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value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="homotopy_theory">Homotopy theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>, <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+theory">(∞,1)-category theory</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a></strong></p> <p>flavors: <a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable</a>, <a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+theory">equivariant</a>, <a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rational</a>, <a class="existingWikiWord" href="/nlab/show/p-adic+homotopy+theory">p-adic</a>, <a class="existingWikiWord" href="/nlab/show/proper+homotopy+theory">proper</a>, <a class="existingWikiWord" href="/nlab/show/geometric+homotopy+theory">geometric</a>, <a class="existingWikiWord" href="/nlab/show/cohesive+homotopy+theory">cohesive</a>, <a class="existingWikiWord" href="/nlab/show/directed+homotopy+theory">directed</a>…</p> <p>models: <a class="existingWikiWord" href="/nlab/show/topological+homotopy+theory">topological</a>, <a class="existingWikiWord" href="/nlab/show/simplicial+homotopy+theory">simplicial</a>, <a class="existingWikiWord" href="/nlab/show/localic+homotopy+theory">localic</a>, …</p> <p>see also <strong><a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a></strong></p> <p><strong>Introductions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Topology+--+2">Introduction to Basic Homotopy Theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Homotopy+Theory">Introduction to Abstract Homotopy Theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+homotopy+types">geometry of physics – homotopy types</a></p> </li> </ul> <p><strong>Definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a>, <a class="existingWikiWord" href="/nlab/show/higher+homotopy">higher homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pi-algebra">Pi-algebra</a>, <a class="existingWikiWord" href="/nlab/show/spherical+object+and+Pi%28A%29-algebra">spherical object and Pi(A)-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coherent+category+theory">homotopy coherent category theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopical+category">homotopical category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">category of fibrant objects</a>, <a class="existingWikiWord" href="/nlab/show/cofibration+category">cofibration category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Waldhausen+category">Waldhausen category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ho%28Top%29">Ho(Top)</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/homotopy+category+of+an+%28%E2%88%9E%2C1%29-category">homotopy category of an (∞,1)-category</a></li> </ul> </li> </ul> <p><strong>Paths and cylinders</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cylinder+object">cylinder object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/right+homotopy">right homotopy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/path+object">path object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cocone">mapping cocone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+universal+bundle">universal bundle</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interval+object">interval object</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+localization">homotopy localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infinitesimal+interval+object">infinitesimal interval object</a></p> </li> </ul> </li> </ul> <p><strong>Homotopy groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+group+of+a+topos">fundamental group of a topos</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brown-Grossman+homotopy+group">Brown-Grossman homotopy group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/categorical+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">categorical homotopy groups in an (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">geometric homotopy groups in an (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid">fundamental ∞-groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+groupoid">fundamental groupoid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/path+groupoid">path groupoid</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+in+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+of+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%28%E2%88%9E%2C1%29-category">fundamental (∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+category">fundamental category</a></li> </ul> </li> </ul> <p><strong>Basic facts</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+group+of+the+circle+is+the+integers">fundamental group of the circle is the integers</a></li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+theorem+of+covering+spaces">fundamental theorem of covering spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freudenthal+suspension+theorem">Freudenthal suspension theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Blakers-Massey+theorem">Blakers-Massey theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+homotopy+van+Kampen+theorem">higher homotopy van Kampen theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nerve+theorem">nerve theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitehead%27s+theorem">Whitehead's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hurewicz+theorem">Hurewicz theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Galois+theory">Galois theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis</a>-theorem</p> </li> </ul> </div></div> <h4 id="bundles">Bundles</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/bundles">bundles</a></strong></p> <ul> <li> <p>(<a class="existingWikiWord" href="/nlab/show/parameterized+stable+homotopy+theory">stable</a>) <a class="existingWikiWord" href="/nlab/show/parameterized+homotopy+theory">parameterized homotopy theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+bundles+in+physics">fiber bundles in physics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory">Chern-Weil theory</a></p> </li> </ul> <h2 id="sidebar_context">Context</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/slice+topos">slice topos</a>, <a class="existingWikiWord" href="/nlab/show/slice+%28%E2%88%9E%2C1%29-topos">slice (∞,1)-topos</a></p> </li> <li> <p>(<a class="existingWikiWord" href="/nlab/show/dependent+linear+type+theory">linear</a>) <a class="existingWikiWord" href="/nlab/show/dependent+type+theory">dependent type theory</a></p> </li> </ul> <h2 id="sidebar_classes_of_bundles">Classes of bundles</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/covering+space">covering space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/retractive+space">retractive space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+bundle">fiber bundle</a>, <a class="existingWikiWord" href="/nlab/show/fiber+%E2%88%9E-bundle">fiber ∞-bundle</a></p> <p><a class="existingWikiWord" href="/nlab/show/numerable+bundle">numerable bundle</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/sphere+bundle">sphere bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/projective+bundle">projective bundle</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a>, <a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+2-bundle">principal 2-bundle</a>, <a class="existingWikiWord" href="/nlab/show/principal+3-bundle">principal 3-bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/circle+bundle">circle bundle</a>, <a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle n-bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orientation+bundle">orientation bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spinor+bundle">spinor bundle</a>, <a class="existingWikiWord" href="/nlab/show/stringor+bundle">stringor bundle</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/associated+bundle">associated bundle</a>, <a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundle">associated ∞-bundle</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/gerbe">gerbe</a>, <a class="existingWikiWord" href="/nlab/show/2-gerbe">2-gerbe</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-gerbe">∞-gerbe</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+coefficient+bundle">local coefficient bundle</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vector+bundle">vector bundle</a>, <a class="existingWikiWord" href="/nlab/show/2-vector+bundle">2-vector bundle</a>, <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-vector+bundle">(∞,1)-vector bundle</a></p> <p><a class="existingWikiWord" href="/nlab/show/real+vector+bundle">real</a>, <a class="existingWikiWord" href="/nlab/show/complex+vector+bundle">complex</a>/<a class="existingWikiWord" href="/nlab/show/holomorphic+vector+bundle">holomorphic</a>, <a class="existingWikiWord" href="/nlab/show/quaternionic+vector+bundle">quaternionic</a></p> <p><a class="existingWikiWord" href="/nlab/show/topological+vector+bundle">topological</a>, <a class="existingWikiWord" href="/nlab/show/differentiable+vector+bundle">differentiable</a>, <a class="existingWikiWord" href="/nlab/show/algebraic+vector+bundle">algebraic</a></p> <p><a class="existingWikiWord" href="/nlab/show/connection+on+a+vector+bundle">with connection</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/line+bundle">line bundle</a></p> <p><a class="existingWikiWord" href="/nlab/show/complex+line+bundle">complex</a>, <a class="existingWikiWord" href="/nlab/show/holomorphic+line+bundle">holomorphic</a>, <a class="existingWikiWord" href="/nlab/show/algebraic+line+bundle">algebraic</a></p> <p><a class="existingWikiWord" href="/nlab/show/cubical+structure+on+a+line+bundle">cubical structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory">Chern-Weil theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+K-theory">topological K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tensor+category">tensor category</a><a class="existingWikiWord" href="/nlab/show/Vect%28X%29">of vector bundles</a></p> <p>(<a class="existingWikiWord" href="/nlab/show/VectBund">VectBund</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/direct+sum+of+vector+bundles">direct sum</a>, <a class="existingWikiWord" href="/nlab/show/tensor+product+of+vector+bundles">tensor product</a>, <a class="existingWikiWord" href="/nlab/show/external+tensor+product+of+vector+bundles">external tensor product</a>, <a class="existingWikiWord" href="/nlab/show/inner+product+of+vector+bundles">inner product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dual+vector+bundle">dual vector bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+vector+bundle">stable vector bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/virtual+vector+bundle">virtual vector bundle</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bundle+of+spectra">bundle of spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/natural+bundle">natural bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+bundle">equivariant bundle</a></p> </li> </ul> <h2 id="sidebar_universal_bundles">Universal bundles</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+principal+bundle">universal principal bundle</a>, <a class="existingWikiWord" href="/nlab/show/universal+principal+%E2%88%9E-bundle">universal principal ∞-bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+vector+bundle">universal vector bundle</a>, <a class="existingWikiWord" href="/nlab/show/universal+complex+line+bundle">universal complex line bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/subobject+classifier">subobject classifier</a>, <a class="existingWikiWord" href="/nlab/show/object+classifier">object classifier</a></p> </li> </ul> <h2 id="sidebar_presentations">Presentations</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/bundle+gerbe">bundle gerbe</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/groupal+model+for+universal+principal+%E2%88%9E-bundles">groupal model for universal principal ∞-bundles</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/microbundle">microbundle</a></p> </li> </ul> <h2 id="sidebar_examples">Examples</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/empty+bundle">empty bundle</a>, <a class="existingWikiWord" href="/nlab/show/zero+bundle">zero bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/trivial+vector+bundle">trivial vector bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a>, <a class="existingWikiWord" href="/nlab/show/normal+bundle">normal bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tautological+line+bundle">tautological line bundle</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/basic+line+bundle+on+the+2-sphere">basic line bundle on the 2-sphere</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hopf+fibration">Hopf fibration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/canonical+line+bundle">canonical line bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/prequantum+circle+bundle">prequantum circle bundle</a>, <a class="existingWikiWord" href="/nlab/show/prequantum+circle+n-bundle">prequantum circle n-bundle</a></p> </li> </ul> <h2 id="sidebar_constructions">Constructions</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/clutching+construction">clutching construction</a></li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#parameterized_pointset_topology'>Parameterized point-set topology</a></li> <ul> <li><a href='#FiberwiseMappingSpaces'>Fiberwise mapping spaces</a></li> <li><a href='#homotopy_theory_of_the_fiberwise_mapping_space'>Homotopy theory of the fiberwise mapping space</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#ParameterizedTopology'>Exponential law for parameterized topological spaces</a></li> <li><a href='#parameterized_fiberwise_homotopy_theory'>Parameterized (“fiberwise”) homotopy theory</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p>Parameterized (<a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable</a>) <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> is (<a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable</a>) <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> of <a class="existingWikiWord" href="/nlab/show/bundles">bundles</a> of <a class="existingWikiWord" href="/nlab/show/homotopy+types">homotopy types</a>/<a class="existingWikiWord" href="/nlab/show/stable+homotopy+types">stable homotopy types</a> over a given base space.</p> <p>For formalizations see also at</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/ex-space">ex-space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/parameterized+spectrum">parameterized spectrum</a>, <a class="existingWikiWord" href="/nlab/show/excisive+functor">excisive functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-module+bundle">(infinity,1)-module bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tangent+%28infinity%2C1%29-topos">tangent (infinity,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted cohomology</a></p> </li> </ul> <h2 id="parameterized_pointset_topology">Parameterized point-set topology</h2> <blockquote> <p>The <a class="existingWikiWord" href="/nlab/show/point-set+topology">point-set topology</a> of parametrized spaces is surprisingly subtle. [<a href="#MaySigurdsson06">May & Sigurdsson 2006, p. 15</a>]</p> </blockquote> <p>Write:</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>kTop</mi></mrow><annotation encoding="application/x-tex">kTop</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/convenient+category+of+topological+spaces">convenient category</a> of <a class="existingWikiWord" href="/nlab/show/compactly+generated+topological+spaces">compactly generated topological spaces</a> (k-spaces)</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>kwHaus</mi><mo>⊂</mo><msub><mi>Top</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">kwHaus \subset Top_k</annotation></semantics></math> for that of <a class="existingWikiWord" href="/nlab/show/compactly+generated+weak+Hausdorff+spaces">compactly generated weak Hausdorff spaces</a>.</p> </li> </ul> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mspace width="thinmathspace"></mspace><mo>∈</mo><mspace width="thinmathspace"></mspace><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">B \,\in\, \mathcal{C}</annotation></semantics></math> an <a class="existingWikiWord" href="/nlab/show/object">object</a> in any <a class="existingWikiWord" href="/nlab/show/category">category</a>, write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mi>B</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\mathcal{C}_{/B}</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/slice+category">slice category</a> over it.</p> <p>In the following all bases spaces are assumed (as in <a href="#MaySigurdsson06">MaSi06, p. 19</a>) to be <a class="existingWikiWord" href="/nlab/show/compactly+generated+weak+Hausdorff+spaces">compactly generated weak Hausdorff spaces</a> regarded among k-spaces:</p> <div class="maruku-equation" id="eq:kwHausBaseSpace"><span class="maruku-eq-number">(1)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>B</mi><mspace width="thickmathspace"></mspace><mo>∈</mo><mspace width="thickmathspace"></mspace><mi>kwHaus</mi><mover><mo>↪</mo><mspace width="thickmathspace"></mspace></mover><mi>kTop</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> B \;\in\; kwHaus \xhookrightarrow{\;} kTop \,. </annotation></semantics></math></div> <p>Notice that for <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><msub><mi>p</mi> <mi>X</mi></msub><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><msub><mi>p</mi> <mi>Y</mi></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>∈</mo><mspace width="thinmathspace"></mspace><msub><mi>kTop</mi> <mrow><mo stretchy="false">/</mo><mi>B</mi></mrow></msub></mrow><annotation encoding="application/x-tex">(X,p_X), (Y,p_Y) \,\in\, kTop_{/B}</annotation></semantics></math>, their <a class="existingWikiWord" href="/nlab/show/Cartesian+product">Cartesian product</a> in the <a class="existingWikiWord" href="/nlab/show/slice+category">slice category</a> is given by the <a class="existingWikiWord" href="/nlab/show/fiber+product">fiber product</a> in <a class="existingWikiWord" href="/nlab/show/compactly+generated+topological+space"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>kTop</mi> </mrow> <annotation encoding="application/x-tex">kTop</annotation> </semantics> </math></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><mi>X</mi><mo>,</mo><msub><mi>p</mi> <mi>X</mi></msub><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><msub><mi>p</mi> <mi>Y</mi></msub><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>X</mi><msub><mo>×</mo> <mi>B</mi></msub><mi>Y</mi><mo>,</mo><mspace width="thinmathspace"></mspace><msub><mi>p</mi> <mi>X</mi></msub><mo>∘</mo><msub><mi>pr</mi> <mi>X</mi></msub><mo>=</mo><msub><mi>p</mi> <mi>Y</mi></msub><mo>∘</mo><msub><mi>pr</mi> <mi>Y</mi></msub><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>∈</mo><mspace width="thickmathspace"></mspace><msub><mi>kTop</mi> <mrow><mo 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x="115.421" y="49.986"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#2_7r0-yNYVvI1uLtdVZGawmPnyo=-glyph4-1" x="120.506" y="51.231"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#2_7r0-yNYVvI1uLtdVZGawmPnyo=-glyph5-1" x="117.175" y="20.72"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#2_7r0-yNYVvI1uLtdVZGawmPnyo=-glyph5-1" x="117.175" y="40.646"></use> </g> </g> </svg> <p>where</p> <ol> <li> <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^\ast</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>kTop</mi></mrow><annotation encoding="application/x-tex">kTop</annotation></semantics></math> 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> </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></mrow><annotation encoding="application/x-tex">f_!</annotation></semantics></math> is post-<a class="existingWikiWord" href="/nlab/show/composition">composition</a> 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>;</p> </li> </ol> <p>Notice the “<a class="existingWikiWord" href="/nlab/show/Frobenius+reciprocity">Frobenius reciprocity</a> law” (in its <a class="existingWikiWord" href="/nlab/show/cartesian+monoidal+category">cartesian</a> version <a href="Frobenius+reciprocity#InCategoryTheory">here</a>) which follows immediately by the <a class="existingWikiWord" href="/nlab/show/pasting+law">pasting law</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>kTop</mi></mrow><annotation encoding="application/x-tex">kTop</annotation></semantics></math>, namely the following <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural isomorphism</a>:</p> <div class="maruku-equation" id="eq:CartesianFrobeniusReciprocity"><span class="maruku-eq-number">(3)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mo>!</mo></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><msub><mi>p</mi> <mi>U</mi></msub><mo stretchy="false">)</mo><mo>×</mo><msup><mi>f</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>p</mi> <mi>X</mi></msub><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mi>f</mi> <mo>!</mo></msub><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><msub><mi>p</mi> <mi>U</mi></msub><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mo>×</mo><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>p</mi> <mi>X</mi></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> f_! \big( (U,p_U) \times f^\ast (X,p_X) \big) \;\simeq\; \big( f_!(U,p_U) \big) \times (X,p_X) \,. </annotation></semantics></math></div> <p>In the special case where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal</a> map to the <a class="existingWikiWord" href="/nlab/show/point+space">point space</a>, which we denote</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mi>B</mi></msub><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>B</mi><mover><mo>→</mo><mspace width="thickmathspace"></mspace></mover><mo>*</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> p_B \,\colon\, B \xrightarrow{\;} \ast \,. </annotation></semantics></math></div> <p>we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>kTop</mi> <mrow><mo stretchy="false">/</mo><mo>*</mo></mrow></msub><mspace width="thinmathspace"></mspace><mo>≃</mo><mspace width="thinmathspace"></mspace><mi>kTop</mi></mrow><annotation encoding="application/x-tex">kTop_{/\ast} \,\simeq\, kTop</annotation></semantics></math> and the above <a class="existingWikiWord" href="/nlab/show/base+change">base change</a> <a class="existingWikiWord" href="/nlab/show/adjoint+triple">adjoint triple</a> becomes</p> <div class="maruku-equation" id="eq:TerminalBaseChange"><span class="maruku-eq-number">(4)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math></div><svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="224.093pt" height="56.247pt" viewBox="0 0 224.093 56.247" version="1.1"> <defs> <g> <symbol overflow="visible" id="i2Ju496_HLEQMyPzwfv76FGoBvA=-glyph0-0"> <path style="stroke:none;" d=""></path> </symbol> <symbol overflow="visible" 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xlink:href="#i2Ju496_HLEQMyPzwfv76FGoBvA=-glyph3-1" x="102.534" y="50.539"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#i2Ju496_HLEQMyPzwfv76FGoBvA=-glyph1-3" x="106.65025" y="50.539"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#i2Ju496_HLEQMyPzwfv76FGoBvA=-glyph4-1" x="111.979" y="52.214"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#i2Ju496_HLEQMyPzwfv76FGoBvA=-glyph3-2" x="119.88275" y="50.539"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#i2Ju496_HLEQMyPzwfv76FGoBvA=-glyph6-1" x="123.999" y="51.78525"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#i2Ju496_HLEQMyPzwfv76FGoBvA=-glyph7-1" x="112.477" y="21.274"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#i2Ju496_HLEQMyPzwfv76FGoBvA=-glyph7-1" x="112.477" y="41.199"></use> </g> </g> </svg> <p>In this case</p> <ol> <li> <p>the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>p</mi> <mi>B</mi></msub><msup><mo stretchy="false">)</mo> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">(p_B)^\ast</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/Cartesian+product">Cartesian product</a> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>, regarded as the trivial fibration:</p> <div class="maruku-equation" id="eq:TrivialFibration"><span class="maruku-eq-number">(5)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msubsup><mi>p</mi> <mi>B</mi> <mo>*</mo></msubsup><mo stretchy="false">)</mo><msub><mi>X</mi> <mn>0</mn></msub><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mi>B</mi><mo>×</mo><msub><mi>X</mi> <mn>0</mn></msub><mover><mo>→</mo><mrow><mspace width="thickmathspace"></mspace><msub><mi>pr</mi> <mi>B</mi></msub><mspace width="thickmathspace"></mspace></mrow></mover><mi>B</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> (p_B^\ast) X_0 \;=\; B \times X_0 \xrightarrow{\; pr_B \;} B \,, </annotation></semantics></math></div></li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>p</mi> <mi>B</mi></msub><msub><mo stretchy="false">)</mo> <mo>!</mo></msub></mrow><annotation encoding="application/x-tex">(p_B)_!</annotation></semantics></math> gives the total space of a fibration:</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>B</mi></msub><msub><mo stretchy="false">)</mo> <mo>!</mo></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>X</mi><mo>→</mo><mi>B</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mi>X</mi></mrow><annotation encoding="application/x-tex"> (p_B)_! \big( X \to B\big) \;=\; X </annotation></semantics></math></div></li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>p</mi> <mi>B</mi></msub><msub><mo stretchy="false">)</mo> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">(p_B)_*</annotation></semantics></math> gives the <a class="existingWikiWord" href="/nlab/show/space+of+sections">space of sections</a> of a fibration.</p> </li> </ol> <p>Eventually we consider <a class="existingWikiWord" href="/nlab/show/pointed+objects">pointed objects</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><mi>X</mi><mo>,</mo><msub><mi>p</mi> <mi>X</mi></msub><mo>,</mo><msub><mi>σ</mi> <mi>X</mi></msub><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>∈</mo><mspace width="thickmathspace"></mspace><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mi>kTop</mi> <mrow><mo stretchy="false">/</mo><mi>B</mi></mrow></msub><msup><mo maxsize="1.2em" minsize="1.2em">)</mo> <mrow><mi>B</mi><mo stretchy="false">/</mo></mrow></msup><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><msubsup><mi>kTop</mi> <mrow><mo stretchy="false">/</mo><mi>B</mi></mrow> <mrow><mi>B</mi><mo stretchy="false">/</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex"> (X, p_X, \sigma_X) \;\in\; \big( kTop_{/B}\big)^{B/} \;=\; kTop_{/B}^{B/} </annotation></semantics></math></div> <p>in the <a class="existingWikiWord" href="/nlab/show/slice+category">slice category</a> of such a base space – hence topological “<a class="existingWikiWord" href="/nlab/show/bundles">bundles</a>” <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mover><mo>→</mo><mrow><msub><mi>p</mi> <mi>X</mi></msub></mrow></mover><mi>B</mi></mrow><annotation encoding="application/x-tex">X \xrightarrow{p_X} B</annotation></semantics></math> (in the most general sense, without any condition on the bundle projection, except <a class="existingWikiWord" href="/nlab/show/continuous+function">continuity</a>) equipped with a fixed <a class="existingWikiWord" href="/nlab/show/section">section</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>σ</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">\sigma_X</annotation></semantics></math> (sometimes called “ex-spaces”, see [<a href="#MaySigurdsson06">May & Sigurdsson 2006, p. 19, footnote 1</a>]).</p> <h3 id="FiberwiseMappingSpaces">Fiberwise mapping spaces</h3> <blockquote> <p>Parametrized mapping spaces are especially delicate [<a href="#MaySigurdsson06">May & Sigurdsson 2006, p. 15</a>, see Rem. <a class="maruku-ref" href="#FibMapSpaceDoesNotPreserveWeakHausdorffness"></a> below]</p> </blockquote> <p> <div class='num_defn' id='PartialMapClassifierSpace'> <h6>Definition</h6> <p><strong>(partial map classifier space)</strong> <br /> For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mspace width="thinmathspace"></mspace><mo>∈</mo><mspace width="thinmathspace"></mspace><mi>kTop</mi></mrow><annotation encoding="application/x-tex">X \,\in\, kTop</annotation></semantics></math>, write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>X</mi><mo>˜</mo></mover><mspace width="thinmathspace"></mspace><mo>∈</mo><mspace width="thinmathspace"></mspace><mi>kTop</mi></mrow><annotation encoding="application/x-tex">\widetilde X \,\in\, kTop</annotation></semantics></math> for its continuous <a class="existingWikiWord" href="/nlab/show/partial+map+classifier">partial map classifier</a>: The result of forming the <a class="existingWikiWord" href="/nlab/show/disjoint+union">disjoint union</a> of the <a class="existingWikiWord" href="/nlab/show/underlying">underlying</a> <a class="existingWikiWord" href="/nlab/show/set">set</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> with a <a class="existingWikiWord" href="/nlab/show/singleton+set">singleton set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mo>⊥</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{\bot\}</annotation></semantics></math> and declaring the <a class="existingWikiWord" href="/nlab/show/closed+subsets">closed subsets</a> on the result to be those 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> under the defining <a class="existingWikiWord" href="/nlab/show/injection">injection</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mover><mo>↪</mo><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msub><mi>ι</mi> <mi>X</mi></msub><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow></mover><mover><mi>X</mi><mo>˜</mo></mover></mrow><annotation encoding="application/x-tex"> X \xhookrightarrow{\;\; \iota_X \;\;} \widetilde{X} </annotation></semantics></math></div> <p>together with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>X</mi><mo>˜</mo></mover></mrow><annotation encoding="application/x-tex">\widetilde{X}</annotation></semantics></math> itself.</p> <p></p> </div> </p> <p>(<a href="#MaySigurdsson06">MaSi06, Def. 1.3.6</a>)</p> <p> <div class='num_defn' id='FiberwiseMappingSpace'> <h6>Definition</h6> <p><strong>(fiberwise mapping space)</strong> <br /> For</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>X</mi><mo>,</mo><msub><mi>p</mi> <mi>X</mi></msub><mo stretchy="false">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><msub><mi>p</mi> <mi>A</mi></msub><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>∈</mo><mspace width="thickmathspace"></mspace><msub><mi>kTop</mi> <mi>B</mi></msub></mrow><annotation encoding="application/x-tex"> (X, p_X), \, (A,p_A) \;\in\; kTop_{B} </annotation></semantics></math></div> <p>a <a class="existingWikiWord" href="/nlab/show/pair">pair</a> of <a class="existingWikiWord" href="/nlab/show/k-spaces">k-spaces</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> <a class="maruku-eqref" href="#eq:kwHausBaseSpace">(1)</a> their <em>fiberwise mapping space</em> is the <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> (in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>kTop</mi></mrow><annotation encoding="application/x-tex">kTop</annotation></semantics></math>):</p> <div class="maruku-equation" id="eq:FiberwiseMappingSpaceAsPullback"><span class="maruku-eq-number">(6)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mphantom><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>−</mo><mo lspace="verythinmathspace" rspace="0em">−</mo></mrow></mphantom><mpadded width="0" lspace="-50%width"><mrow><mi>Map</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>p</mi> <mi>X</mi></msub><mo stretchy="false">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><msub><mi>p</mi> <mi>A</mi></msub><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mpadded><mphantom><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>−</mo><mo lspace="verythinmathspace" rspace="0em">−</mo></mrow></mphantom></mtd> <mtd><mover><mo>→</mo><mphantom><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>−</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>−</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>−</mo><mo lspace="verythinmathspace" rspace="0em">−</mo></mrow></mphantom></mover></mtd> <mtd><mi>Map</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>X</mi><mo>,</mo><mover><mi>A</mi><mo>˜</mo></mover><mo maxsize="1.2em" minsize="1.2em">)</mo></mtd></mtr> <mtr><mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo></mtd> <mtd><msub><mrow></mrow> <mrow><msup><mrow></mrow> <mrow><mo stretchy="false">(</mo><mi>pb</mi><mo stretchy="false">)</mo></mrow></msup></mrow></msub></mtd> <mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo><msup><mrow></mrow> <mpadded width="0"><mrow><mi>Map</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>X</mi><mo>,</mo><mover><mrow><msub><mi>p</mi> <mi>A</mi></msub></mrow><mo>˜</mo></mover><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>B</mi></mtd> <mtd><munder><mo>⟶</mo><mrow><mi>b</mi><mo>↦</mo><mrow><mo>(</mo><mi>x</mi><mo>↦</mo><mrow><mo>{</mo><mtable displaystyle="false" rowspacing="0.5ex" columnalign="left left"><mtr><mtd><mi>b</mi></mtd> <mtd><mi>if</mi><mspace width="thinmathspace"></mspace><mi>x</mi><mo>∈</mo><msub><mi>X</mi> <mi>b</mi></msub></mtd></mtr> <mtr><mtd><mo>⊥</mo></mtd> <mtd><mi>otherwise</mi></mtd></mtr></mtable></mrow><mo>)</mo></mrow></mrow></munder></mtd> <mtd><mi>Map</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>X</mi><mo>,</mo><mover><mi>B</mi><mo>˜</mo></mover><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \phantom{---} \mathclap{ Map \big( (X,p_X) ,\, (A,p_A) \big) } \phantom{---} &\xrightarrow{\phantom{-------}}& Map \big( X, \widetilde{A} \big) \\ \big\downarrow &{}_{{}^{(pb)}}& \big\downarrow{}^{\mathrlap{ Map \big( X, \widetilde{p_A} \big) }} \\ B & \underset{ b \mapsto \left( x \mapsto \left\{ \begin{array}{ll} b & if\, x \in X_b \\ \bot & otherwise \end{array} \right. \right) }{\longrightarrow} & Map \big( X, \widetilde{B} \big) \,, } </annotation></semantics></math></div> <p>regarded as an object of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>kTop</mi> <mrow><mo stretchy="false">/</mo><mi>B</mi></mrow></msub></mrow><annotation encoding="application/x-tex">kTop_{/B}</annotation></semantics></math>.</p> <p>Here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Map</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mover><mi>A</mi><mo>˜</mo></mover><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Map(X,\widetilde{A})</annotation></semantics></math> denotes the ordinary <a class="existingWikiWord" href="/nlab/show/compact-open+topology">mapping space</a> into the continuous partial map classifier from Def. <a class="maruku-ref" href="#PartialMapClassifierSpace"></a>.</p> </div> </p> <p>(This is <a href="#MaySigurdsson06">May & Sigurdsson 2006, Def. 1.3.7</a>, following <a href="#BoothBrown78a">Booth & Brown 1978a</a>).</p> <p> <div class='num_remark'> <h6>Remark</h6> <p><strong>(on notation)</strong> <br /> Contrary to most references, Def. <a class="maruku-ref" href="#FiberwiseMappingSpace"></a> is intentionally not using a subsript “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mrow></mrow> <mi>B</mi></msub></mrow><annotation encoding="application/x-tex">{}_B</annotation></semantics></math>” in the notation for the fiberwise mapping space: This is because “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Map</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><msub><mo stretchy="false">)</mo> <mi>B</mi></msub></mrow><annotation encoding="application/x-tex">Map(X,A)_B</annotation></semantics></math>” is also standard notation for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Map</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><munder><mo>×</mo><mrow><mi>Map</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo></mrow></munder><mo stretchy="false">{</mo><msub><mi>p</mi> <mi>B</mi></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">Map(X,A) \underset{Map(X,B)}{\times} \{p_B\}</annotation></semantics></math> (see e.g. at <em><a class="existingWikiWord" href="/nlab/show/space+of+sections">space of sections</a></em>), which is crucially different. Instead, with the above notation, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Map</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Map(-,-)</annotation></semantics></math> is always of the same <a class="existingWikiWord" href="/nlab/show/type">type</a> as its arguments, as befits an <a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom</a>.</p> </div> </p> <p> <div class='num_prop' id='FiberwiseMappingSpaceSatisfiesExponentialLaw'> <h6>Proposition</h6> <p><strong>(fiberwise mapping space satisfies the exponential law)</strong> <br /> With <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> as above <a class="maruku-eqref" href="#eq:kwHausBaseSpace">(1)</a>, the fiberwise mapping space (Def. <a class="maruku-ref" href="#FiberwiseMappingSpace"></a>) is an <a class="existingWikiWord" href="/nlab/show/exponential+object">exponential object</a> (satisfies the <a class="existingWikiWord" href="/nlab/show/exponential+law+for+spaces">exponential law</a>) in that there is a <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural isomorphism</a> of <a class="existingWikiWord" href="/nlab/show/hom-sets">hom-sets</a></p> <div class="maruku-equation" id="eq:ExponentialLawForFiberwiseMappingSpace"><span class="maruku-eq-number">(7)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>kTop</mi> <mrow><mo stretchy="false">/</mo><mi>B</mi></mrow></msub><mo maxsize="1.8em" minsize="1.8em">(</mo><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>p</mi> <mi>X</mi></msub><mo stretchy="false">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mi>Map</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><msub><mi>p</mi> <mi>Y</mi></msub><mo stretchy="false">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><msub><mi>p</mi> <mi>A</mi></msub><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mo maxsize="1.8em" minsize="1.8em">)</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msub><mi>kTop</mi> <mrow><mo stretchy="false">/</mo><mi>B</mi></mrow></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>p</mi> <mi>X</mi></msub><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><msub><mi>p</mi> <mi>Y</mi></msub><mo stretchy="false">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><msub><mi>p</mi> <mi>A</mi></msub><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex"> kTop_{/B} \Big( (X,p_X) ,\, Map \big( (Y,p_Y) ,\, (A,p_A) \big) \Big) \;\; \simeq \;\; kTop_{/B} \big( (X,p_X) \times (Y,p_Y) ,\, (A,p_A) \big) </annotation></semantics></math></div> <p>(where on the right we have the <a class="existingWikiWord" href="/nlab/show/Cartesian+product">Cartesian product</a> in the <a class="existingWikiWord" href="/nlab/show/slice+category">slice</a>, given by the <a class="existingWikiWord" href="/nlab/show/fiber+product">fiber product</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><msub><mo>×</mo> <mi>B</mi></msub><mi>Y</mi></mrow><annotation encoding="application/x-tex">X \times_B Y</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>kTop</mi></mrow><annotation encoding="application/x-tex">kTop</annotation></semantics></math>).</p> </div> (<a href="#BoothBrown78a">Booth & Brown 1978a, Thm. 3.5</a>, see <a href="#MaySigurdsson06">May & Sigurdsson 2006, (1.3.9)</a>)</p> <p> <div class='num_remark' id='FiberwiseMappingSpaceBetweenTrivialFibrations'> <h6>Example</h6> <p><strong>(fiberwise mapping space between trivial fibrations)</strong> <br /> The fiberwise mapping space (Def. <a class="maruku-ref" href="#FiberwiseMappingSpace"></a>) between trivial fibrations <a class="maruku-eqref" href="#eq:TrivialFibration">(5)</a> is the trivial fibration with fiber the ordinary <a class="existingWikiWord" href="/nlab/show/compact-open+topology">mapping space</a> between the fibers:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Map</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><msubsup><mi>p</mi> <mi>B</mi> <mo>*</mo></msubsup><msub><mi>X</mi> <mn>0</mn></msub><mo>,</mo><mspace width="thinmathspace"></mspace><msubsup><mi>p</mi> <mi>B</mi> <mo>*</mo></msubsup><msub><mi>A</mi> <mn>0</mn></msub><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><msubsup><mi>p</mi> <mi>B</mi> <mo>*</mo></msubsup><mi>Map</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mi>X</mi> <mn>0</mn></msub><mo>,</mo><mspace width="thinmathspace"></mspace><msub><mi>A</mi> <mn>0</mn></msub><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Map \big( p_B^\ast X_0 ,\, p_B^\ast A_0 \big) \;\simeq\; p_B^\ast Map\big(X_0,\, A_0\big) \,. </annotation></semantics></math></div> <p></p> </div> <div class='proof'> <h6>Proof</h6> <p>This may be gleaned concretely from <a class="existingWikiWord" href="/nlab/show/point-set+topology">point-set</a>-analysis of the defining pullback diagram <a class="maruku-eqref" href="#eq:FiberwiseMappingSpaceAsPullback">(6)</a>, but it also follows abstractly by <a class="existingWikiWord" href="/nlab/show/adjunction">adjointness</a> from the <a class="existingWikiWord" href="/nlab/show/exponential+law">exponential law</a> (Prop. <a class="maruku-ref" href="#FiberwiseMappingSpaceSatisfiesExponentialLaw"></a>):</p> <p>For any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><msub><mi>p</mi> <mi>U</mi></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>∈</mo><mspace width="thinmathspace"></mspace><msub><mi>kTop</mi> <mrow><mo stretchy="false">/</mo><mi>B</mi></mrow></msub></mrow><annotation encoding="application/x-tex">(U, p_U) \,\in\, kTop_{/B}</annotation></semantics></math> we have the following sequence of <a class="existingWikiWord" href="/nlab/show/natural+isomorphisms">natural isomorphisms</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></mtd> <mtd><msub><mi>kTop</mi> <mrow><mo stretchy="false">/</mo><mi>B</mi></mrow></msub><mo maxsize="1.8em" minsize="1.8em">(</mo><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><msub><mi>p</mi> <mi>U</mi></msub><mo stretchy="false">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mo lspace="0em" rspace="thinmathspace">Map</mo><mo maxsize="1.2em" minsize="1.2em">(</mo><msubsup><mi>p</mi> <mi>B</mi> <mo>*</mo></msubsup><msub><mi>X</mi> <mn>0</mn></msub><mo>,</mo><mspace width="thinmathspace"></mspace><msubsup><mi>p</mi> <mi>B</mi> <mo>*</mo></msubsup><msub><mi>A</mi> <mn>0</mn></msub><mo maxsize="1.2em" minsize="1.2em">)</mo><mo maxsize="1.8em" minsize="1.8em">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><msub><mi>kTop</mi> <mrow><mo stretchy="false">/</mo><mi>B</mi></mrow></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><msub><mi>p</mi> <mi>U</mi></msub><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">(</mo><mi>B</mi><mo>×</mo><msub><mi>X</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>pr</mi> <mi>B</mi></msub><mo stretchy="false">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><msubsup><mi>p</mi> <mi>B</mi> <mo>*</mo></msubsup><msub><mi>A</mi> <mn>0</mn></msub><mo maxsize="1.2em" minsize="1.2em">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><msub><mi>kTop</mi> <mrow><mo stretchy="false">/</mo><mi>B</mi></mrow></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mo stretchy="false">(</mo><mi>U</mi><mo>×</mo><msub><mi>X</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>p</mi> <mi>U</mi></msub><mo>∘</mo><msub><mi>pr</mi> <mi>U</mi></msub><mo stretchy="false">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><msubsup><mi>p</mi> <mi>B</mi> <mo>*</mo></msubsup><msub><mi>A</mi> <mn>0</mn></msub><mo maxsize="1.2em" minsize="1.2em">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mi>kTop</mi><mo maxsize="1.8em" minsize="1.8em">(</mo><mi>U</mi><mo>×</mo><msub><mi>X</mi> <mn>0</mn></msub><mo>,</mo><mspace width="thinmathspace"></mspace><msub><mi>A</mi> <mn>0</mn></msub><mo maxsize="1.2em" minsize="1.2em">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mi>kTop</mi><mo maxsize="1.8em" minsize="1.8em">(</mo><mi>U</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mi>Map</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mi>X</mi> <mn>0</mn></msub><mo>,</mo><mspace width="thinmathspace"></mspace><msub><mi>A</mi> <mn>0</mn></msub><mo maxsize="1.2em" minsize="1.2em">)</mo><mo maxsize="1.8em" minsize="1.8em">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mi>kTop</mi><mo maxsize="1.8em" minsize="1.8em">(</mo><mo stretchy="false">(</mo><msub><mi>p</mi> <mi>B</mi></msub><msub><mo stretchy="false">)</mo> <mo>!</mo></msub><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><msub><mi>p</mi> <mi>U</mi></msub><mo stretchy="false">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mi>Map</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mi>X</mi> <mn>0</mn></msub><mo>,</mo><mspace width="thinmathspace"></mspace><msub><mi>A</mi> <mn>0</mn></msub><mo maxsize="1.2em" minsize="1.2em">)</mo><mo maxsize="1.8em" minsize="1.8em">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><msub><mi>kTop</mi> <mrow><mo stretchy="false">/</mo><mi>B</mi></mrow></msub><mo maxsize="1.8em" minsize="1.8em">(</mo><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><msub><mi>p</mi> <mi>U</mi></msub><mo stretchy="false">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><msubsup><mi>p</mi> <mi>B</mi> <mo>*</mo></msubsup><mi>Map</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mi>X</mi> <mn>0</mn></msub><mo>,</mo><mspace width="thinmathspace"></mspace><msub><mi>A</mi> <mn>0</mn></msub><mo maxsize="1.2em" minsize="1.2em">)</mo><mo maxsize="1.8em" minsize="1.8em">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} & kTop_{/B} \Big( (U, p_U) ,\, \Map \big( p_B^\ast X_0 ,\, p_B^\ast A_0 \big) \Big) \\ & \;\simeq\; kTop_{/B} \big( (U, p_U) \times (B \times X_0, pr_{B}) ,\, p_B^\ast A_0 \big) \\ & \;\simeq\; kTop_{/B} \big( (U \times X_0, p_U \circ pr_U) ,\, p_B^\ast A_0 \big) \\ & \;\simeq\; kTop \Big( U \times X_0 ,\, A_0 \big) \\ & \;\simeq\; kTop \Big( U ,\, Map \big( X_0 ,\, A_0 \big) \Big) \\ & \;\simeq\; kTop \Big( (p_B)_! (U,p_U) ,\, Map \big( X_0 ,\, A_0 \big) \Big) \\ & \;\simeq\; kTop_{/B} \Big( (U,p_U) ,\, p_B^\ast Map \big( X_0 ,\, A_0 \big) \Big) \end{aligned} </annotation></semantics></math></div> <p>Here most steps are <a href="adjoint+functor#InTermsOfHomIsomorphism">Hom-isomorphisms</a> of the various <a class="existingWikiWord" href="/nlab/show/adjoint+functors">adjoint functors</a>: <a class="maruku-eqref" href="#eq:TerminalBaseChange">(4)</a> and <a class="maruku-eqref" href="#eq:ExponentialLawForFiberwiseMappingSpace">(7)</a>. Since this holds naturally for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><msub><mi>p</mi> <mi>U</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(U, p_U)</annotation></semantics></math>, the claim follows by the <a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a> (over the <a class="existingWikiWord" href="/nlab/show/large+category">large category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mi>kTop</mi> <mrow><mo stretchy="false">/</mo><mi>B</mi></mrow></msub><msup><mo maxsize="1.2em" minsize="1.2em">)</mo> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">\big(kTop_{/B}\big)^{op}</annotation></semantics></math>).</p> </div> </p> <p>Similarly:</p> <p> <div class='num_prop' id='PullbackOfFiberwiseMappingSpace'> <h6>Proposition</h6> <p><strong>(pullback of fiberwise mapping space)</strong> <br /> For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>B</mi><mo>′</mo><mo>⟶</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">f \,\colon\, B' \longrightarrow B</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/map">map</a> of base spaces <a class="maruku-eqref" href="#eq:kwHausBaseSpace">(1)</a>, the <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> <a class="maruku-eqref" href="#eq:BaseChangeAdjointTriple">(2)</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> of the fiberwise mapping space (Def. <a class="maruku-ref" href="#FiberwiseMappingSpace"></a>) is the fiberwise mapping space of the pullback of the arguments:</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><mi>Map</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>p</mi> <mi>X</mi></msub><mo stretchy="false">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><msub><mi>p</mi> <mi>A</mi></msub><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>Map</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><msup><mi>f</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>p</mi> <mi>X</mi></msub><mo stretchy="false">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><msup><mi>f</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><msub><mi>p</mi> <mi>A</mi></msub><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> f^\ast Map \big( (X,p_X) ,\, (A, p_A) \big) \;\; \simeq \;\; Map \big( f^\ast (X,p_X) ,\, f^\ast (A, p_A) \big) \,. </annotation></semantics></math></div> <p>In other words: Pullback <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/closed+functor">closed functor</a> with respect to fiberwise mapping spaces.</p> </div> <div class='proof'> <h6>Proof</h6> <p>For any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><msub><mi>p</mi> <mi>U</mi></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>∈</mo><mspace width="thinmathspace"></mspace><msub><mi>kTop</mi> <mrow><mo stretchy="false">/</mo><mi>B</mi><mo>′</mo></mrow></msub></mrow><annotation encoding="application/x-tex">(U, p_U) \,\in\, kTop_{/B'}</annotation></semantics></math> we have the following sequence of <a class="existingWikiWord" href="/nlab/show/natural+isomorphisms">natural isomorphisms</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></mtd> <mtd><msub><mi>kTop</mi> <mrow><mo stretchy="false">/</mo><mi>B</mi><mo>′</mo></mrow></msub><mo maxsize="1.8em" minsize="1.8em">(</mo><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><msub><mi>p</mi> <mi>U</mi></msub><mo stretchy="false">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><msup><mi>f</mi> <mo>*</mo></msup><mi>Map</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>p</mi> <mi>X</mi></msub><mo stretchy="false">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><msub><mi>p</mi> <mi>A</mi></msub><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mo maxsize="1.8em" minsize="1.8em">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><msub><mi>kTop</mi> <mrow><mo stretchy="false">/</mo><mi>B</mi></mrow></msub><mo maxsize="1.8em" minsize="1.8em">(</mo><msub><mi>f</mi> <mo>!</mo></msub><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><msub><mi>p</mi> <mi>U</mi></msub><mo stretchy="false">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mi>Map</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>p</mi> <mi>X</mi></msub><mo stretchy="false">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><msub><mi>p</mi> <mi>A</mi></msub><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mo maxsize="1.8em" minsize="1.8em">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><msub><mi>kTop</mi> <mrow><mo stretchy="false">/</mo><mi>B</mi></mrow></msub><mo maxsize="1.8em" minsize="1.8em">(</mo><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mi>f</mi> <mo>!</mo></msub><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><msub><mi>p</mi> <mi>U</mi></msub><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mo>×</mo><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>p</mi> <mi>X</mi></msub><mo stretchy="false">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><msub><mi>p</mi> <mi>A</mi></msub><mo stretchy="false">)</mo><mo maxsize="1.8em" minsize="1.8em">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><msub><mi>kTop</mi> <mrow><mo stretchy="false">/</mo><mi>B</mi></mrow></msub><mo maxsize="1.8em" minsize="1.8em">(</mo><msub><mi>f</mi> <mo>!</mo></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><msub><mi>p</mi> <mi>U</mi></msub><mo stretchy="false">)</mo><mo>×</mo><msup><mi>f</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>p</mi> <mi>X</mi></msub><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><msub><mi>p</mi> <mi>A</mi></msub><mo stretchy="false">)</mo><mo maxsize="1.8em" minsize="1.8em">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><msub><mi>kTop</mi> <mrow><mo stretchy="false">/</mo><mi>B</mi><mo>′</mo></mrow></msub><mo maxsize="1.8em" minsize="1.8em">(</mo><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><msub><mi>p</mi> <mi>U</mi></msub><mo stretchy="false">)</mo><mo>×</mo><msup><mi>f</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>p</mi> <mi>X</mi></msub><mo stretchy="false">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><msup><mi>f</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><msub><mi>p</mi> <mi>A</mi></msub><mo stretchy="false">)</mo><mo maxsize="1.8em" minsize="1.8em">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><msub><mi>kTop</mi> <mrow><mo stretchy="false">/</mo><mi>B</mi><mo>′</mo></mrow></msub><mo maxsize="1.8em" minsize="1.8em">(</mo><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><msub><mi>p</mi> <mi>U</mi></msub><mo stretchy="false">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mi>Map</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><msup><mi>f</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>p</mi> <mi>X</mi></msub><mo stretchy="false">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><msup><mi>f</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><msub><mi>p</mi> <mi>A</mi></msub><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mo maxsize="1.8em" minsize="1.8em">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} & kTop_{/B'} \Big( (U,p_U) ,\, f^\ast Map \big( (X,p_X) ,\, (A,p_A) \big) \Big) \\ & \;\simeq\; kTop_{/B} \Big( f_! (U,p_U) ,\, Map \big( (X,p_X) ,\, (A,p_A) \big) \Big) \\ & \;\simeq\; kTop_{/B} \Big( \big( f_!(U, p_U) \big) \times (X, p_X) ,\, (A,p_A) \Big) \\ & \;\simeq\; kTop_{/B} \Big( f_! \big( (U, p_U) \times f^\ast (X, p_X) \big) ,\, (A,p_A) \Big) \\ & \;\simeq\; kTop_{/B'} \Big( (U, p_U) \times f^\ast (X, p_X) ,\, f^\ast (A,p_A) \Big) \\ & \;\simeq\; kTop_{/B'} \Big( (U, p_U) ,\, Map \big( f^\ast (X, p_X) ,\, f^\ast (A,p_A) \big) \Big) \end{aligned} </annotation></semantics></math></div> <p>Here the crucial step, besides various <a href="adjoint+functor#InTermsOfHomIsomorphism">Hom-isomorphisms</a>, is the use of Cartesian “<a class="existingWikiWord" href="/nlab/show/Frobenius+reciprocity">Frobenius reciprocity</a>” <a class="maruku-eqref" href="#eq:CartesianFrobeniusReciprocity">(3)</a>.</p> <p>Since these isomorphism are <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><msub><mi>p</mi> <mi>U</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(U,p_U)</annotation></semantics></math>, the claim follows by the <a class="existingWikiWord" href="/nlab/show/Yoneda+embedding">Yoneda embedding</a> (for the <a class="existingWikiWord" href="/nlab/show/large+category">large category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mi>kTop</mi> <mrow><mo stretchy="false">/</mo><mi>B</mi><mo>′</mo></mrow></msub><msup><mo maxsize="1.2em" minsize="1.2em">)</mo> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">\big( kTop_{/B'}\big)^{op}</annotation></semantics></math>).</p> </div> ,</p> <p> <div class='num_remark' id='FiberOfFiberwiseMappingSpace'> <h6>Example</h6> <p><strong>(fiber of fiberwise mapping space is mapping space of fibers)</strong> <br /> For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∈</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">b \in B</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/fiber">fiber</a> of the fiberwise mapping space fibration (Def. <a class="maruku-ref" href="#FiberwiseMappingSpace"></a>) is <a class="existingWikiWord" href="/nlab/show/homeomorphism">homemorphic</a> to the ordinary <a class="existingWikiWord" href="/nlab/show/compact-open+topology">mapping space</a> betwee the <a class="existingWikiWord" href="/nlab/show/fibers">fibers</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Map</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>p</mi> <mi>x</mi></msub><mo stretchy="false">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><msub><mi>p</mi> <mi>A</mi></msub><mo stretchy="false">)</mo><msub><mo maxsize="1.2em" minsize="1.2em">)</mo> <mi>b</mi></msub><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>Map</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mi>X</mi> <mi>b</mi></msub><mo>,</mo><mspace width="thinmathspace"></mspace><msub><mi>A</mi> <mi>b</mi></msub><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Map \big( (X,p_x) ,\, (A,p_A) \big)_b \;\; \simeq \;\; Map \big( X_b ,\, A_b \big) \,. </annotation></semantics></math></div> <p></p> </div> (e.g. <a href="#MaySigurdsson06">May & Sigurdsson 2006, p. 21</a>) <div class='proof'> <h6>Proof</h6> <p>This is immediate from concrete analysis of the defining <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a>-diagram <a class="maruku-eqref" href="#eq:FiberwiseMappingSpaceAsPullback">(6)</a> in Def. <a class="maruku-ref" href="#FiberwiseMappingSpace"></a>, but it is also the special case of Prop. <a class="maruku-ref" href="#PullbackOfFiberwiseMappingSpace"></a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>=</mo><mo stretchy="false">{</mo><mi>b</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">B = \{b\}</annotation></semantics></math>.</p> </div> </p> <h3 id="homotopy_theory_of_the_fiberwise_mapping_space">Homotopy theory of the fiberwise mapping space</h3> <p> <div class='num_prop' id='FiberwiseMappingSpacePreservesHFibrations'> <h6>Proposition</h6> <p><strong>(fiberwise mapping space preserves h-fibrations)</strong> <br /> If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mi>X</mi></msub><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">p_X \colon X \to B</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mi>A</mi></msub><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">p_A \colon A \to B</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/Hurewicz+fibrations">Hurewicz fibrations</a>, then so is the map <a class="maruku-eqref" href="#eq:FiberwiseMappingSpaceAsPullback">(6)</a> out of their fiberwise mapping space (Def. <a class="maruku-ref" href="#FiberwiseMappingSpace"></a>):</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mi>X</mi></msub><mo>,</mo><msub><mi>p</mi> <mi>A</mi></msub><mspace width="thinmathspace"></mspace><mo>∈</mo><mspace width="thinmathspace"></mspace><mi>hFib</mi><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><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><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msub><mi>p</mi> <mrow><mi>Map</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>p</mi> <mi>X</mi></msub><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><msub><mi>p</mi> <mi>A</mi></msub><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></msub><mspace width="thickmathspace"></mspace><mo>∈</mo><mspace width="thickmathspace"></mspace><mi>hFib</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> p_X, p_A \,\in\, hFib \;\;\;\;\;\; \Rightarrow \;\;\;\;\;\; p_{ Map\big( (X,p_X) , (A,p_A)\big) } \;\in\; hFib \,. </annotation></semantics></math></div> <p></p> </div> (This is due to <a href="#Booth70">Booth 1970, §6.1</a>, see <a href="#MaySigurdsson06">MaSi06, Prop. 1.3.11</a>.)</p> <p> <div class='num_remark' id='FibMapSpaceDoesNotPreserveWeakHausdorffness'> <h6>Remark</h6> <p><strong>(fiberwise mapping space does not preserve weak Hausdorffness)</strong> <br /> Even if <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> and <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> are <a class="existingWikiWord" href="/nlab/show/weak+Hausdorff+spaces">weak Hausdorff spaces</a> over the weak Hausdorff space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> <a class="maruku-eqref" href="#eq:kwHausBaseSpace">(1)</a>, their fiberwise mapping space (Def. <a class="maruku-ref" href="#FiberwiseMappingSpace"></a>) need not be weak Hausdorff (<a href="#BoothBrown74a">Booth & Brown 1974a</a>). Sufficient conditions for this to be the case are given in <a href="#Lewis85">Lewis 1985, Prop. 1.5</a></p> <p>On the other hand, the suitable <a class="existingWikiWord" href="/nlab/show/cofibrant+resolution">cofibrant resolution</a> of the fiberwise mapping space will again be weak Hausdorff (see <a href="#MaySigurdsson06">MaSi06, p. 19</a>).</p> </div> </p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/retractive+space">retractive space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/parameterized+spectrum">parameterized spectrum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+parameterized+spectra">model structure for parameterized spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/rational+parametrized+homotopy+theory">rational parametrized homotopy theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/rational+fibration+lemma">rational fibration lemma</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/rational+parameterized+stable+homotopy+theory">rational parameterized stable homotopy theory</a></p> </li> </ul> <h2 id="references">References</h2> <div> <h3 id="ParameterizedTopology">Exponential law for parameterized topological spaces</h3> <p>On <a class="existingWikiWord" href="/nlab/show/exponential+law+for+spaces">exponential objects</a> (<a class="existingWikiWord" href="/nlab/show/internal+homs">internal homs</a>) in <a class="existingWikiWord" href="/nlab/show/slice+categories">slice categories</a> of (<a class="existingWikiWord" href="/nlab/show/compactly+generated+topological+spaces">compactly generated</a>) <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a> – see at <em><a class="existingWikiWord" href="/nlab/show/parameterized+homotopy+theory">parameterized homotopy theory</a></em>):</p> <ul> <li id="Booth70"> <p><a class="existingWikiWord" href="/nlab/show/Peter+I.+Booth">Peter I. Booth</a>, <em>The Exponential Law of Maps I</em>, Proceedings of the London Mathematical Society <strong>s3-20</strong> 1 (1970) 179-192 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://doi.org/10.1112/plms/s3-20.1.179">doi:10.1112/plms/s3-20.1.179</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></p> </li> <li id="Booth71"> <p><a class="existingWikiWord" href="/nlab/show/Peter+I.+Booth">Peter I. Booth</a>, <em>The exponential law of maps. II</em>, Mathematische Zeitschrift <strong>121</strong> (1971) 311–319 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://doi.org/10.1007/BF01109977">doi:10.1007/BF01109977</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></p> </li> <li id="BoothBrown78a"> <p><a class="existingWikiWord" href="/nlab/show/Peter+I.+Booth">Peter I. Booth</a>, <a class="existingWikiWord" href="/nlab/show/Ronnie+Brown">Ronnie Brown</a>, <em>Spaces of partial maps, fibred mapping spaces and the compact-open topology</em>, General Topology and its Applications <strong>8</strong> 2 (1978) 181-195 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://doi.org/10.1016/0016-660X(78)90049-1">doi:10.1016/0016-660X(78)90049-1</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></p> </li> <li id="BoothBrown78b"> <p><a class="existingWikiWord" href="/nlab/show/Peter+I.+Booth">Peter I. Booth</a>, <a class="existingWikiWord" href="/nlab/show/Ronnie+Brown">Ronnie Brown</a>, <em>On the application of fibred mapping spaces to exponential laws for bundles, ex-spaces and other categories of maps</em>, General Topology and its Applications <strong>8</strong> 2 (1978) 165-179 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://doi.org/10.1016/0016-660X(78)90048-X">doi:10.1016/0016-660X(78)90048-X</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></p> </li> <li id="Lewis85"> <p><a class="existingWikiWord" href="/nlab/show/L.+Gaunce+Lewis%2C+Jr.">L. Gaunce Lewis, Jr.</a>, §1 of: <em>Open maps, colimits, and a convenient category of fibre spaces</em>, Topology and its Applications <strong>19</strong> 1 (1985) 75-89 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://doi.org/10.1016/0166-8641(85)90087-2">doi.org/10.1016/0166-8641(85)90087-2</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></p> </li> </ul> <p>And with an eye towards <a class="existingWikiWord" href="/nlab/show/parameterized+homotopy+theory">parameterized homotopy theory</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Ioan+Mackenzie+James">Ioan Mackenzie James</a>: §II.9 in: <em>Fibrewise topology</em>, Cambridge Tracts in Mathematics, Cambridge University Press (1989) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math>ISBN:9780521360906<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></p> </li> <li id="MaySigurdsson06"> <p><a class="existingWikiWord" href="/nlab/show/Peter+May">Peter May</a>, <a class="existingWikiWord" href="/nlab/show/Johann+Sigurdsson">Johann Sigurdsson</a>, §1.3.7-§1.3.9 in: <em><a class="existingWikiWord" href="/nlab/show/Parametrized+Homotopy+Theory">Parametrized Homotopy Theory</a></em>, Mathematical Surveys and Monographs, vol. 132, AMS 2006 (<a href="https://bookstore.ams.org/surv-132">ISBN:978-0-8218-3922-5</a>, <a href="https://arxiv.org/abs/math/0411656">arXiv:math/0411656</a>, <a href="http://www.math.uchicago.edu/~may/EXTHEORY/MaySig.pdf">pdf</a>)</p> </li> </ul> </div> <h3 id="parameterized_fiberwise_homotopy_theory">Parameterized (“fiberwise”) homotopy theory</h3> <p>On the <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> of such parameterized topological spaces:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Ioan+Mackenzie+James">Ioan Mackenzie James</a>, §IV of: <em>Fibrewise topology</em>, Cambridge Tracts in Mathematics, Cambridge University Press (1989) [ISBN:9780521360906]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Ioan+Mackenzie+James">Ioan Mackenzie James</a>, <em><a class="existingWikiWord" href="/nlab/show/Introduction+to+fibrewise+homotopy+theory">Introduction to fibrewise homotopy theory</a></em>, in <a class="existingWikiWord" href="/nlab/show/Ioan+Mackenzie+James">Ioan Mackenzie James</a> (ed.), <em><a class="existingWikiWord" href="/nlab/show/Handbook+of+Algebraic+Topology">Handbook of Algebraic Topology</a></em> (1995)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Michael+C.+Crabb">Michael C. Crabb</a>, <a class="existingWikiWord" href="/nlab/show/Ioan+Mackenzie+James">Ioan Mackenzie James</a>: <em>Fiberwise homotopy theory</em>, Springer Monographs in Mathematics, Springer (1998) [<a href="https://doi.org/10.1007/978-1-4471-1265-5">doi:10.1007/978-1-4471-1265-5</a>, <a href="https://www.maths.ed.ac.uk/~v1ranick/papers/crabbjames.pdf">pdf</a> ,<a href="https://www.gbv.de/dms/ilmenau/toc/243540361.PDF">pdf</a>]</p> </li> </ul> <p>On <a class="existingWikiWord" href="/nlab/show/model+structures+for+parameterized+spectra">model structures for parameterized spectra</a>:</p> <ul> <li id="MaySigurdsson06"> <p><a class="existingWikiWord" href="/nlab/show/Peter+May">Peter May</a>, <a class="existingWikiWord" href="/nlab/show/Johann+Sigurdsson">Johann Sigurdsson</a>, <em><a class="existingWikiWord" href="/nlab/show/Parametrized+Homotopy+Theory">Parametrized Homotopy Theory</a></em>, Mathematical Surveys and Monographs, vol. 132, AMS 2006 (<a href="https://bookstore.ams.org/surv-132">ISBN:978-0-8218-3922-5</a>, <a href="https://arxiv.org/abs/math/0411656">arXiv:math/0411656</a>, <a href="http://www.math.uchicago.edu/~may/EXTHEORY/MaySig.pdf">pdf</a>)</p> </li> <li id="BraunackMayer19"> <p><a class="existingWikiWord" href="/nlab/show/Vincent+Braunack-Mayer">Vincent Braunack-Mayer</a>, <em>Combinatorial parametrised spectra</em>, Algebr. Geom. Topol. <strong>21</strong> (2021) 801-891 [<a href="https://arxiv.org/abs/1907.08496">arXiv:1907.08496</a>, <a href="https://doi.org/10.2140/agt.2021.21.801">doi:10.2140/agt.2021.21.801</a>]</p> <blockquote> <p>(based on the <a class="existingWikiWord" href="/schreiber/show/thesis+Braunack-Mayer">PhD thesis</a>)</p> </blockquote> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fabian+Hebestreit">Fabian Hebestreit</a>, <a class="existingWikiWord" href="/nlab/show/Steffen+Sagave">Steffen Sagave</a>, <a class="existingWikiWord" href="/nlab/show/Christian+Schlichtkrull">Christian Schlichtkrull</a>, <em>Multiplicative parametrized homotopy theory via symmetric spectra in retractive spaces</em>, Forum of Mathematics, Sigma <strong>8</strong> (2020) e16 [<a href="https://arxiv.org/abs/1904.01824">arXiv:1904.01824</a>, <a href="https://doi.org/10.1017/fms.2020.11">doi:10.1017/fms.2020.11</a>]</p> </li> <li id="Malkiewich19"> <p><a class="existingWikiWord" href="/nlab/show/Cary+Malkiewich">Cary Malkiewich</a>, <em>Parametrized spectra, a low-tech approach</em> [<a href="https://arxiv.org/abs/1906.04773">arXiv:1906.04773</a>, user guide: <a href="https://people.math.binghamton.edu/malkiewich/users_guide_parametrized.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Malkiewich-ParametrizedSpectraUserGuide.pdf" title="pdf">pdf</a>]</p> </li> <li id="Malkiewich23"> <p><a class="existingWikiWord" href="/nlab/show/Cary+Malkiewich">Cary Malkiewich</a>, <em>A convenient category of parametrized spectra</em> [<a href="https://arxiv.org/abs/2305.15327">arXiv:2305.15327</a>]</p> </li> </ul> <p>Discussion in <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+theory">(∞,1)-category theory</a>:</p> <ul> <li id="AndoBlumbergGepner11"><a class="existingWikiWord" href="/nlab/show/Matthew+Ando">Matthew Ando</a>, <a class="existingWikiWord" href="/nlab/show/Andrew+Blumberg">Andrew Blumberg</a>, <a class="existingWikiWord" href="/nlab/show/David+Gepner">David Gepner</a>, <em>Parametrized spectra, multiplicative Thom spectra, and the twisted Umkehr map</em>, Geom. Topol. 22 (2018) 3761-3825 (<a href="http://arxiv.org/abs/1112.2203">arXiv:1112.2203</a>)</li> </ul> <p>Discussion as a <a class="existingWikiWord" href="/nlab/show/linear+homotopy+type+theory">linear homotopy type theory</a>:</p> <ul> <li id="Schreiber14"> <p><a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/schreiber/show/Quantization+via+Linear+homotopy+types">Quantization via Linear homotopy types</a></em> [<a href="http://arxiv.org/abs/1402.7041">arXiv:1402.7041</a>]</p> <blockquote> <p>(intended <a class="existingWikiWord" href="/nlab/show/categorical+semantics">categorical semantics</a>)</p> </blockquote> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Mitchell+Riley">Mitchell Riley</a>, <a class="existingWikiWord" href="/nlab/show/Eric+Finster">Eric Finster</a>, <a class="existingWikiWord" href="/nlab/show/Daniel+R.+Licata">Daniel R. Licata</a>, <em>Synthetic Spectra via a Monadic and Comonadic Modality</em> [<a href="https://arxiv.org/abs/2102.04099">arXiv:2102.04099</a>]</p> <blockquote> <p>(<a class="existingWikiWord" href="/nlab/show/inference+rules">inference rules</a> for the <a class="existingWikiWord" href="/nlab/show/classical+modality">classical modality</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>♮</mo></mrow><annotation encoding="application/x-tex">\natural</annotation></semantics></math>)</p> </blockquote> </li> <li id="Riley22Thesis"> <p><a class="existingWikiWord" href="/nlab/show/Mitchell+Riley">Mitchell Riley</a>, <em>A Bunched Homotopy Type Theory for Synthetic Stable Homotopy Theory</em>, PhD Thesis (2022) [<a href="https://doi.org/10.14418/wes01.3.139">doi:10.14418/wes01.3.139</a>, <a href="https://digitalcollections.wesleyan.edu/object/ir%3A3269">ir:3269</a>, <a href="https://mvr.hosting.nyu.edu/pubs/thesis.pdf">pdf</a>]</p> <blockquote> <p>(including <a class="existingWikiWord" href="/nlab/show/inference+rules">inference rules</a> for the <a class="existingWikiWord" href="/nlab/show/multiplicative+conjunction">multiplicative conjunction</a> by bringing in the required <a class="existingWikiWord" href="/nlab/show/bunched+logic">bunched logic</a>)</p> </blockquote> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on May 25, 2023 at 09:40:08. 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