Carboni, G. Kelly, R. Wood, <em>A 2-categorical approach to change of base and geometric morphisms I</em> (<a href="http://archive.numdam.org/ARCHIVE/CTGDC/CTGDC_1991__32_1/CTGDC_1991__32_1_47_0/CTGDC_1991__32_1_47_0.pdf">numdam</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on July 12, 2022 at 08:10:36. 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