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href="/nlab/show/%28%E2%88%9E%2C1%29-category+theory">(∞,1)-category theory</a></strong></p> <p><strong>Background</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28n%2Cr%29-category">(n,r)-category</a></p> </li> </ul> <p><strong>Basic concepts</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hom-object+in+a+quasi-category">hom-objects</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivalence+in+a+quasi-category">equivalences in</a>/<a class="existingWikiWord" href="/nlab/show/equivalence+of+quasi-categories">of</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-categories</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sub-quasi-category">sub-(∞,1)-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/reflective+sub-%28%E2%88%9E%2C1%29-category">reflective sub-(∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/localization+of+an+%28%E2%88%9E%2C1%29-category">reflective localization</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/opposite+quasi-category">opposite (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/over+quasi-category">over (∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/join+of+quasi-categories">join of quasi-categories</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/exact+%28%E2%88%9E%2C1%29-functor">exact (∞,1)-functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-functors">(∞,1)-category of (∞,1)-functors</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-presheaves">(∞,1)-category of (∞,1)-presheaves</a></p> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/fibrations+of+quasi-categories">fibrations</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/inner+fibration">inner fibration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/left+fibration">left/right fibration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cartesian+fibration">Cartesian fibration</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Cartesian+morphism">Cartesian morphism</a></li> </ul> </li> </ul> </li> </ul> <p><strong>Universal constructions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/limit+in+quasi-categories">limit</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/terminal+object+in+a+quasi-category">terminal object</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+%28%E2%88%9E%2C1%29-functor">adjoint functors</a></p> </li> </ul> <p><strong>Local presentation</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+presentable+%28%E2%88%9E%2C1%29-category">locally presentable</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/essentially+small+%28%E2%88%9E%2C1%29-category">essentially small</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+small+%28%E2%88%9E%2C1%29-category">locally small</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/accessible+%28%E2%88%9E%2C1%29-category">accessible</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/idempotent-complete+%28%E2%88%9E%2C1%29-category">idempotent-complete</a></p> </li> </ul> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-Yoneda+lemma">(∞,1)-Yoneda lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-Grothendieck+construction">(∞,1)-Grothendieck construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+%28%E2%88%9E%2C1%29-functor+theorem">adjoint (∞,1)-functor theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monadicity+theorem">(∞,1)-monadicity theorem</a></p> </li> </ul> <p><strong>Extra stuff, structure, properties</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a></p> </li> </ul> <p><strong>Models</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category+with+weak+equivalences">category with weak equivalences</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derivator">derivator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quasi-category">quasi-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+quasi-categories">model structure for quasi-categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+Cartesian+fibrations">model structure for Cartesian fibrations</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relation+between+quasi-categories+and+simplicial+categories">relation to simplicial categories</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coherent+nerve">homotopy coherent nerve</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/simplicial+model+category">simplicial model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/presentable+%28%E2%88%9E%2C1%29-category">presentable quasi-category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+sets">model structure for Kan complexes</a></li> </ul> </li> </ul> </div></div> <h4 id="stable_homotopy_theory">Stable homotopy theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable homotopy theory</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/homological+algebra">homological algebra</a>, <a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a></li> </ul> <p><em><a class="existingWikiWord" href="/nlab/show/Introduction+to+Stable+Homotopy+Theory">Introduction</a></em></p> <h1 id="ingredients">Ingredients</h1> <ul> <li><a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a></li> </ul> <h1 id="contents">Contents</h1> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/loop+space+object">loop space object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/suspension+object">suspension object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/looping+and+delooping">looping and delooping</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/stabilization">stabilization</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/spectrum+object">spectrum object</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+derivator">stable derivator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/triangulated+category">triangulated category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category+of+spectra">stable (∞,1)-category of spectra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/spectrum">spectrum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+homotopy+category">stable homotopy category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smash+product+of+spectra">smash product of spectra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+smash+product+of+spectra">symmetric smash product of spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Spanier-Whitehead+duality">Spanier-Whitehead duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+ring">A-∞ ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+ring">E-∞ ring</a></p> </li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/stable+homotopy+theory+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="mapping_space">Mapping space</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom</a>/<a class="existingWikiWord" href="/nlab/show/mapping+space">mapping space</a></strong></p> <h3 id="general_abstract">General abstract</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/hom-set">hom-set</a>, <a class="existingWikiWord" href="/nlab/show/hom-object">hom-object</a>, <a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom</a>, <a class="existingWikiWord" href="/nlab/show/exponential+object">exponential object</a>, <a class="existingWikiWord" href="/nlab/show/derived+hom-space">derived hom-space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/loop+space+object">loop space object</a>, <a class="existingWikiWord" href="/nlab/show/free+loop+space+object">free loop space object</a>, <a class="existingWikiWord" href="/nlab/show/derived+loop+space">derived loop space</a></p> </li> </ul> <h3 id="topology">Topology</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+space">mapping space</a> (<a class="existingWikiWord" href="/nlab/show/compact-open+topology">compact-open topology</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topology+of+mapping+spaces">topology of mapping spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/evaluation+fibration+of+mapping+spaces">evaluation fibration of mapping spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/loop+space">loop space</a>, <a class="existingWikiWord" href="/nlab/show/free+loop+space">free loop space</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/free+loop+space+of+a+classifying+space">free loop space of a classifying space</a></li> </ul> </li> </ul> <h3 id="simplicial_homotopy_theory">Simplicial homotopy theory</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/simplicial+mapping+complex">simplicial mapping complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/inertia+groupoid">inertia groupoid</a></p> </li> </ul> <h3 id="differential_topology">Differential topology</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+topology+of+mapping+spaces">differential topology of mapping spaces</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/C-k+topology">C-k topology</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/manifold+structure+of+mapping+spaces">manifold structure of mapping spaces</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/tangent+spaces+of+mapping+spaces">tangent spaces of mapping spaces</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+loop+space">smooth loop space</a></p> </li> </ul> <h3 id="stable_homotopy_theory">Stable homotopy theory</h3> <ul> <li><a class="existingWikiWord" href="/nlab/show/mapping+spectrum">mapping spectrum</a></li> </ul> <h3 id="geometric_homotopy_theory">Geometric homotopy theory</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+stack">mapping stack</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/inertia+stack">inertia stack</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/free+loop+stack">free loop stack</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/mapping+space+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <ul> <li><a href='#in_higher_category_theory'>In higher category theory</a></li> <li><a href='#InHomotopyTypeTheory'>In homotopy type theory</a></li> </ul> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#HomPullModel'>Models by homotopy pullbacks</a></li> <li><a href='#RelationToBasedLoops'>Relation to based loop space object</a></li> <li><a href='#SliceAction'>Action on objects of the slice</a></li> <li><a href='#InATopos'>In an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-topos</a></li> <ul> <li><a href='#AsMappingSpaceObject'>As a mapping space object</a></li> <li><a href='#CircleAction'>Intrinsic circle action</a></li> </ul> <li><a href='#hochschild_cohomology_and_cyclic_cohomology'>Hochschild cohomology and cyclic cohomology</a></li> </ul> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#free_topological_loop_spaces'>Free topological loop spaces</a></li> <li><a href='#LoopsInBG'>Details for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathcal{L} \mathbf{B}G</annotation></semantics></math></a></li> <li><a href='#chern_character'>Chern character</a></li> <li><a href='#isotropy_of_a_topos'>Isotropy of a topos</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#References'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>In an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/pullbacks">pullbacks</a>, the <em>free loop space object</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">\mathcal{L}X</annotation></semantics></math> of any object <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> – also called the <em><a class="existingWikiWord" href="/nlab/show/inertia+groupoid">inertia groupoid</a></em> – is an object that behaves as if its <a class="existingWikiWord" href="/nlab/show/generalized+elements">generalized elements</a> are loops in <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>, morphisms between generalized elements <a class="existingWikiWord" href="/nlab/show/homotopy">homotopies</a> of loops, and so on.</p> <p>For the case that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">C = </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Top">Top</a> this reproduces the ordinary notion of <a class="existingWikiWord" href="/nlab/show/free+loop+spaces">free loop spaces</a> of <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a>.</p> <p>Over each fixed element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x \in X</annotation></semantics></math>, the free loop space object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">\mathcal{L}X</annotation></semantics></math> looks like the based <a class="existingWikiWord" href="/nlab/show/loop+space+object">loop space object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Ω</mi> <mi>x</mi></msub><mi>X</mi></mrow><annotation encoding="application/x-tex">\Omega_x X</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> <p>Free loop space objects come naturally equipped with various structures of interest, such as a categorical circle action. The <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">\mathcal{L}X</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Hochschild cohomology</a> or <a class="existingWikiWord" href="/nlab/show/cyclic+cohomology">cyclic cohomology</a> of function algebras <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C(X)</annotation></semantics></math> on <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>. The categorical circle action induces <a class="existingWikiWord" href="/nlab/show/differential">differential</a>s on these cohomologies, identifying them, in suitable cases, with algebras of <a class="existingWikiWord" href="/nlab/show/K%C3%A4hler+differential">Kähler</a> <a class="existingWikiWord" href="/nlab/show/differential+form">differential form</a>s on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> <h2 id="definition">Definition</h2> <h3 id="in_higher_category_theory">In higher category theory</h3> <p>There are various equivalent ways to define the free loop space object.</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> be an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a>. Recall that every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-category is <a class="existingWikiWord" href="/nlab/show/enriched+category">enriched</a> over <span class="newWikiWord">?Gpd<a href="/nlab/new/%3FGpd">?</a></span>, in that there is a hom-space <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>Y</mi><mo stretchy="false">)</mo><mo>∈</mo><mn>∞</mn><mi>Gpd</mi></mrow><annotation encoding="application/x-tex">Map(X,Y) \in \infty Gpd</annotation></semantics></math> for any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>,</mo><mi>Y</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">X,Y\in C</annotation></semantics></math>. This enables us to define the <a class="existingWikiWord" href="/nlab/show/power">power</a> of an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">X\in C</annotation></semantics></math> by any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoid <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>, as an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>X</mi> <mi>K</mi></msup><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">X^K \in C</annotation></semantics></math> together with a map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo>→</mo><mi>Map</mi><mo stretchy="false">(</mo><msup><mi>X</mi> <mi>K</mi></msup><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K \to Map(X^K,X)</annotation></semantics></math> inducing equivalences <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Map</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>X</mi><msup><mo stretchy="false">)</mo> <mi>K</mi></msup><mo>≃</mo><mi>Map</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><msup><mi>X</mi> <mi>K</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Map(Y,X)^K \simeq Map(Y,X^K)</annotation></semantics></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">Y\in C</annotation></semantics></math> (where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Map</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>X</mi><msup><mo stretchy="false">)</mo> <mi>K</mi></msup></mrow><annotation encoding="application/x-tex">Map(Y,X)^K</annotation></semantics></math> denotes the mapping-space from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Map</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Map(Y,X)</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn><mi>Gpd</mi></mrow><annotation encoding="application/x-tex">\infty Gpd</annotation></semantics></math>), if such exists.</p> <div class="num_defn"> <h6 id="definition_2">Definition</h6> <p>The <strong>free loop space object</strong> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">X\in C</annotation></semantics></math> is the power <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi><mi>X</mi><mo>=</mo><msup><mi>X</mi> <mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow></msup></mrow><annotation encoding="application/x-tex">\mathcal{L}X = X^{S^1}</annotation></semantics></math>, if it exists, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">S^1</annotation></semantics></math> denotes the homotopical <a class="existingWikiWord" href="/nlab/show/circle">circle</a>.</p> </div> <p>This can also be written in terms of “conical” limits in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>. Most commonly, if we note that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">S^1</annotation></semantics></math> is the pushout of two copies of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo></mrow><annotation encoding="application/x-tex">\ast</annotation></semantics></math> under <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo><mo>⊔</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">\ast\sqcup \ast</annotation></semantics></math>, and that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Map</mi><mo stretchy="false">(</mo><mo>*</mo><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">Map(\ast,X) \simeq X</annotation></semantics></math> while <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Map</mi><mo stretchy="false">(</mo><mo>*</mo><mo>⊔</mo><mo>*</mo><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>X</mi><mo>×</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">Map(\ast\sqcup \ast ,X) \simeq X\times X</annotation></semantics></math>, we find that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>X</mi> <mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow></msup></mrow><annotation encoding="application/x-tex">X^{S^1}</annotation></semantics></math> is the pullback 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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">X\times X</annotation></semantics></math>:</p> <div class="num_defn"> <h6 id="definition_3">Definition</h6> <p>In an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> with <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-pullbacks">(∞,1)-pullbacks</a>, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">X \in C</annotation></semantics></math> an <a class="existingWikiWord" href="/nlab/show/object">object</a>, its <strong>free loop space object</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">\mathcal{L}X</annotation></semantics></math> is the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-pullback of the <a class="existingWikiWord" href="/nlab/show/diagonal">diagonal</a> along itself</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>ℒ</mi><mi>X</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo stretchy="false">(</mo><mi>Id</mi><mo>,</mo><mi>Id</mi><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mrow><mo stretchy="false">(</mo><mi>Id</mi><mo>,</mo><mi>Id</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mi>X</mi><mo>×</mo><mi>X</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \mathcal{L} X &\to& X \\ \downarrow && \downarrow^{\mathrlap{(Id,Id)}} \\ X &\stackrel{(Id,Id)}{\to}& X \times X } \,. </annotation></semantics></math></div></div> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>This is the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-categorical <a class="existingWikiWord" href="/nlab/show/span+trace">span trace</a> of the identity-<a class="existingWikiWord" href="/nlab/show/span">span</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi><mi>X</mi><mo>=</mo><mi>Tr</mi><mrow><mo>(</mo><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>Id</mi></mpadded></msup><mo>↙</mo></mtd> <mtd></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mi>Id</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>X</mi></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathcal{L}X = Tr \left( \array{ && X \\ & {}^{\mathllap{Id}}\swarrow && \searrow^{\mathrlap{Id}} \\ X &&&& X } \right) \,. </annotation></semantics></math></div></div> <p>We can also use the fact that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">S^1</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/homotopy+coequalizer">homotopy coequalizer</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo><mo>⇉</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">\ast \rightrightarrows \ast</annotation></semantics></math>:</p> <div class="num_defn"> <h6 id="definition_4">Definition</h6> <p>The <strong>free loop space object</strong> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">X\in C</annotation></semantics></math> is the homotopy equalizer of two copies of the identity map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>⇉</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">X \rightrightarrows X</annotation></semantics></math>.</p> </div> <h3 id="InHomotopyTypeTheory">In homotopy type theory</h3> <p>In the literature (see <a href="#References">below</a>) when the free loop space object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">\mathcal{L}X</annotation></semantics></math> is defined as the homotopy pullback <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><msub><mo>×</mo> <mrow><mi>X</mi><mo>×</mo><mi>X</mi></mrow></msub><mi>X</mi></mrow><annotation encoding="application/x-tex">X\times_{X\times X} X</annotation></semantics></math>, it is sometimes described heuristically as: “a point of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">\mathcal{L}X</annotation></semantics></math> is a choice of making two points 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> equal in two ways.” In terms of <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a> this heuristics becomes a theorem. In that higher categorical logic we have the expression</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><mi>ℒ</mi><mi>X</mi></mtd> <mtd><mo>≔</mo><mrow><mo>{</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>:</mo><mi>X</mi><mspace width="thickmathspace"></mspace><mo stretchy="false">|</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><mi>x</mi><mo>=</mo><mi>y</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mi>and</mi><mspace width="thinmathspace"></mspace><mo stretchy="false">(</mo><mi>x</mi><mo>=</mo><mi>y</mi><mo stretchy="false">)</mo><mo>}</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mrow><mo>{</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>:</mo><mi>X</mi><mspace width="thickmathspace"></mspace><mo stretchy="false">|</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>y</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>}</mo></mrow></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} \mathcal{L}X & \coloneqq \left\{ x,y : X \;|\; (x = y) \, and\, (x = y) \right\} \\ & = \left\{ x,y : X \;|\; (x,x) = (y,y) \right\} \end{aligned} \,. </annotation></semantics></math></div> <p>Here on the right we have</p> <ol> <li> <p>the <a class="existingWikiWord" href="/nlab/show/dependent+sum">dependent sum</a>;</p> </li> <li> <p>over the <a class="existingWikiWord" href="/nlab/show/identity+type">identity type</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Id</mi><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Id (X \times X)</annotation></semantics></math>;</p> </li> <li> <p>of the <a class="existingWikiWord" href="/nlab/show/product+type">product type</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">X \times X</annotation></semantics></math>.</p> </li> </ol> <p>See the discussion at <em><a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a></em> in the section <em><a href="http://ncatlab.org/nlab/show/homotopy+pullback#ConstructionInHomotopyTypeTheory">Construction in homotopy type theory</a></em> for how this is equivalent to the previous definition.</p> <p>Now since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>y</mi><mo>:</mo><mi>X</mi></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>=</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\sum_{y:X} (x=y)</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/contractible+type">contractible</a>, the above type is equivalent to</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>x</mi><mo>:</mo><mi>X</mi></mrow></munder><mo stretchy="false">(</mo><mi>x</mi><mo>=</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \sum_{x:X} (x=x) </annotation></semantics></math></div> <p>i.e. the type of points that are equal to themselves (in a specified, not necessarily reflexivity, way). This corresponds to the other definitions, as a homotopy equalizer or a powering by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">S^1</annotation></semantics></math>.</p> <h2 id="properties">Properties</h2> <h3 id="HomPullModel">Models by homotopy pullbacks</h3> <p>To see what the definition of a free loop space object amounts to in more detail, assume that the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> is modeled by a <a class="existingWikiWord" href="/nlab/show/homotopical+category">homotopical category</a>, say for simplicity a <a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">category of fibrant objects</a>, for instance the full subcategory on fibrant objects of a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a>.</p> <p>Then following the discussion at <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a> and <a class="existingWikiWord" href="/nlab/show/generalized+universal+bundle">generalized universal bundle</a> we can compute the about <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-pullback <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><msubsup><mo>×</mo> <mrow><mi>X</mi><mo>×</mo><mi>X</mi></mrow> <mi>h</mi></msubsup><mi>X</mi></mrow><annotation encoding="application/x-tex">X\times_{X\times X}^h X</annotation></semantics></math> as the ordinary <a class="existingWikiWord" href="/nlab/show/limit">limit</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>ℒ</mi><mi>X</mi></mtd> <mtd><mo>→</mo></mtd> <mtd></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo stretchy="false">(</mo><mi>Id</mi><mo>,</mo><mi>Id</mi><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><mi>X</mi><msup><mo stretchy="false">)</mo> <mi>I</mi></msup></mtd> <mtd><mover><mo>→</mo><mrow><mo stretchy="false">(</mo><msub><mi>d</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>d</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mi>X</mi><mo>×</mo><mi>X</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo stretchy="false">(</mo><msub><mi>d</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>d</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mrow><mo stretchy="false">(</mo><mi>Id</mi><mo>,</mo><mi>Id</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mi>X</mi><mo>×</mo><mi>X</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathcal{L}X &\to& &\to& X \\ \downarrow && && \downarrow^{\mathrlap{(Id,Id)}} \\ && (X \times X)^I &\stackrel{(d_0,d_0)}{\to}& X \times X \\ \downarrow && {}^{\mathllap{(d_1,d_1)}}\downarrow \\ X &\stackrel{(Id,Id)}{\to} & X \times X \,, } </annotation></semantics></math></div> <p>where <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><mi>X</mi><msup><mo stretchy="false">)</mo> <mi>I</mi></msup></mrow><annotation encoding="application/x-tex">(X\times X)^I</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/path+space+object">path space object</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">X \times X</annotation></semantics></math>. At least if we have the structure of a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> we may take <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><mi>X</mi><msup><mo stretchy="false">)</mo> <mi>I</mi></msup><mo>=</mo><msup><mi>X</mi> <mi>I</mi></msup><mo>×</mo><msup><mi>X</mi> <mi>I</mi></msup></mrow><annotation encoding="application/x-tex">(X \times X)^I = X^I \times X^I</annotation></semantics></math> for a <a class="existingWikiWord" href="/nlab/show/path+space+object">path space object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>X</mi> <mi>I</mi></msup></mrow><annotation encoding="application/x-tex">X^I</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> <p>From this description one sees that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">\mathcal{L}X</annotation></semantics></math> is built from <em>pairs of paths</em> in <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 coinciding endpoints, that are <em>glued at their coinciding endpoint</em> . So the loops here are all built from two semi-ciricle paths.</p> <h3 id="RelationToBasedLoops">Relation to based loop space object</h3> <p>The fiber of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">\mathcal{L}X</annotation></semantics></math> over a <a class="existingWikiWord" href="/nlab/show/point">point</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>:</mo><mo>*</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x : {*} \to X</annotation></semantics></math> is the corresponding (based) <a class="existingWikiWord" href="/nlab/show/loop+space+object">loop space object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Ω</mi> <mi>x</mi></msub><mi>X</mi></mrow><annotation encoding="application/x-tex">\Omega_x X</annotation></semantics></math> 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>: we have an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>Ω</mi> <mi>x</mi></msub><mi>X</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>ℒ</mi><mi>X</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo stretchy="false">(</mo><mi>Id</mi><mo>,</mo><mi>Id</mi><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd><mover><mo>→</mo><mi>x</mi></mover></mtd> <mtd><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mrow><mo stretchy="false">(</mo><mi>Id</mi><mo>,</mo><mi>Id</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mi>X</mi><mo>×</mo><mi>X</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \Omega_x X &\to& \mathcal{L}X &\to& X \\ \downarrow && \downarrow && \downarrow^{\mathrlap{(Id,Id)}} \\ {*} &\stackrel{x}{\to} & X &\stackrel{(Id,Id)}{\to}& X\times X } \,. </annotation></semantics></math></div> <p>To see this, use that <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a>s paste to homotopy pullbacks, so that the outer pullback is modeled by the ordinary <a class="existingWikiWord" href="/nlab/show/limit">limit</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msubsup><mi>Ω</mi> <mi>x</mi> <mrow><mi>I</mi><mo>∨</mo><mi>I</mi></mrow></msubsup><mi>X</mi></mtd> <mtd><mo>→</mo></mtd> <mtd></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo stretchy="false">(</mo><mi>Id</mi><mo>,</mo><mi>Id</mi><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><mi>X</mi><msup><mo stretchy="false">)</mo> <mi>I</mi></msup></mtd> <mtd><mover><mo>→</mo><mrow><mo stretchy="false">(</mo><msub><mi>d</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>d</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mi>X</mi><mo>×</mo><mi>X</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo stretchy="false">(</mo><msub><mi>d</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>d</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd><mover><mo>→</mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mi>X</mi><mo>×</mo><mi>X</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \Omega_x^{I \vee I}X &\to& &\to& X \\ \downarrow && && \downarrow^{\mathrlap{(Id,Id)}} \\ && (X \times X)^I &\stackrel{(d_0,d_0)}{\to}& X \times X \\ \downarrow && {}^{\mathllap{(d_1,d_1)}}\downarrow \\ {*} &\stackrel{(x,x))}{\to} & X \times X \,, } </annotation></semantics></math></div> <p>which builds based loops on <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> from two consecutive paths, the first starting at the basepoint <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>, the second ending there. This is weakly equivalent <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Ω</mi> <mi>x</mi></msub><mi>X</mi><mo>=</mo><msubsup><mi>Ω</mi> <mi>x</mi> <mi>I</mi></msubsup><mi>X</mi><mover><mo>→</mo><mo>≃</mo></mover><msup><mi>Ω</mi> <mrow><mi>I</mi><mo>∨</mo><mi>I</mi></mrow></msup><mi>X</mi></mrow><annotation encoding="application/x-tex">\Omega_x X = \Omega^I_x X \stackrel{\simeq}{\to} \Omega^{I \vee I} X</annotation></semantics></math> to the based <a class="existingWikiWord" href="/nlab/show/loop+space+object">loop space object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Ω</mi> <mi>x</mi></msub><mi>X</mi></mrow><annotation encoding="application/x-tex">\Omega_x X</annotation></semantics></math> built from just the path space object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>X</mi> <mi>I</mi></msup></mrow><annotation encoding="application/x-tex">X^I</annotation></semantics></math> with a single copy of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math>, by standard arguments as for instance form page 12 on in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kenneth+Brown">Kenneth Brown</a>, <em><a class="existingWikiWord" href="/nlab/show/BrownAHT">Abstract Homotopy Theory and Generalized Sheaf Cohomology</a></em> .</li> </ul> <h3 id="SliceAction">Action on objects of the slice</h3> <p>The free loop space object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">\mathcal{L} X</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/group+object">group object</a> in the <a class="existingWikiWord" href="/nlab/show/slice+%28%E2%88%9E%2C1%29-category">slice (∞,1)-category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">/</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">C/X</annotation></semantics></math>, and has a canonical <a class="existingWikiWord" href="/nlab/show/action">action</a> on all objects of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">/</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">C/X</annotation></semantics></math>.</p> <p>Intuitively, the group structure comes from composition and inversion of loops. When the free loop space is expressed as a <a class="existingWikiWord" href="/nlab/show/power">power</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>X</mi> <mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow></msup></mrow><annotation encoding="application/x-tex">X^{S^1}</annotation></semantics></math>, this group structure comes from the canonical <a class="existingWikiWord" href="/nlab/show/cogroup">cogroup</a> structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">S^1</annotation></semantics></math>. In <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a>, it is literally concatenation of paths. And when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">\mathcal{L}X</annotation></semantics></math> is expressed as the pullback <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><msub><mo>×</mo> <mrow><mi>X</mi><mo>×</mo><mi>X</mi></mrow></msub><mi>X</mi></mrow><annotation encoding="application/x-tex">X\times_{X\times X} X</annotation></semantics></math>, the group multiplication can be obtained by considering the following pasting square:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>ℒ</mi><mi>X</mi><msub><mo>×</mo> <mi>X</mi></msub><mi>ℒ</mi><mi>X</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>ℒ</mi><mi>X</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>ℒ</mi><mi>X</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>X</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>X</mi><mo>×</mo><mi>X</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mi>id</mi></msup></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>X</mi><mo>×</mo><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mi>id</mi></mover></mtd> <mtd><mi>X</mi><mo>×</mo><mi>X</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{ \mathcal{L}X \times_X \mathcal{L}X & \to & \mathcal{L} X & \to & X\\ \downarrow && \downarrow && \downarrow \\ \mathcal{L} X & \to & X & \to & X\times X \\ \downarrow && \downarrow && \downarrow^{id} \\ X & \to & X\times X & \xrightarrow{id} & X\times X } </annotation></semantics></math></div> <p>Here the bottom-left and top-right squares are the pullback defining <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">\mathcal{L}X</annotation></semantics></math>, the top-left square is the pullback defining <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi><mi>X</mi><msub><mo>×</mo> <mi>X</mi></msub><mi>ℒ</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">\mathcal{L}X \times_X\mathcal{L}X</annotation></semantics></math>, and the bottom-right square commutes but is not a pullback. By the universal property of the pullback square defining <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">\mathcal{L}X</annotation></semantics></math>, this square factors through it uniquely, giving the composition map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi><mi>X</mi><msub><mo>×</mo> <mi>X</mi></msub><mi>ℒ</mi><mi>X</mi><mo>→</mo><mi>ℒ</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">\mathcal{L}X \times_X \mathcal{L}X\to \mathcal{L}X</annotation></semantics></math>. Similarly but more simply, the inversion map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi><mi>X</mi><mo>→</mo><mi>ℒ</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">\mathcal{L}X\to \mathcal{L}X</annotation></semantics></math> comes from transposing its defining pullback square and factoring it through itself.</p> <p>Now consider <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">Y\to X</annotation></semantics></math> an object of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">/</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">C/X</annotation></semantics></math>. There is a canonical projection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi><mi>X</mi><msub><mo>×</mo> <mi>X</mi></msub><mi>Y</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">\mathcal{L}X \times_X Y \to Y</annotation></semantics></math>, which is not the action of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">\mathcal{L}X</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math>, but it almost is. In fact since this projection is a morphism in the “homotopy” slice <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">/</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">C/X</annotation></semantics></math>, it consists not just of a morphism in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> but a homotopy witnessing that a certain triangle commutes, which is equivalently the homotopy in the left-hand commutative square below (which is the pullback defining <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi><mi>X</mi><msub><mo>×</mo> <mi>X</mi></msub><mi>Y</mi></mrow><annotation encoding="application/x-tex">\mathcal{L}X \times_X Y</annotation></semantics></math>):</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>ℒ</mi><mi>X</mi><msub><mo>×</mo> <mi>X</mi></msub><mi>Y</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>ℒ</mi><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mi>id</mi></mover></mtd> <mtd><mi>ℒ</mi><mi>X</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>Y</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mi>id</mi></mover></mtd> <mtd><mi>X</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{ \mathcal{L}X \times_X Y & \to & \mathcal{L} X & \xrightarrow{id} & \mathcal{L}X \\ \downarrow && \downarrow && \downarrow \\ Y & \to & X & \xrightarrow{id} & X} </annotation></semantics></math></div> <p>Pasting this with the right-hand commutative square, which is the canonical automorphism of the projection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi><mi>X</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\mathcal{L}X\to X</annotation></semantics></math>, we obtain a different homotopy witnessing a different morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi><mi>X</mi><msub><mo>×</mo> <mi>X</mi></msub><mi>Y</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">\mathcal{L}X \times_X Y \to Y</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">/</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">C/X</annotation></semantics></math> (with the same underlying morphism in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>), and <em>this</em> is the action of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">\mathcal{L}X</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math>.</p> <p>The definitiong of the right-hand commutative square above may not be obvious. It is clear when we write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi><mi>X</mi><mo>=</mo><msup><mi>X</mi> <mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow></msup></mrow><annotation encoding="application/x-tex">\mathcal{L}X = X^{S^1}</annotation></semantics></math>; when we write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi><mi>X</mi><mo>=</mo><mi>X</mi><msub><mo>×</mo> <mrow><mi>X</mi><mo>×</mo><mi>X</mi></mrow></msub><mi>X</mi></mrow><annotation encoding="application/x-tex">\mathcal{L}X = X\times_{X\times X} X</annotation></semantics></math> it can be obtained as the following pasting square, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi></mrow><annotation encoding="application/x-tex">q</annotation></semantics></math> are the two projections in the defining pullback of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">\mathcal{L}X</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>ℒ</mi><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mi>p</mi></mover></mtd> <mtd><mi>X</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>X</mi><mo>×</mo><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>π</mi> <mn>1</mn></msub></mrow></mover></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mo></mo><mi>id</mi></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mi>id</mi></msup></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>ℒ</mi><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mi>q</mi></mover></mtd> <mtd><mi>X</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>X</mi><mo>×</mo><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>π</mi> <mn>1</mn></msub></mrow></mover></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mo></mo><mi>id</mi></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mi>id</mi></msup></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>ℒ</mi><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mi>q</mi></mover></mtd> <mtd><mi>X</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>X</mi><mo>×</mo><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>π</mi> <mn>2</mn></msub></mrow></mover></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mo></mo><mi>id</mi></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mi>id</mi></msup></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>ℒ</mi><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mi>p</mi></mover></mtd> <mtd><mi>X</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>X</mi><mo>×</mo><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>π</mi> <mn>2</mn></msub></mrow></mover></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mo></mo><mi>id</mi></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mi>id</mi></msup></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>ℒ</mi><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mi>p</mi></mover></mtd> <mtd><mi>X</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>X</mi><mo>×</mo><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>π</mi> <mn>1</mn></msub></mrow></mover></mtd> <mtd><mi>X</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathcal{L}X & \xrightarrow{p} & X & \to & X\times X & \xrightarrow{\pi_1} & X\\ ^{id}\downarrow && && \downarrow^{id} && \downarrow \\ \mathcal{L}X & \xrightarrow{q} & X & \to & X\times X & \xrightarrow{\pi_1} & X\\ ^{id}\downarrow && \downarrow^{id} && && \downarrow \\ \mathcal{L}X & \xrightarrow{q} & X & \to & X\times X & \xrightarrow{\pi_2} & X\\ ^{id}\downarrow && && \downarrow^{id} && \downarrow \\ \mathcal{L}X & \xrightarrow{p} & X & \to & X\times X & \xrightarrow{\pi_2} & X\\ ^{id}\downarrow && \downarrow^{id} && && \downarrow \\ \mathcal{L}X & \xrightarrow{p} & X & \to & X\times X & \xrightarrow{\pi_1} & X } </annotation></semantics></math></div> <p>To extend these structures to a coherent <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-group structure and a coherent <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-action, see for instance <a href="https://mathoverflow.net/a/281937">this MO answer</a>.</p> <h3 id="InATopos">In an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-topos</h3> <p>We consider now the case that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>=</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">C = \mathbf{H}</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> (of <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-sheaves">(∞,1)-sheaves</a>/<a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stack">∞-stack</a>s). This comes canonically with its <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal</a> <a class="existingWikiWord" href="/nlab/show/global+section">global section</a>s <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-geometric+morphism">(∞,1)-geometric morphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>LConst</mi><mo>⊣</mo><mi>Γ</mi><mo stretchy="false">)</mo><mo>:</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mover><munder><mo>→</mo><mi>Γ</mi></munder><mover><mo>←</mo><mi>LConst</mi></mover></mover><mn>∞</mn><mi>Grpd</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (LConst \dashv \Gamma) : \mathbf{H} \stackrel{\overset{LConst}{\leftarrow}}{\underset{\Gamma}{\to}} \infty Grpd \,. </annotation></semantics></math></div> <p>In this case we can reformulate the power definition <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi><mi>X</mi><mo>=</mo><msup><mi>X</mi> <mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow></msup></mrow><annotation encoding="application/x-tex">\mathcal{L}X = X^{S^1}</annotation></semantics></math> using a version of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">S^1</annotation></semantics></math> that is an object of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> itself.</p> <h4 id="AsMappingSpaceObject">As a mapping space object</h4> <div class="num_defn"> <h6 id="definitionproposition">Definition/Proposition</h6> <p>Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup><mo>∈</mo><mi>Top</mi><mo>≃</mo><mn>∞</mn><mi>Grpd</mi><mover><mo>↪</mo><mi>LConst</mi></mover><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex"> S^1 \in Top \simeq \infty Grpd \stackrel{LConst}{\hookrightarrow} \mathbf{H} </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/circle">circle</a>. In <a class="existingWikiWord" href="/nlab/show/Top">Top</a> this is the usual topological circle. In <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a> this is (the <a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a> of) the <a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid">fundamental ∞-groupoid</a> of the topological circle. We may think of this as the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-pushout">(∞,1)-pushout</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup><mo>≃</mo><mo>*</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mrow><mo>*</mo><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mo>*</mo></mrow></munder><mo>*</mo></mrow><annotation encoding="application/x-tex"> S^1 \simeq * \coprod_{* \coprod *} * </annotation></semantics></math></div> <p>hence as the universal <a class="existingWikiWord" href="/nlab/show/cocone">cocone</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo>*</mo><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mo>*</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><msub><mo>⇙</mo> <mo>≃</mo></msub></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><msup><mi>S</mi> <mn>1</mn></msup></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ * \coprod * &\to& * \\ \downarrow &\swArrow_{\simeq}& \downarrow \\ * &\to& S^1 } </annotation></semantics></math></div> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">\infty Grpd</annotation></semantics></math>.</p> <p>In <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> we still write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">S^1</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/constant+%E2%88%9E-stack">constant ∞-stack</a> on this, the image of this under <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>LConst</mi></mrow><annotation encoding="application/x-tex">LConst</annotation></semantics></math>. Since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>LConst</mi></mrow><annotation encoding="application/x-tex">LConst</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> and hence preserves this poushout, there is no risk of confusion.</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>To see that the given <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-pushout indeed produces the circle, we use the standard <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+sets">model structure on simplicial sets</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>sSet</mi> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">sSet_{Quillen}</annotation></semantics></math> to <a class="existingWikiWord" href="/nlab/show/presentable+%28%E2%88%9E%2C1%29-category">present</a> <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a>. In <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>sSet</mi> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">sSet_{Quillen}</annotation></semantics></math> the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-pushout is computed by the <a class="existingWikiWord" href="/nlab/show/homotopy+pushout">homotopy pushout</a>. By general facts about this, it may be computed as an ordinary <a class="existingWikiWord" href="/nlab/show/pushout">pushout</a> in <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a> once we pass to an equivalent pushout diagram in which at least one morphism is a <a class="existingWikiWord" href="/nlab/show/monomorphism">monomorphism</a>. This is the case for</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo>*</mo><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mo>*</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo stretchy="false">/</mo><mo>∂</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ * \coprod * &\to& * \\ \downarrow && \downarrow \\ \Delta[1] &\to& \Delta[1]/\partial \Delta[1] } \,. </annotation></semantics></math></div></div> <div class="num_remark"> <h6 id="remark_2">Remark</h6> <p>Informally the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-pushout <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mrow><mo>*</mo><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mo>*</mo></mrow></msub><mo>*</mo></mrow><annotation encoding="application/x-tex">* \coprod_{* \coprod *} *</annotation></semantics></math> may be thought of as</p> <ul> <li> <p>the disjoint union of two points <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>*</mo> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">*_1</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>*</mo> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">*_2</annotation></semantics></math>;</p> </li> <li> <p>equipped with <em>two</em> non-equivalent abstract <a class="existingWikiWord" href="/nlab/show/homotopies">homotopies</a> between them</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd></mtd> <mtd><mo>↗</mo><msup><mo>↘</mo> <mpadded width="0"><mo>≃</mo></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mo>*</mo> <mn>1</mn></msub></mtd> <mtd></mtd> <mtd><msub><mo>*</mo> <mn>2</mn></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo><msub><mo>↗</mo> <mpadded width="0" lspace="-100%width"><mo>≃</mo></mpadded></msub></mtd></mtr></mtable></mrow><mo>}</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> S^1 = \left\{ \array{ & \nearrow \searrow^{\mathrlap{\simeq}} \\ *_1 && *_2 \\ & \searrow \nearrow_{\mathllap{\simeq}} } \right\} \,. </annotation></semantics></math></div></li> </ul> <p>This equivalent way of modelling the circle not as a single point with an automorphism, but as two points with two isomorphisms is what connects directly to the definition of the free loop space object. This we now come to. It is also the fundamenal source of the basic structure of <a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Hochschild (co)homology</a> (as discussed there).</p> </div> <p> <div class='num_lemma'> <h6>Lemma</h6> <p></p> <p>Every <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> is a <a class="existingWikiWord" href="/nlab/show/cartesian+closed+%28%E2%88%9E%2C1%29-category">cartesian closed (∞,1)-category</a>: we have for every object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">X \in \mathbf{H}</annotation></semantics></math> an <a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom</a>-<a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</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><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo><mo>:</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>→</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [X,-] : \mathbf{H} \to \mathbf{H} \,. </annotation></semantics></math></div> <p></p> </div> </p> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>This is discussed at <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> in the section <a href="http://ncatlab.org/nlab/show/(infinity,1)-topos#ClosedMonoidalStructure">Closed monoidal structure</a>.</p> </div> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>There is a natural <a class="existingWikiWord" href="/nlab/show/equivalence+in+an+%28%E2%88%9E%2C1%29-category">equivalence</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi><mi>X</mi><mo>≃</mo><mo stretchy="false">[</mo><msup><mi>S</mi> <mn>1</mn></msup><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathcal{L}X \simeq [S^1 , X] \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_3">Proof</h6> <p>This follows from the above by the fact (see <a href="http://ncatlab.org/nlab/show/(infinity%2C1)-topos#ClosedMonoidalStructure">closed monoidal structure on (∞,1)-toposes</a>) that the internal hom in an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-topos preserves finite colimits in its first argument and satisfies</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo>*</mo><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo><mo>≃</mo><mi>X</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [*,X] \simeq X \,. </annotation></semantics></math></div> <p>This yields</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><mo stretchy="false">[</mo><msup><mi>S</mi> <mn>1</mn></msup><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mtd> <mtd><mo>≃</mo><mo stretchy="false">[</mo><mo>*</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mrow><mo>*</mo><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mo>*</mo></mrow></munder><mo>*</mo><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><mo stretchy="false">[</mo><mo>*</mo><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo><msub><mo>×</mo> <mrow><mo stretchy="false">[</mo><mo>*</mo><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mo>*</mo><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mrow></msub><mo stretchy="false">[</mo><mo>*</mo><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><mi>X</mi><msub><mo>×</mo> <mrow><mi>X</mi><mo>×</mo><mi>X</mi></mrow></msub><mi>X</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><mi>ℒ</mi><mi>X</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} [S^1 ,X] &\simeq [* \coprod_{* \coprod *} *, X] \\ & \simeq [*,X] \times_{[* \coprod *, X]} [*,X] \\ & \simeq X \times_{X \times X} X \\ & \simeq \mathcal{L}X \,. \end{aligned} </annotation></semantics></math></div></div> <div class="num_prop"> <h6 id="corollary">Corollary</h6> <p>We have that the free loop space object of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">X \in \mathbf{H}</annotation></semantics></math> is equivalently the <a href="http://ncatlab.org/nlab/show/limit+in+a+quasi-category#Tensoring">powering</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> by the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoid <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">S^1</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi><mi>X</mi><mo>≃</mo><msup><mi>X</mi> <mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow></msup></mrow><annotation encoding="application/x-tex"> \mathcal{L} X \simeq X^{S^1} </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_4">Proof</h6> <p>Follows by the above from the equivalence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>LConst</mi><msup><mi>S</mi> <mn>1</mn></msup><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo><mo>≃</mo><msup><mi>X</mi> <mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow></msup></mrow><annotation encoding="application/x-tex">[LConst S^1 , X] \simeq X^{S^1}</annotation></semantics></math> discussed at <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a>.</p> </div> <h4 id="CircleAction">Intrinsic circle action</h4> <p>By precomposition, the <a class="existingWikiWord" href="/nlab/show/automorphism+2-group">automorphism 2-group</a> of the <a class="existingWikiWord" href="/nlab/show/circle">circle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">S^1</annotation></semantics></math> acts on free loop space of an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">X \in \mathbf{H}</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Aut</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">[</mo><msup><mi>S</mi> <mn>1</mn></msup><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo><mo>→</mo><mo stretchy="false">[</mo><msup><mi>S</mi> <mn>1</mn></msup><mo>,</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">]</mo><mo>×</mo><mo stretchy="false">[</mo><msup><mi>S</mi> <mn>1</mn></msup><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo><mover><mo>→</mo><mo>∘</mo></mover><mo stretchy="false">[</mo><msup><mi>S</mi> <mn>1</mn></msup><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Aut(S^1) \times [S^1, X] \to [S^1, S^1] \times [S^1, X] \stackrel{\circ}{\to} [S^1, X] \,. </annotation></semantics></math></div> <div class="num_prop"> <h6 id="proposition_2">Proposition</h6> <p>The connected component of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>S</mi> <mn>1</mn></msup><mo>,</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[S^1,S^1]</annotation></semantics></math> on the identity is equivalent to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">S^1</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>S</mi> <mn>1</mn></msup><mo>,</mo><msup><mi>S</mi> <mn>1</mn></msup><msub><mo stretchy="false">]</mo> <mi>Id</mi></msub><mo>≃</mo><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex"> [S^1 , S^1]_{Id} \simeq S^1 </annotation></semantics></math></div></div> <div class="num_defn"> <h6 id="definition_5">Definition</h6> <p>We say that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup><mo>×</mo><mo stretchy="false">[</mo><msup><mi>S</mi> <mn>1</mn></msup><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo><mo>≃</mo><mo stretchy="false">[</mo><msup><mi>S</mi> <mn>1</mn></msup><mo>,</mo><msup><mi>S</mi> <mn>1</mn></msup><msub><mo stretchy="false">]</mo> <mi>Id</mi></msub><mo>×</mo><mo stretchy="false">[</mo><msup><mi>S</mi> <mn>1</mn></msup><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo><mover><mo>→</mo><mo>∘</mo></mover><mo stretchy="false">[</mo><msup><mi>S</mi> <mn>1</mn></msup><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> S^1 \times [S^1, X] \simeq [S^1, S^1]_{Id} \times [S^1, X] \stackrel{\circ}{\to} [S^1, X] </annotation></semantics></math></div> <p>is the intrinsic <strong>circle action</strong> on the free loop space object.</p> </div> <div class="proof"> <h6 id="proof_5">Proof</h6> <p>We spell out in detail what this action looks like. The reader should thoughout keep the <a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis</a>-equivalence, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo stretchy="false">|</mo><mo>−</mo><mo stretchy="false">|</mo><mo>⊣</mo><mi>Π</mi><mo stretchy="false">)</mo><mo>:</mo><mi>Top</mi><mo>≃</mo><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">(|-| \dashv \Pi) : Top \simeq \infty Grpd</annotation></semantics></math> in mind.</p> <p>We may realize the <a class="existingWikiWord" href="/nlab/show/circle">circle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup><mo>∈</mo></mrow><annotation encoding="application/x-tex">S^1 \in </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Top">Top</a> under <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi><mo>:</mo><mi>Top</mi><mo>≃</mo><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">\Pi : Top \simeq \infty Grpd</annotation></semantics></math> as the [delooping]] <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}\mathbb{Z}</annotation></semantics></math> of the additive <a class="existingWikiWord" href="/nlab/show/group">group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/integer">integer</a>s</p> <p>The <a class="existingWikiWord" href="/nlab/show/automorphism+2-group">automorphism 2-group</a> of this object is the <a class="existingWikiWord" href="/nlab/show/functor+category">functor groupoid</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Aut</mi> <mi>Grpd</mi></msub><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Aut_{Grpd}(\mathbf{B}\mathbb{Z}) </annotation></semantics></math></div> <p>whose <a class="existingWikiWord" href="/nlab/show/object">object</a>s are invertible <a class="existingWikiWord" href="/nlab/show/functor">functor</a>s <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>ℤ</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}\mathbb{Z} \to \mathbf{B}\mathbb{Z}</annotation></semantics></math> and whose <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a>s are <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a>s between these.</p> <p>The functors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>ℤ</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}\mathbb{Z} \to \mathbf{B}\mathbb{Z}</annotation></semantics></math> correspond bijectively to group <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a>s <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo>→</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z} \to \mathbb{Z}</annotation></semantics></math>, hence to multiplication by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">n\in\mathbb{Z}</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>:</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>ℤ</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>ℤ</mi></mrow><annotation encoding="application/x-tex"> [n] : \mathbf{B}\mathbb{Z} \to \mathbf{B}\mathbb{Z} </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo>•</mo><mover><mo>→</mo><mi>k</mi></mover><mo>•</mo><mo stretchy="false">)</mo><mo>↦</mo><mo stretchy="false">(</mo><mo>•</mo><mover><mo>→</mo><mrow><mi>n</mi><mo>⋅</mo><mi>k</mi></mrow></mover><mo>•</mo><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex"> (\bullet \stackrel{k}{\to} \bullet) \mapsto (\bullet \stackrel{n\cdot k}{\to} \bullet). </annotation></semantics></math></div> <p>Natural transformations between two such endomorphisms are given by a component <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℓ</mi><mo>∈</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\ell \in \mathbb{Z}</annotation></semantics></math> such that all diagrams</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo>•</mo></mtd> <mtd><mover><mo>→</mo><mi>ℓ</mi></mover></mtd> <mtd><mo>•</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mi>n</mi><mo>⋅</mo><mi>k</mi></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>n</mi><mo>′</mo><mo>⋅</mo><mi>k</mi></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mo>•</mo></mtd> <mtd><mover><mo>→</mo><mi>ℓ</mi></mover></mtd> <mtd><mo>•</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \bullet &\stackrel{\ell}{\to}& \bullet \\ {}^{\mathllap{n\cdot k}}\downarrow && \downarrow^{\mathrlap{n' \cdot k}} \\ \bullet &\stackrel{\ell}{\to}& \bullet } </annotation></semantics></math></div> <p>commute in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}\mathbb{Z}</annotation></semantics></math>. This can happen only for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mi>n</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">n = n'</annotation></semantics></math>, but then it happens for arbitrary <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℓ</mi></mrow><annotation encoding="application/x-tex">\ell</annotation></semantics></math>.</p> <p>In other words we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Aut</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>ℤ</mi><mo stretchy="false">)</mo><mo>≃</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mrow><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>∈</mo><msup><mi>ℤ</mi> <mo>×</mo></msup></mrow></munder><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>ℤ</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Aut(\mathbf{B}\mathbb{Z}) \simeq \coprod_{[n] \in \mathbb{Z}^\times}\mathbf{B}\mathbb{Z} \,. </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Aut</mi> <mi>Id</mi></msub><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>ℤ</mi><mo stretchy="false">)</mo><mo>≃</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>ℤ</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Aut_{Id}(\mathbf{B}\mathbb{Z}) \simeq \mathbf{B}\mathbb{Z} \,. </annotation></semantics></math></div> <p>The object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[n]</annotation></semantics></math> corresponds to the self-mapping of the circle that fixes the basepoint and has winding number <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">n\in\mathbb{Z}</annotation></semantics></math>. The transformation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℓ</mi></mrow><annotation encoding="application/x-tex">\ell</annotation></semantics></math> corresponds then to a rigid rotation of the loop by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℓ</mi></mrow><annotation encoding="application/x-tex">\ell</annotation></semantics></math> full circles</p> <p>Notably for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n = 1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">k = 1</annotation></semantics></math> we may think of the diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo>•</mo></mtd> <mtd><mover><mo>→</mo><mi>ℓ</mi></mover></mtd> <mtd><mo>•</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mn>1</mn></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mn>1</mn></mpadded></msup></mtd></mtr> <mtr><mtd><mo>•</mo></mtd> <mtd><mover><mo>→</mo><mi>ℓ</mi></mover></mtd> <mtd><mo>•</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \bullet &\stackrel{\ell}{\to}& \bullet \\ {}^{\mathllap{1}}\downarrow && \downarrow^{\mathrlap{1}} \\ \bullet &\stackrel{\ell}{\to}& \bullet } </annotation></semantics></math></div> <p>as depicting the unit loop around the circle (on the left, say) and the result of translating its basepoint <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℓ</mi></mrow><annotation encoding="application/x-tex">\ell</annotation></semantics></math>-times around the circle (the rest of the diagram). Of course since we are using a model of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">S^1</annotation></semantics></math> with a single object here, every rotation of the loop is a full circle rotation, which is a bit hard to see.</p> </div> <blockquote> <p>Exercise: spell out the above discussion analogously for the equivalent model given by the <a class="existingWikiWord" href="/nlab/show/fundamental+groupoid">fundamental groupoid</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Pi_1(S^1)</annotation></semantics></math> of the standard circle. The is the groupoid with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>S</mi> <mi>Top</mi> <mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">S^1_{Top}</annotation></semantics></math> as its set of objects homotopy classes of paths in the circle as morphisms. In this model things look more like one might expect from a circle action. Notice that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}\mathbb{Z}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/skeleton">skeleton</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Pi_1(S^1)</annotation></semantics></math>.</p> </blockquote> <div class="num_example"> <h6 id="example">Example</h6> <p>Consider <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>=</mo></mrow><annotation encoding="application/x-tex">\mathbf{H} = </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/group">group</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>=</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">X = \mathbf{B}G</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/delooping+groupoid">delooping groupoid</a>. Then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi><mi>X</mi><mo>=</mo><mi>G</mi><mo stretchy="false">/</mo><msub><mo stretchy="false">/</mo> <mi>Ad</mi></msub><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathcal{L}X = G//_{Ad}G</annotation></semantics></math> (as discussed in detail <a href="#LoopsInBG">below</a>). A morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>g</mi><mover><mo>→</mo><mi>h</mi></mover><msub><mi>Ad</mi> <mi>h</mi></msub><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(g \stackrel{h}{\to} Ad_h a)</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">G//G</annotation></semantics></math> corresponds to a natural transformation</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd><mo>↗</mo><msup><mo>↘</mo> <mpadded width="0"><mi>g</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>ℤ</mi></mtd> <mtd><msup><mo>⇓</mo> <mi>h</mi></msup></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo><msub><mo>↗</mo> <mpadded width="0"><mrow><msub><mi>Ad</mi> <mi>h</mi></msub><mi>g</mi></mrow></mpadded></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ & \nearrow \searrow^{\mathrlap{g}} \\ \mathbf{B}\mathbb{Z} &\Downarrow^{h}& \mathbf{B}G \\ & \searrow \nearrow_{\mathrlap{Ad_h g}} } \,. </annotation></semantics></math></div> <p>Precomposing this with the automorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℓ</mi></mrow><annotation encoding="application/x-tex">\ell</annotation></semantics></math> of the object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[n]</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>END</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">END(\mathbf{B}\mathbb{Z})</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd><mo>↗</mo><msup><mo>↘</mo> <mpadded width="0"><mi>n</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>ℤ</mi></mtd> <mtd><msup><mo>⇓</mo> <mi>ℓ</mi></msup></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>ℤ</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo><msub><mo>↗</mo> <mpadded width="0"><mi>n</mi></mpadded></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ & \nearrow \searrow^{\mathrlap{n}} \\ \mathbf{B}\mathbb{Z} &\Downarrow^{\ell}& \mathbf{B}\mathbb{Z} \\ & \searrow \nearrow_{\mathrlap{n}} } </annotation></semantics></math></div> <p>produces the new transformation</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd><mo>↗</mo><msup><mo>↘</mo> <mpadded width="0"><mi>n</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><mo>↗</mo><msup><mo>↘</mo> <mpadded width="0"><mi>g</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>ℤ</mi></mtd> <mtd><msup><mo>⇓</mo> <mi>ℓ</mi></msup></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>ℤ</mi></mtd> <mtd><msup><mo>⇓</mo> <mi>h</mi></msup></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo><msub><mo>↗</mo> <mpadded width="0"><mi>n</mi></mpadded></msub></mtd> <mtd></mtd> <mtd><mo>↘</mo><msub><mo>↗</mo> <mpadded width="0"><mrow><msub><mi>Ad</mi> <mi>h</mi></msub><mi>g</mi></mrow></mpadded></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ & \nearrow \searrow^{\mathrlap{n}} && \nearrow \searrow^{\mathrlap{g}} \\ \mathbf{B}\mathbb{Z} &\Downarrow^{\ell}& \mathbf{B}\mathbb{Z} &\Downarrow^{h}& \mathbf{B}G \\ & \searrow \nearrow_{\mathrlap{n}} && \searrow \nearrow_{\mathrlap{Ad_h g}} } \,. </annotation></semantics></math></div> <p>By the rules of horizontal composition of <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a>s, this is the transformation whose component naturality square on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo>•</mo><mover><mo>→</mo><mn>1</mn></mover><mo>•</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\bullet \stackrel{1}{\to} \bullet)</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}\mathbb{Z}</annotation></semantics></math> is the diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo>•</mo></mtd> <mtd><mover><mo>→</mo><mrow><msup><mi>g</mi> <mi>ℓ</mi></msup></mrow></mover></mtd> <mtd><mo>•</mo></mtd> <mtd><mover><mo>→</mo><mi>h</mi></mover></mtd> <mtd><mo>•</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msup><mi>g</mi> <mi>n</mi></msup></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mrow></mrow> <mrow><msup><mi>g</mi> <mi>n</mi></msup></mrow></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>Ad</mi> <mi>h</mi></msub><msup><mi>g</mi> <mi>n</mi></msup></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mo>•</mo></mtd> <mtd><munder><mo>→</mo><mrow><msup><mi>g</mi> <mi>ℓ</mi></msup></mrow></munder></mtd> <mtd><mo>•</mo></mtd> <mtd><munder><mo>→</mo><mi>h</mi></munder></mtd> <mtd><mo>•</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \bullet &\stackrel{g^\ell}{\to}& \bullet &\stackrel{h}{\to}&\bullet \\ {}^{\mathllap{g^{n}}}\downarrow && {}^{g^n}\downarrow && \downarrow^{\mathrlap{Ad_h g^n}} \\ \bullet &\underset{g^\ell}{\to}& \bullet &\underset{h}{\to}&\bullet } </annotation></semantics></math></div> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}\mathbb{Z}</annotation></semantics></math>, hence the morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>g</mi> <mi>n</mi></msup><mover><mo>→</mo><mrow><msup><mi>g</mi> <mi>ℓ</mi></msup><mi>h</mi></mrow></mover><msub><mi>Ad</mi> <mi>h</mi></msub><msup><mi>g</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(g^n \stackrel{g^{\ell} h}{\to} Ad_h g^n)</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo stretchy="false">/</mo><msub><mo stretchy="false">/</mo> <mi>Ad</mi></msub><mi>G</mi></mrow><annotation encoding="application/x-tex">G//_{Ad}G</annotation></semantics></math>. In particular, the categorical circle action is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ℓ</mi><mo>:</mo><mo stretchy="false">(</mo><mi>g</mi><mover><mo>→</mo><mi>h</mi></mover><msub><mi>Ad</mi> <mi>h</mi></msub><mi>g</mi><mo stretchy="false">)</mo><mo>↦</mo><mo stretchy="false">(</mo><mi>g</mi><mover><mo>→</mo><mrow><msup><mi>g</mi> <mi>ℓ</mi></msup><mi>h</mi></mrow></mover><msub><mi>Ad</mi> <mi>h</mi></msub><mi>g</mi><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex"> \ell:(g \stackrel{h}{\to} Ad_h g)\mapsto (g \stackrel{g^{\ell} h}{\to} Ad_h g). </annotation></semantics></math></div></div> <h3 id="hochschild_cohomology_and_cyclic_cohomology">Hochschild cohomology and cyclic cohomology</h3> <p><a class="existingWikiWord" href="/nlab/show/quasicoherent+%E2%88%9E-stack">quasicoherent ∞-stack</a>s on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">\mathcal{L}X</annotation></semantics></math> form the <a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Hochschild homology</a> object 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> (if the axioms of <a class="existingWikiWord" href="/nlab/show/geometric+function+theory">geometric function theory</a> are met) as described there. The circle acton on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">\mathcal{L}X</annotation></semantics></math> induces differentials on these.</p> <blockquote> <p>… details to be written, but see <a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Hochschild cohomology</a> and <a class="existingWikiWord" href="/nlab/show/cyclic+cohomology">cyclic cohomology</a> for more.</p> </blockquote> <h2 id="examples">Examples</h2> <h3 id="free_topological_loop_spaces">Free topological loop spaces</h3> <p>In <a class="existingWikiWord" href="/nlab/show/Top">Top</a> the notion of free loop space objects reproduces the standard notion of topological <a class="existingWikiWord" href="/nlab/show/free+loop+space">free loop space</a>s.</p> <h3 id="LoopsInBG">Details for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathcal{L} \mathbf{B}G</annotation></semantics></math></h3> <p>Let the ambient <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> be <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a>, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> be an ordinary <a class="existingWikiWord" href="/nlab/show/group">group</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math> its one-object <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a>.</p> <div class="num_prop"> <h6 id="proposition_3">Proposition</h6> <p>We have that the <a class="existingWikiWord" href="/nlab/show/loop+groupoid">loop groupoid</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>≃</mo><mi>G</mi><mo stretchy="false">/</mo><msub><mo stretchy="false">/</mo> <mi>Ad</mi></msub><mi>G</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \mathcal{L} \mathbf{B}G \simeq G//_{Ad} G \,, </annotation></semantics></math></div> <p>the <a class="existingWikiWord" href="/nlab/show/action+groupoid">action groupoid</a> of the adjoint action of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> on itself.</p> </div> <div class="proof"> <h6 id="proof_6">Proof</h6> <p>We spell this out in full pedestrian detail, as a little exercise in computing <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a>s.</p> <p>We have that the <a class="existingWikiWord" href="/nlab/show/path+space+object">path space object</a> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><msup><mi>G</mi> <mi>I</mi></msup><mo>=</mo><mo stretchy="false">[</mo><mi>I</mi><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}G^I = [I,\mathbf{B}G]</annotation></semantics></math> – the <a class="existingWikiWord" href="/nlab/show/functor+category">functor groupoid</a>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> is the free groupoid <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>=</mo><mo stretchy="false">{</mo><mi>a</mi><mover><mo>→</mo><mo>≃</mo></mover><mi>b</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">I = \{a \stackrel{\simeq}{\to} b\}</annotation></semantics></math> on the standard <a class="existingWikiWord" href="/nlab/show/interval+object">interval object</a> – which is (by the definition of <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a>) the <a class="existingWikiWord" href="/nlab/show/action+groupoid">action groupoid</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><msup><mi>G</mi> <mi>I</mi></msup><mo>=</mo><mi>G</mi><mo>\</mo><mo>\</mo><mi>G</mi><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi></mrow><annotation encoding="application/x-tex"> \mathbf{B}G^I = G\backslash \backslash G // G </annotation></semantics></math></div> <p>for the action of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> on itself, by <em>inverse</em> left and direct right multiplication separately: the naturality square of a <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a> defining a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mover><mo>→</mo><mrow><msub><mi>h</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>h</mi> <mn>2</mn></msub></mrow></mover><msubsup><mi>h</mi> <mn>1</mn> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mi>g</mi><msub><mi>h</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">g \stackrel{h_1,h_2}{\to} h_1^{-1} g h_2</annotation></semantics></math> in this groupoid is the commuting square</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo>•</mo></mtd> <mtd><mover><mo>→</mo><mi>g</mi></mover></mtd> <mtd><mo>•</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>h</mi> <mn>1</mn></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>h</mi> <mn>2</mn></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mo>•</mo></mtd> <mtd><mover><mo>→</mo><mrow><msubsup><mi>h</mi> <mn>1</mn> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mi>g</mi><msub><mi>h</mi> <mn>2</mn></msub></mrow></mover></mtd> <mtd><mo>•</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \bullet &\stackrel{g}{\to}& \bullet \\ {}^{\mathllap{h_1}}\downarrow && \downarrow^{\mathrlap{h_2}} \\ \bullet &\stackrel{h_1^{-1}g h_2}{\to}& \bullet } </annotation></semantics></math></div> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>=</mo><mo>*</mo><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G = {*}//G</annotation></semantics></math>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> of the top right corner of the above defining limit diagram is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo stretchy="false">(</mo><mi>G</mi><mo>\</mo><mo>\</mo><mi>G</mi><mo>×</mo><mi>G</mi><mo>\</mo><mo>\</mo><mi>G</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo stretchy="false">(</mo><mi>Id</mi><mo>,</mo><mi>Id</mi><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mo stretchy="false">(</mo><mi>G</mi><mo>\</mo><mo>\</mo><mi>G</mi><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">(</mo><mi>G</mi><mo>\</mo><mo>\</mo><mi>G</mi><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>×</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ (G\backslash\backslash G \times G\backslash\backslash G)//G &\to& \mathbf{B}G \\ \downarrow && \downarrow^{\mathrlap{(Id,Id)}} \\ (G\backslash\backslash G//G) \times (G\backslash\backslash G//G) &\to& \mathbf{B}G \times \mathbf{B}G } </annotation></semantics></math></div> <p>identifying the two actions from the right, and then the remaining pullback completing the limit diagram is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>G</mi><mo>\</mo><mo>\</mo><mo stretchy="false">(</mo><mi>G</mi><mo>×</mo><mi>G</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>G</mi><mo>\</mo><mo>\</mo><mi>G</mi><mo>×</mo><mi>G</mi><mo>\</mo><mo>\</mo><mi>G</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd> <mtd><mover><mo>→</mo><mrow><mo stretchy="false">(</mo><mi>Id</mi><mo>,</mo><mi>Id</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>×</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ G\backslash\backslash (G\times G)//G &\to& (G\backslash\backslash G \times G\backslash\backslash G)//G \\ \downarrow && \downarrow \\ \mathbf{B}G &\stackrel{(Id,Id)}{\to}& \mathbf{B}G \times \mathbf{B}G } </annotation></semantics></math></div> <p>now identifying also the two actions from the left, so that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>\</mo><mo>\</mo><mo stretchy="false">(</mo><mi>G</mi><mo>×</mo><mi>G</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">G\backslash\backslash (G\times G)//G</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/action+groupoid">action groupoid</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> acting diagonally on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>×</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">G \times G</annotation></semantics></math> by multiplication from the left and from the right, separately.</p> <p>To see better what this is, we pass to an equivalent smaller groupoid (the <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a> is defined, of course, only up to weak equivalence). Notice that every morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>g</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><msub><mi>h</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>h</mi> <mn>2</mn></msub></mrow></mover><mo stretchy="false">(</mo><mi>g</mi><msub><mo>′</mo> <mn>1</mn></msub><mo>,</mo><mi>g</mi><msub><mo>′</mo> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(g_1,g_2) \stackrel{h_1,h_2}{\to} (g'_1, g'_2)</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>\</mo><mo>\</mo><mo stretchy="false">(</mo><mi>G</mi><mo>×</mo><mi>G</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">G\backslash\backslash (G\times G)//G</annotation></semantics></math> corresponding to a <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo>•</mo></mtd> <mtd><mover><mo>→</mo><mrow><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>g</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mo>•</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo stretchy="false">(</mo><msub><mi>h</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>h</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo stretchy="false">(</mo><msub><mi>h</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>h</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mo>•</mo></mtd> <mtd><mover><mo>→</mo><mrow><mo stretchy="false">(</mo><msubsup><mi>h</mi> <mn>1</mn> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><msub><mi>g</mi> <mn>1</mn></msub><msub><mi>h</mi> <mn>2</mn></msub><mo>,</mo><msubsup><mi>h</mi> <mn>1</mn> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><msub><mi>g</mi> <mn>2</mn></msub><msub><mi>h</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mo>•</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \bullet &\stackrel{(g_1,g_2)}{\to}& \bullet \\ {}^{\mathllap{(h_1,h_1)}}\downarrow && \downarrow^{\mathrlap{(h_2,h_2)}} \\ \bullet &\stackrel{(h_1^{-1} g_1 h_2, h_1^{-1} g_2 h_2)}{\to} & \bullet } </annotation></semantics></math></div> <p>between functors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>×</mo><mi>I</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>×</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">I\times I \to \mathbf{B}G \times \mathbf{B}G</annotation></semantics></math> may always be decomposed as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo>•</mo></mtd> <mtd><mover><mo>→</mo><mrow><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>g</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mo>•</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo stretchy="false">(</mo><msubsup><mi>g</mi> <mn>2</mn> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo>,</mo><msubsup><mi>g</mi> <mn>2</mn> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mo>•</mo></mtd> <mtd><mover><mo>→</mo><mrow><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><msubsup><mi>g</mi> <mn>2</mn> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mo>•</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo stretchy="false">(</mo><msub><mi>h</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>h</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo stretchy="false">(</mo><msub><mi>h</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>h</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mo>•</mo></mtd> <mtd><mover><mo>→</mo><mrow><mo stretchy="false">(</mo><msubsup><mi>h</mi> <mn>1</mn> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><msubsup><mi>g</mi> <mn>2</mn> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">)</mo><msub><mi>h</mi> <mn>1</mn></msub><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mo>•</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo stretchy="false">(</mo><mi>g</mi><msub><mo>′</mo> <mn>2</mn></msub><mo>,</mo><mi>g</mi><msub><mo>′</mo> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mo>•</mo></mtd> <mtd><mover><mo>→</mo><mrow><mo stretchy="false">(</mo><msubsup><mi>h</mi> <mn>1</mn> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><msubsup><mi>g</mi> <mn>2</mn> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">)</mo><msub><mi>h</mi> <mn>1</mn></msub><mi>g</mi><msub><mo>′</mo> <mn>2</mn></msub><mo>,</mo><mi>g</mi><msub><mo>′</mo> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mo>•</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \bullet &\stackrel{(g_1,g_2)}{\to}& \bullet \\ {}^{\mathllap{(e,e)}}\downarrow && \downarrow^{\mathrlap{(g_2^{-1}, g_2^{-1})}} \\ \bullet &\stackrel{(g_1 g_2^{-1}, e)}{\to}& \bullet \\ {}^{\mathllap{(h_1,h_1)}}\downarrow && \downarrow^{\mathrlap{(h_1,h_1)}} \\ \bullet &\stackrel{(h_1^{-1}(g_1 g_2^{-1})h_1, e)}{\to}& \bullet \\ {}^{\mathllap{(e,e)}}\downarrow && \downarrow^{\mathrlap{(g'_2,g'_2)}} \\ \bullet &\stackrel{(h_1^{-1}(g_1 g_2^{-1})h_1 g'_2, g'_2)}{\to}& \bullet } \,. </annotation></semantics></math></div> <p>Staring at this for a moment shows that this is a unique factorization of every morphism through one of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo>•</mo></mtd> <mtd><mover><mo>→</mo><mrow><mo stretchy="false">(</mo><mi>g</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mo>•</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>k</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>k</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mo>•</mo></mtd> <mtd><mover><mo>→</mo><mrow><mo stretchy="false">(</mo><msub><mi>Ad</mi> <mi>k</mi></msub><mi>g</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mo>•</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ \bullet & \stackrel{(g,e)}{\to} & \bullet \\ {}^{\mathllap{k}}\downarrow && \downarrow^{\mathrlap{k}} \\ \bullet & \stackrel{(Ad_k g,e)}{\to} & \bullet } \,, </annotation></semantics></math></div> <p>which is naturally identified with a morphism in the <a class="existingWikiWord" href="/nlab/show/action+groupoid">action groupoid</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo stretchy="false">/</mo><msub><mo stretchy="false">/</mo> <mi>Ad</mi></msub><mi>G</mi></mrow><annotation encoding="application/x-tex">G//_{Ad} G</annotation></semantics></math> of the adjoint action of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> on itself.</p> <p>This means that the inclusion</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo stretchy="false">/</mo><msub><mo stretchy="false">/</mo> <mi>Ad</mi></msub><mi>G</mi><mover><mo>↪</mo><mrow></mrow></mover><mi>G</mi><mo>\</mo><mo>\</mo><mo stretchy="false">(</mo><mi>G</mi><mo>×</mo><mi>G</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi></mrow><annotation encoding="application/x-tex"> G//_{Ad} G \stackrel{}{\hookrightarrow} G\backslash\backslash(G \times G)//G </annotation></semantics></math></div> <p>given by this identification is <a class="existingWikiWord" href="/nlab/show/essentially+surjective+functor">essentially surjective</a> and <a class="existingWikiWord" href="/nlab/show/full+and+faithful+functor">full and faithful</a>, and hence an <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence of groupoids</a>.</p> <p>So in conclusion we have that the free loop space object of the <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math> of a group is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>≃</mo><mi>G</mi><mo stretchy="false">/</mo><msub><mo stretchy="false">/</mo> <mi>Ad</mi></msub><mi>G</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathcal{L} \mathbf{B}G \simeq G//_{Ad}G \,. </annotation></semantics></math></div></div> <h3 id="chern_character">Chern character</h3> <p>We describe how the <a class="existingWikiWord" href="/nlab/show/Chern+character">Chern character</a> of <a class="existingWikiWord" href="/nlab/show/vector+bundle">vector bundle</a>s over <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> may be realized in terms of the <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> of the free loop space object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">\mathcal{L}X</annotation></semantics></math>.</p> <p>Assume now <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is a nice category of smooth spaces, and let <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> be an object of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>.</p> <p>Consider a <a class="existingWikiWord" href="/nlab/show/group+object">group object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> and a <a class="existingWikiWord" href="/nlab/show/representation">representation</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> given by a group homomorphism to the <a class="existingWikiWord" href="/nlab/show/general+linear+group">general linear group</a> (in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>): <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo>:</mo><mi>G</mi><mo>→</mo><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo>;</mo><mi>ℂ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\rho:G\to GL(n;\mathbb{C})</annotation></semantics></math>. For instance <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> could be <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GL(n)</annotation></semantics></math> itself and this morphism the identity.</p> <p>The <a class="existingWikiWord" href="/nlab/show/trace">trace</a> of the representation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math> is invariant under conjugation in the group and so defnes a map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Tr</mi><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo><mo>:</mo><mi>G</mi><mo stretchy="false">/</mo><msub><mo stretchy="false">/</mo> <mi>Ad</mi></msub><mi>G</mi><mo>→</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">Tr(\rho): G//_{Ad}G\to \mathbb {C}</annotation></semantics></math> – a <a class="existingWikiWord" href="/nlab/show/class+function">class function</a>. By the equivalence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>≃</mo><mi>G</mi><mo stretchy="false">/</mo><msub><mo stretchy="false">/</mo> <mi>Ad</mi></msub><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathcal{L}\mathbf{B}G \simeq G//_{Ad} G</annotation></semantics></math> discussed above, this may be regarded as a <a class="existingWikiWord" href="/nlab/show/characteristic+class">characteristic class</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Tr</mi><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>:</mo><mi>ℒ</mi><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>→</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex"> Tr(\rho(-)) : \mathcal{L}\mathbf{B}G\to \mathbb {C} </annotation></semantics></math></div> <p>on the free loop space of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>:</mo><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">g : X\to \mathbf{B}G</annotation></semantics></math> of a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a> on <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> transgresses to a cocycle</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi><mi>g</mi><mo>:</mo><mi>ℒ</mi><mi>X</mi><mo>→</mo><mi>ℒ</mi><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex"> \mathcal{L} g : \mathcal{L}X \to \mathcal{L}\mathbf{B}G </annotation></semantics></math></div> <p>on the free loop space, by the functoriality of the free loop space object construction.</p> <p>The above <a class="existingWikiWord" href="/nlab/show/characteristic+class">characteristic class</a> of this cocycle is the composite morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Tr</mi><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">(</mo><mi>ℒ</mi><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>:</mo><mi>ℒ</mi><mi>X</mi><mo>→</mo><mi>ℒ</mi><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>→</mo><mi>ℂ</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> Tr(\rho(\mathcal{L}g)) : \mathcal{L}X \to \mathcal{L} \mathbf{B}G \to \mathbb{C} \,, </annotation></semantics></math></div> <p>which by the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ad</mi></mrow><annotation encoding="application/x-tex">Ad</annotation></semantics></math>-invariance of the <a class="existingWikiWord" href="/nlab/show/trace">trace</a> is now <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">S^1</annotation></semantics></math>-invariant and hence defines an element in the <a class="existingWikiWord" href="/nlab/show/cyclic+cohomology">cyclic cohomology</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>ℒ</mi><mi>X</mi><mo>,</mo><mi>ℂ</mi><msup><mo stretchy="false">)</mo> <mrow><msubsup><mi>S</mi> <mi>C</mi> <mn>1</mn></msubsup></mrow></msup></mrow><annotation encoding="application/x-tex">C(\mathcal{L}X,\mathbb{C})^{S^1_C}</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> <p>The Hom-space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>ℒ</mi><mi>X</mi><mo>,</mo><mi>ℂ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C(\mathcal{L}X,\mathbb{C})</annotation></semantics></math> is a model for the graded commutative algebra of complex-valued <a class="existingWikiWord" href="/nlab/show/differential+form">differential form</a>s on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, with the categorical circle action corresponding to the <a class="existingWikiWord" href="/nlab/show/de+Rham+complex">de Rham differential</a>. Hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>ℒ</mi><mi>X</mi><mo>,</mo><mi>ℂ</mi><msup><mo stretchy="false">)</mo> <mrow><msubsup><mi>S</mi> <mi>C</mi> <mn>1</mn></msubsup></mrow></msup></mrow><annotation encoding="application/x-tex">C(\mathcal{L}X,\mathbb{C})^{S^1_C}</annotation></semantics></math> is a model for closed forms and maps to <a class="existingWikiWord" href="/nlab/show/de+Rham+cohomology">de Rham cohomology</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>H</mi> <mi>dR</mi> <mo>•</mo></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H_{dR}^\bullet(X)</annotation></semantics></math> 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>. If the <a class="existingWikiWord" href="/nlab/show/de+Rham+theorem">de Rham theorem</a> holds for <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> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>, then this may be identified with the real cohomology <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℝ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^\bullet(X,\mathbb{R})</annotation></semantics></math>.</p> <p>In the case that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>=</mo><mi>GL</mi><mo stretchy="false">(</mo><mn>∞</mn><mo>;</mo><mi>ℂ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G=GL(\infty;\mathbb{C})</annotation></semantics></math>, the compatibility of the trace with <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a>s and <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a>s of <a class="existingWikiWord" href="/nlab/show/vector+bundle">vector bundle</a>s over <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> makes the above construction a <a class="existingWikiWord" href="/nlab/show/ring">ring</a> homomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>H</mi> <mi>dR</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(X)\to H_{dR}(X)</annotation></semantics></math> from the <a class="existingWikiWord" href="/nlab/show/topological+K-theory">topological K-theory</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> to <a class="existingWikiWord" href="/nlab/show/de+Rham+cohomology">de Rham cohomology</a>, hence a very good candidate to being the <a class="existingWikiWord" href="/nlab/show/Chern+character">Chern character</a></p> <p>( <em>to be completed…</em> )</p> <h3 id="isotropy_of_a_topos">Isotropy of a topos</h3> <p>The <a class="existingWikiWord" href="/nlab/show/isotropy+group+of+a+topos">isotropy group of a topos</a> is its free loop space object in the 2-category <a class="existingWikiWord" href="/nlab/show/Topos">Topos</a>.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/loop+space+object">loop space object</a>, <strong>free loop space object</strong>,</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/delooping">delooping</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/loop+space">loop space</a>, <a class="existingWikiWord" href="/nlab/show/free+loop+space">free loop space</a>, <a class="existingWikiWord" href="/nlab/show/derived+loop+space">derived loop space</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/suspension+object">suspension object</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/suspension">suspension</a></li> </ul> </li> </ul> <h2 id="References">References</h2> <p>Free loop space objects in the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> of <a class="existingWikiWord" href="/nlab/show/derived+stack">derived stack</a>s on the site of <a class="existingWikiWord" href="/nlab/show/differential+graded+algebra">differential graded algebra</a>s are discussed in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/David+Ben-Zvi">David Ben-Zvi</a>, <a class="existingWikiWord" href="/nlab/show/David+Nadler">David Nadler</a>, <em>Loop Spaces and Connections</em> (<a href="http://arxiv.org/abs/1002.3636">arXiv</a>)</li> </ul> <p>More information in the topological case is given in:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ronnie+Brown">Ronnie Brown</a><em>Crossed modules and the homotopy 2-type of a free loop space</em>, (<a href="http://arxiv.org/abs/1003.5617">arXiv</a>)</li> </ul> <p>which gives complete information on the 2-type of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">\mathcal{L}X</annotation></semantics></math> for a space <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> which is the classifying space of a crossed module of groups. This generalises the above example of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex"> \mathcal{L} \mathbf{B}G</annotation></semantics></math>.</p> </body></html> </div> <div class="revisedby"> <p> Last revised on October 31, 2021 at 05:15:25. 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