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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="stabe_homotopy_theory">Stabe 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="loop_space_objects">Loop space objects</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <ul> <li><a href='#remarks'>Remarks</a></li> </ul> <li><a href='#explicit_constructions'>Explicit constructions</a></li> <ul> <li><a href='#FreeLoopSpaceObject'>Free loop space objects</a></li> <li><a href='#based_loop_space_objects'>Based loop space objects</a></li> <ul> <li><a href='#remarks_2'>Remarks</a></li> </ul> </ul> <li><a href='#remarks_3'>Remarks</a></li> <li><a href='#examples'>Examples</a></li> <li><a href='#related_concepts'>Related concepts</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>In the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> <a class="existingWikiWord" href="/nlab/show/Top">Top</a> the construction of a <a class="existingWikiWord" href="/nlab/show/loop+space">loop space</a> of a given <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> is familiar.</p> <p>This construction may be generalized to any other <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> and in fact to any other <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> with <a class="existingWikiWord" href="/nlab/show/homotopy+pullbacks">homotopy pullbacks</a>.</p> <h2 id="definition">Definition</h2> <p>Loop space objects are defined in any <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><mstyle mathvariant="bold"><mi>C</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{C}</annotation></semantics></math> with <a class="existingWikiWord" href="/nlab/show/homotopy+pullbacks">homotopy pullbacks</a>: 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> any <a class="existingWikiWord" href="/nlab/show/pointed+object">pointed object</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>C</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{C}</annotation></semantics></math> with point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">{*} \to X</annotation></semantics></math>, its <a class="existingWikiWord" href="/nlab/show/loop+space+object">loop space object</a> is <a class="existingWikiWord" href="/nlab/show/generalized+the">the</a> <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">\Omega X</annotation></semantics></math> of this point 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>Ω</mi><mi>X</mi></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><mo>*</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>X</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \Omega X &\to& {*} \\ \downarrow && \downarrow \\ {*} &\to& X } \,. </annotation></semantics></math></div> <p>A (<a class="existingWikiWord" href="/nlab/show/generalised+element">generalised</a>) <a class="existingWikiWord" href="/nlab/show/global+point">element</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">\Omega X</annotation></semantics></math> may be thought of as a <a class="existingWikiWord" href="/nlab/show/loop">loop</a> 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> at the <a class="existingWikiWord" href="/nlab/show/base+point">base point</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo></mrow><annotation encoding="application/x-tex">*</annotation></semantics></math>.</p> <p>When the point <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 not clear from context, we can write <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> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Omega(X,x)</annotation></semantics></math> to indicate the point.</p> <h3 id="remarks">Remarks</h3> <p>Since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>C</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{C}(X,-)</annotation></semantics></math> commutes with homotopy limits, one has a natural homotopy equivalence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Ω</mi> <mover><mi>y</mi><mo stretchy="false">¯</mo></mover></msub><mstyle mathvariant="bold"><mi>C</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>≃</mo><mstyle mathvariant="bold"><mi>C</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>Ω</mi> <mi>y</mi></msub><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Omega_{\bar{y}}\mathbf{C}(X,Y)\simeq \mathbf{C}(X,\Omega_y Y)</annotation></semantics></math>, for any objects <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 pointed object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(Y,y)</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>C</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{C}</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>y</mi><mo stretchy="false">¯</mo></mover></mrow><annotation encoding="application/x-tex">\bar{y}</annotation></semantics></math> denotes the morphism <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>Y</mi></mrow><annotation encoding="application/x-tex">X \to * \to Y</annotation></semantics></math>.</p> <p>See also</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fibration+sequence">fibration sequence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/delooping">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> </li> </ul> <h2 id="explicit_constructions">Explicit constructions</h2> <p>Usually the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> in question is <a class="existingWikiWord" href="/nlab/show/presentable+%28infinity%2C1%29-category">presented</a> by concrete 1-categorical data, such as that of a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a>. In that case the above <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a> has various realizations as an ordinary <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a>.</p> <p>Notably it may be expressed using <a class="existingWikiWord" href="/nlab/show/path+objects">path objects</a> which may come from <a class="existingWikiWord" href="/nlab/show/interval+objects">interval objects</a>. Even if the context is not (or not manifestly) that of a <a class="existingWikiWord" href="/nlab/show/homotopical+category">homotopical category</a>, an <a class="existingWikiWord" href="/nlab/show/interval+object">interval object</a> may still exist and may be used as indicated in the following to construct loop space objects.</p> <h3 id="FreeLoopSpaceObject">Free loop space objects</h3> <p>In a category with <a class="existingWikiWord" href="/nlab/show/interval+object">interval object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo><mover><mo>→</mo><mn>0</mn></mover><mi>I</mi><mover><mo>←</mo><mn>1</mn></mover><mo>*</mo></mrow><annotation encoding="application/x-tex"> * \xrightarrow{0} I \xleftarrow{1} * </annotation></semantics></math> the <strong><a class="existingWikiWord" href="/nlab/show/free+loop+space+object">free loop space object</a></strong> is the part of the <a class="existingWikiWord" 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fill-opacity="1"> <use xlink:href="#xE3obliC0rM9-k3p083iqgMl65U=-glyph-6-1" x="64.56725" y="75.416"></use> <use xlink:href="#xE3obliC0rM9-k3p083iqgMl65U=-glyph-6-2" x="68.37895" y="75.416"></use> </g> </svg> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex"> d_0 </annotation></semantics></math> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">d_1</annotation></semantics></math> resp.) is the composition of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mi>B</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> [0,B] </annotation></semantics></math> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mn>1</mn><mo>,</mo><mi>B</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[1,B]</annotation></semantics></math> resp.) with the canonical identification of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo>*</mo><mo>,</mo><mi>B</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[*, B]</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>.</p> <p>This is the same as the image of the <a class="existingWikiWord" href="/nlab/show/co-span+co-trace">co-span co-trace</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>cotr</mi><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">cotr(I)</annotation></semantics></math> of the interval object (which is the interval object closed to a loop!, see the examples at <a class="existingWikiWord" href="/nlab/show/co-span+co-trace">co-span co-trace</a>) in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>[</mo><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>cotr</mi><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↗</mo></mtd> <mtd></mtd> <mtd><mo>↖</mo></mtd></mtr> <mtr><mtd><mi>pt</mi></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>I</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mrow><mi>Id</mi><mo>⊔</mo><mi>Id</mi></mrow></msub><mo>↖</mo></mtd> <mtd></mtd> <mtd><msub><mo>↗</mo> <mrow><mi>in</mi><mo>⊔</mo><mi>out</mi></mrow></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>pt</mi><mo>⊔</mo><mi>pt</mi></mtd></mtr></mtable></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>,</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>B</mi><mo>]</mo></mrow><mspace width="thickmathspace"></mspace><mo>,</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mo>,</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>Λ</mi><mi>B</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><mi>B</mi></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">[</mo><mi>I</mi><mo>,</mo><mi>B</mi><mo stretchy="false">]</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mrow><mi>Id</mi><mo>×</mo><mi>Id</mi></mrow></msub><mo>↘</mo></mtd> <mtd></mtd> <mtd><msub><mo>↙</mo> <mrow><msub><mi>d</mi> <mn>0</mn></msub><mo>×</mo><msub><mi>d</mi> <mn>1</mn></msub></mrow></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>B</mi><mo>×</mo><mi>B</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \left[ \array{ && cotr(I) \\ & \nearrow && \nwarrow \\ pt &&&& I \\ & {}_{Id \sqcup Id}\nwarrow && \nearrow_{in \sqcup out} \\ && pt \sqcup pt } \;\;\;\;,\;\;\;\; B \right] \;,\;\;\;\; \simeq \;,\;\;\;\; \array{ && \Lambda B \\ & \swarrow && \searrow \\ B &&&& [I,B] \\ & {}_{Id \times Id}\searrow && \swarrow_{d_0 \times d_1} \\ && B \times B } </annotation></semantics></math></div> <h3 id="based_loop_space_objects">Based loop space objects</h3> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/pointed+object">pointed object</a> with point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>pt</mi><mover><mo>→</mo><mrow><msub><mi>pt</mi> <mi>B</mi></msub></mrow></mover><mi>B</mi></mrow><annotation encoding="application/x-tex">pt \stackrel{pt_B}{\to} B</annotation></semantics></math> then the <strong>based loop space object</strong> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> is the pullback <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Ω</mi> <mi>pt</mi></msub><mi>B</mi></mrow><annotation encoding="application/x-tex">\Omega_{pt} B</annotation></semantics></math> in</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>pt</mi></msub><mi>B</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mo stretchy="false">[</mo><mi>I</mi><mo>,</mo><mi>B</mi><mo stretchy="false">]</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mrow><msub><mi>d</mi> <mn>0</mn></msub><mo>×</mo><msub><mi>d</mi> <mn>1</mn></msub></mrow></msup></mtd></mtr> <mtr><mtd><mi>pt</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>pt</mi> <mi>B</mi></msub><mo>×</mo><msub><mi>pt</mi> <mi>B</mi></msub></mrow></mover></mtd> <mtd><mi>B</mi><mo>×</mo><mi>B</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \Omega_{pt}B &\to& [I,B] \\ \downarrow && \downarrow^{d_0 \times d_1} \\ pt &\stackrel{pt_B \times pt_B}{\to}& B \times B } \,. </annotation></semantics></math></div> <h4 id="remarks_2">Remarks</h4> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Ω</mi> <mi>pt</mi></msub><mi>B</mi></mrow><annotation encoding="application/x-tex">\Omega_{pt}B</annotation></semantics></math> is the fiber of the <a class="existingWikiWord" href="/nlab/show/generalized+universal+bundle">generalized universal bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>E</mi></mstyle> <mi>pt</mi></msub><mi>B</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">\mathbf{E}_{pt}B \to B</annotation></semantics></math>.</p> </li> <li> <p>the based loop space object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Ω</mi> <mi>pt</mi></msub><mi>B</mi></mrow><annotation encoding="application/x-tex">\Omega_{pt} B</annotation></semantics></math> is the pullback of the free loop space object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Λ</mi><mi>B</mi></mrow><annotation encoding="application/x-tex">\Lambda B</annotation></semantics></math> to the point</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>pt</mi></msub><mi>B</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>Λ</mi><mi>B</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>pt</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>pt</mi> <mi>B</mi></msub></mrow></mover></mtd> <mtd><mi>B</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \Omega_{pt} B &\to& \Lambda B \\ \downarrow && \downarrow \\ pt &\stackrel{pt_B}{\to}& B } \,. </annotation></semantics></math></div></li> </ul> <h2 id="remarks_3">Remarks</h2> <ul> <li> <p>The loop space object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> can be regarded as the homotopy trace on the identity span on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>, as described at <a class="existingWikiWord" href="/nlab/show/span+trace">span trace</a>.</p> </li> <li> <p>The free loop space object inherits the structure of an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">A_\infty</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/A-infinity-category">category</a> from that of the <a class="existingWikiWord" href="/nlab/show/path+object">path object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>I</mi><mo>,</mo><mi>B</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[I,B]</annotation></semantics></math>.</p> </li> <li> <p>In a <a class="existingWikiWord" href="/nlab/show/generalized+smooth+space">suitable extension</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="0em" rspace="thinmathspace">Diff</mo></mrow><annotation encoding="application/x-tex">\operatorname{Diff}</annotation></semantics></math>, this construction does <strong>not</strong> give the usual <em>smooth</em> loop space (free or based). It gives the space of paths with coincident endpoints rather than the space of smooth maps from the circle. Thus the <a class="existingWikiWord" href="/nlab/show/smooth+loop+space">smooth loop space</a> is not a loop space object.</p> </li> </ul> <h2 id="examples">Examples</h2> <ul> <li> <p>Let <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> with the standard <a class="existingWikiWord" href="/nlab/show/interval+object">interval object</a>. Then for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>=</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">B= X</annotation></semantics></math> a topological space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Λ</mi><mi>B</mi><mo>=</mo><mi>Λ</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">\Lambda B = \Lambda X</annotation></semantics></math> is the ordinary free <a class="existingWikiWord" href="/nlab/show/loop+space">loop space</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> <p>The generalization of this to <em>smooth</em> spaces is discussed at <a class="existingWikiWord" href="/nlab/show/smooth+loop+space">smooth loop space</a>.</p> </li> <li> <p>Let <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/Grpd">Grpd</a> with the standard interval object <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> and let <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> be the one-object groupoid corresponding to a group <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>, then</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Λ</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"> \Lambda \mathbf{B}G = G//_{Ad}G </annotation></semantics></math></div> <p>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 on itself by its adjoint action. Notice the example at <a class="existingWikiWord" href="/nlab/show/co-span+co-trace">co-span co-trace</a> which says that the cotrace on <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 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>cotr</mi><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">cotr(I) = \mathbf{B}\mathbb{Z}</annotation></semantics></math>, and indeed</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Λ</mi><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>=</mo><mo stretchy="false">[</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>ℤ</mi><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Lambda \mathbf{B}G = [\mathbf{B}\mathbb{Z}, \mathbf{B}G] \,. </annotation></semantics></math></div> <p>The role of this <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Λ</mi><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\Lambda \mathbf{B}G</annotation></semantics></math> as a loop object is amplified in particular in</p> <ul> <li> <p>Simon Willerton, <em>The twisted Drinfeld double of a finite group via gerbes and finite groupoids</em> (<a href="http://arxiv.org/abs/math.QA/0503266">arXiv</a>)</p> </li> <li> <p>Bruce Bartlett, <em>On unitary 2-representations of finite groups and topological quantum field theory</em> (<a href="http://arxiv.org/abs/0901.3975">arXiv</a>)</p> </li> </ul> </li> <li> <p>On the other hand, the <em>based</em> loop object 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> is just <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>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>=</mo><mi>G</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Omega \mathbf{B}G = G \,. </annotation></semantics></math></div></li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><strong>loop space object</strong>, <a class="existingWikiWord" href="/nlab/show/free+loop+space+object">free loop space object</a></p> <p><a class="existingWikiWord" href="/nlab/show/loop+space+type">loop space type</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/delooping">delooping</a>, <a class="existingWikiWord" href="/nlab/show/looping+and+delooping">looping and 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> <li> <p><a class="existingWikiWord" href="/nlab/show/inertia+orbifold">inertia orbifold</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>, <a class="existingWikiWord" href="/nlab/show/reduced+suspension">reduced suspension</a></li> </ul> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on December 5, 2022 at 04:20:14. 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