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delooping in nLab

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<div id="Content"> <h1 id="pageName"> <span style="float: left; margin: 0.5em 0.25em -0.25em 0"> <svg xmlns="http://www.w3.org/2000/svg" width="1.872em" height="1.8em" viewBox="0 0 190 181"> <path fill="#226622" d="M72.8 145c-1.6 17.3-15.7 10-23.6 20.2-5.6 7.3 4.8 15 11.4 15 11.5-.2 19-13.4 26.4-20.3 3.3-3 8.2-4 11.2-7.2a14 14 0 0 0 2.9-11.1c-1.4-9.6-12.4-18.6-16.9-27.2-5-9.6-10.7-27.4-24.1-27.7-17.4-.3-.4 26 4.7 30.7 2.4 2.3 5.4 4.1 7.3 6.9 1.6 2.3 2.1 5.8-1 7.2-5.9 2.6-12.4-6.3-15.5-10-8.8-10.6-15.5-23-26.2-31.8-5.2-4.3-11.8-8-18-3.7-7.3 4.9-4.2 12.9.2 18.5a81 81 0 0 0 30.7 23c3.3 1.5 12.8 5.6 10 10.7-2.5 5.2-11.7 3-15.6 1.1-8.4-3.8-24.3-21.3-34.4-13.7-3.5 2.6-2.3 7.6-1.2 11.1 2.8 9 12.2 17.2 20.9 20.5 17.3 6.7 34.3-8 50.8-12.1z"/> <path fill="#a41e32" d="M145.9 121.3c-.2-7.5 0-19.6-4.5-26-5.4-7.5-12.9-1-14.1 5.8-1.4 7.8 2.7 14.1 4.8 21.3 3.4 12 5.8 29-.8 40.1-3.6-6.7-5.2-13-7-20.4-2.1-8.2-12.8-13.2-15.1-1.9-2 9.7 9 21.2 12 30.1 1.2 4 2 8.8 6.4 10.3 6.9 2.3 13.3-4.7 17.7-8.8 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href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/15872/#Item_2" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="category_theory"><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 theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" 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> </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> <li><a href='#remarks'>Remarks</a></li> <li><a href='#characterization_of_deloopable_objects'>Characterization of deloopable objects</a></li> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#topological_loop_spaces'>Topological loop spaces</a></li> <li><a href='#delooping_of_a_group_to_a_groupoid'>Delooping of a group to a groupoid</a></li> </ul> <li><a href='#DeloopingOfHigherCategoricalStructures'>Deloopings of higher categorical structures</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>The <em>delooping</em> of an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is, if it exists, a uniquely <a class="existingWikiWord" href="/nlab/show/pointed+object">pointed object</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>A</mi></mrow><annotation encoding="application/x-tex">\mathbf{B} A</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/loop+space+object">loop space object</a> 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>A</mi></mrow><annotation encoding="application/x-tex">\mathbf{B} A</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>A</mi><mo>≃</mo><mi>Ω</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> A \simeq \Omega(\mathbf{B} A) </annotation></semantics></math></div> <p>In particular, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>=</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">A = G</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/group">group</a> then its delooping</p> <ul> <li> <p>in the context <a class="existingWikiWord" href="/nlab/show/Top">Top</a> is the <a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℬ</mi><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathcal{B}G</annotation></semantics></math></p> </li> <li> <p>in the context <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Grpd">∞-Grpd</a> is the one-object groupoid <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> </li> </ul> <p>Under the <a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis</a> these two objects are identified: the <a class="existingWikiWord" href="/nlab/show/geometric+realization">geometric realization</a> of the groupoid <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 the classifying space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℬ</mi><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathcal{B}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><mo stretchy="false">|</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo stretchy="false">|</mo><mo>≃</mo><mi>ℬ</mi><mi>G</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> |\mathbf{B}G| \simeq \mathcal{B}G \,. </annotation></semantics></math></div> <p>See <a class="existingWikiWord" href="/nlab/show/looping+and+delooping">looping and delooping</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 &amp;\to&amp; {*} \\ \downarrow &amp;&amp; \downarrow \\ {*} &amp;\to&amp; X } \,. </annotation></semantics></math></div> <p>Conversely, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is given and a <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy 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><mi>A</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><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>A</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ A &amp;\to&amp; {*} \\ \downarrow &amp;&amp; \downarrow \\ {*} &amp;\to&amp; \mathbf{B}A } </annotation></semantics></math></div> <p>exists, with the point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>A</mi></mrow><annotation encoding="application/x-tex">{*} \to \mathbf{B} A</annotation></semantics></math> being essentially unique, by the above <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> has been realized as the <a class="existingWikiWord" href="/nlab/show/loop+space+object">loop space object</a> 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>A</mi></mrow><annotation encoding="application/x-tex">\mathbf{B} A</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>A</mi><mo>=</mo><mi>Ω</mi><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>A</mi></mrow><annotation encoding="application/x-tex"> A = \Omega \mathbf{B} A </annotation></semantics></math></div> <p>and we say 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>A</mi></mrow><annotation encoding="application/x-tex">\mathbf{B} A</annotation></semantics></math> is the <strong>delooping</strong> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>.</p> <p>See the section <a href="http://ncatlab.org/nlab/show/groupoid+object+in+an+(infinity%2C1)-category#Delooping">delooping</a> at <a class="existingWikiWord" href="/nlab/show/groupoid+object+in+an+%28%E2%88%9E%2C1%29-category">groupoid object in an (∞,1)-category</a> for more.</p> <h2 id="remarks">Remarks</h2> <p>If <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> is even a <a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a> then all deloopings exist and are then also denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mi>A</mi></mrow><annotation encoding="application/x-tex">\Sigma A</annotation></semantics></math> and called the <strong><a class="existingWikiWord" href="/nlab/show/suspension">suspension</a></strong> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>.</p> <h2 id="characterization_of_deloopable_objects">Characterization of deloopable objects</h2> <p>In section 6.1.3 of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, <a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">Higher Topos Theory</a></li> </ul> <p>a definition of <a class="existingWikiWord" href="/nlab/show/groupoid+object+in+an+%28infinity%2C1%29-category">groupoid object in an (infinity,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> is given as a homotopy simplicial object, i.e. a <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-functor">(infinity,1)-functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>:</mo><msup><mi>Δ</mi> <mi>op</mi></msup><mo>→</mo><mstyle mathvariant="bold"><mi>C</mi></mstyle></mrow><annotation encoding="application/x-tex"> C : \Delta^{op} \to \mathbf{C} </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>⋯</mi><msub><mi>C</mi> <mn>2</mn></msub><mover><mo>⇉</mo><mo>→</mo></mover><msub><mi>C</mi> <mn>1</mn></msub><mo>⇉</mo><msub><mi>C</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex"> \cdots C_2 \stackrel{\to}\rightrightarrows C_1 \rightrightarrows C_0 </annotation></semantics></math></div> <p>satisfying certain conditions (prop. 6.1.2.6) which are such that if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mn>0</mn></msub><mo>=</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">C_0 = {*}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/point">point</a> we have an internal <em>group</em> in a homotopical sense, given by an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">C_1</annotation></semantics></math> equipped with a coherently associative multiplication operation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mn>1</mn></msub><mo>×</mo><msub><mi>C</mi> <mn>1</mn></msub><mo>→</mo><msub><mi>C</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">C_1 \times C_1 \to C_1</annotation></semantics></math> generalizing that of <a class="existingWikiWord" href="/nlab/show/Jim+Stasheff">Stasheff</a> <a class="existingWikiWord" href="/nlab/show/H-space">H-space</a> from 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>-category <a class="existingWikiWord" href="/nlab/show/Top">Top</a> to arbitrary <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> <p>Lurie calls the groupoid object <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> an <em>effective</em> <a class="existingWikiWord" href="/nlab/show/groupoid+object+in+an+%28infinity%2C1%29-category">groupoid object in an (infinity,1)-category</a> precisely if it arises as the delooping, in the above sense, of some object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>C</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}C</annotation></semantics></math>.</p> <p>One of the characterizing properties of an <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-topos">(infinity,1)-topos</a> is that <em>every</em> groupoid object in it is effective.</p> <p>This is the analog of <a class="existingWikiWord" href="/nlab/show/Jim+Stasheff">Stasheff</a>‘s classical result about <a class="existingWikiWord" href="/nlab/show/H-spaces">H-spaces</a>.</p> <p>See the remark at the very end of section 6.1.2 in <a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">HTT</a>.</p> <h2 id="examples">Examples</h2> <h3 id="topological_loop_spaces">Topological loop spaces</h3> <p>For <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> the <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-category">(infinity,1)-category</a> of <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a>, a space is deloopable if it is an <a class="existingWikiWord" href="/nlab/show/A-infinity-space">A-infinity-space</a> and hence homotopy equivalent to a <a class="existingWikiWord" href="/nlab/show/loop+space">loop space</a>.</p> <h3 id="delooping_of_a_group_to_a_groupoid">Delooping of a group to a groupoid</h3> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/group">group</a> regarded as a <a class="existingWikiWord" href="/nlab/show/discrete+category">discrete groupoid</a> in the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a> of <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoids">∞-groupoids</a>.</p> <p>Then <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> exists and is, up to equivalence, the <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a></p> <ul> <li> <p>with a single object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>•</mo></mrow><annotation encoding="application/x-tex">\bullet</annotation></semantics></math>,</p> </li> <li> <p>with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow></msub><mo stretchy="false">(</mo><mo>•</mo><mo>,</mo><mo>•</mo><mo stretchy="false">)</mo><mo>=</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">Hom_{\mathbf{B} G}(\bullet, \bullet) = G</annotation></semantics></math>, or equivalently <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Aut</mi> <mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow></msub><mo stretchy="false">(</mo><mo>•</mo><mo stretchy="false">)</mo><mo>=</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">Aut_{\mathbf{B}G}(\bullet) = G</annotation></semantics></math>,</p> </li> <li> <p>and with composition of morphisms 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></mrow><annotation encoding="application/x-tex">\mathbf{B} G</annotation></semantics></math> being given by the product operation in the group.</p> </li> </ul> <p>More informally but more suggestively we may write</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><mi>G</mi><mo>=</mo><mo stretchy="false">{</mo><mo>•</mo><mover><mo>→</mo><mi>g</mi></mover><mo>•</mo><mo stretchy="false">|</mo><mi>g</mi><mo>∈</mo><mi>G</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex"> \mathbf{B} G = \{ \bullet \stackrel{g}{\to} \bullet | g \in G\} </annotation></semantics></math></div> <p>or</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><mi>G</mi><mo>=</mo><mo stretchy="false">{</mo><mo>•</mo><mo>⟲</mo><mi>g</mi><mspace width="thickmathspace"></mspace><mo stretchy="false">|</mo><mspace width="thickmathspace"></mspace><mi>g</mi><mo>∈</mo><mi>G</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex"> \mathbf{B}G = \{ \bullet \righttoleftarrow g \;|\; g \in G \} </annotation></semantics></math></div> <p>to emphasize that there is really only a single object.</p> <p>Notice how the <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a> works in this simple case:</p> <p>the universal 2-cell <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi></mrow><annotation encoding="application/x-tex">\eta</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>G</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><msup><mo>⇓</mo> <mi>η</mi></msup></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ G &amp;\to&amp; {*} \\ \downarrow &amp;\Downarrow^{\eta}&amp; \downarrow \\ {*} &amp;\to&amp; \mathbf{B}G } </annotation></semantics></math></div> <p>filling this <a class="existingWikiWord" href="/nlab/show/2-limit">2-limit</a> diagram is the <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a> from the constant <a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>→</mo><mo>*</mo><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex"> G \to {*} \to \mathbf{B}G </annotation></semantics></math></div> <p>to itself, whose component map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>η</mi><mo>:</mo><mi>Obj</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Mor</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \eta : Obj(G) \to Mor(\mathbf{B}G) </annotation></semantics></math></div> <p>is just the identity map, using that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Obj</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo>=</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">Obj(G) = G</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Mor</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo stretchy="false">)</mo><mo>=</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">Mor(\mathbf{B}G) = G</annotation></semantics></math>.</p> <h2 id="DeloopingOfHigherCategoricalStructures">Deloopings of higher categorical structures</h2> <p>There is also a notion of delooping which takes a pointed <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n, k+1)</annotation></semantics></math>-category <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> to a pointed <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+1, k)</annotation></semantics></math>-category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>C</mi></mrow><annotation encoding="application/x-tex">\mathbf{B} C</annotation></semantics></math> in which <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>C</mi></mrow><annotation encoding="application/x-tex">\mathbf{B} C</annotation></semantics></math> has a single <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math>-cell <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>•</mo></mrow><annotation encoding="application/x-tex">\bullet</annotation></semantics></math>, and where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>hom</mi><mo stretchy="false">(</mo><mo>•</mo><mo>,</mo><mo>•</mo><mo stretchy="false">)</mo><mo>=</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">\hom(\bullet, \bullet) = C</annotation></semantics></math>. This is a tautological construction if one accepts the <a class="existingWikiWord" href="/nlab/show/delooping+hypothesis">delooping hypothesis</a>, which views a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n, k+1)</annotation></semantics></math>-category <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> as a special type of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+k+1)</annotation></semantics></math>-category, namely a pointed <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>-connected <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+k+1)</annotation></semantics></math>-category: by viewing such as <em>a fortiori</em> a pointed <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(k-1)</annotation></semantics></math>-connected <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+k+1)</annotation></semantics></math>-category, we get the delooping <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>C</mi></mrow><annotation encoding="application/x-tex">\mathbf{B} C</annotation></semantics></math>.</p> <p>This is just a generalization of the fact that a <a class="existingWikiWord" href="/nlab/show/monoid">monoid</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> gives rise to a one-object category (which we are denoting <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>M</mi></mrow><annotation encoding="application/x-tex">\mathbf{B} M</annotation></semantics></math>). For an important example: a <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> has an associated <strong>delooping <a class="existingWikiWord" href="/nlab/show/bicategory">bicategory</a></strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>M</mi></mrow><annotation encoding="application/x-tex">\mathbf{B} M</annotation></semantics></math>, where</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>M</mi></mrow><annotation encoding="application/x-tex">\mathbf{B} M</annotation></semantics></math> has a single <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math>-cell <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>•</mo></mrow><annotation encoding="application/x-tex">\bullet</annotation></semantics></math>,</p> </li> <li> <p>the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math>-cells <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">\bullet \to \bullet</annotation></semantics></math> 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>M</mi></mrow><annotation encoding="application/x-tex">\mathbf{B} M</annotation></semantics></math> are named by objects of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>, and the composite of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>•</mo><mover><mo>→</mo><mi>a</mi></mover><mo>•</mo><mover><mo>→</mo><mi>b</mi></mover><mo>•</mo></mrow><annotation encoding="application/x-tex">\bullet \stackrel{a}{\to} \bullet \stackrel{b}{\to} \bullet</annotation></semantics></math> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>•</mo><mover><mo>→</mo><mrow><mi>a</mi><mo>⊗</mo><mi>b</mi></mrow></mover><mo>•</mo></mrow><annotation encoding="application/x-tex">\bullet \stackrel{a \otimes b}{\to} \bullet</annotation></semantics></math> (using the monoidal product <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊗</mo></mrow><annotation encoding="application/x-tex">\otimes</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>),</p> </li> <li> <p>the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>-cells 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>M</mi></mrow><annotation encoding="application/x-tex">\mathbf{B} M</annotation></semantics></math> are similarly named by morphisms of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>; the vertical composition of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>-cells 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>M</mi></mrow><annotation encoding="application/x-tex">\mathbf{B} M</annotation></semantics></math> is given by composition of morphisms of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>, and the horizontal composition of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>-cells 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>M</mi></mrow><annotation encoding="application/x-tex">\mathbf{B} M</annotation></semantics></math> is given by taking the monoidal product of the morphisms that name them in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>.</p> </li> </ul> <p>Along similar lines, the delooping of a <a class="existingWikiWord" href="/nlab/show/braided+monoidal+category">braided monoidal category</a> produces a <a class="existingWikiWord" href="/nlab/show/monoidal+bicategory">monoidal bicategory</a>, and delooping of <em>that</em> is a <a class="existingWikiWord" href="/nlab/show/tricategory">tricategory</a> or (weak) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>3</mn></mrow><annotation encoding="application/x-tex">3</annotation></semantics></math>-category. See <a class="existingWikiWord" href="/nlab/show/delooping+hypothesis">delooping hypothesis</a> for more.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <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/May+recognition+theorem">May recognition theorem</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>,</p> <ul> <li> <p><strong>delooping</strong></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>, <a class="existingWikiWord" href="/nlab/show/reduced+suspension">reduced suspension</a></li> </ul> </li> </ul> <h2 id="references">References</h2> <p>Discussion in <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a>:</p> <ul> <li id="BCFR23"> <p><a class="existingWikiWord" href="/nlab/show/Ulrik+Buchholtz">Ulrik Buchholtz</a>, <a class="existingWikiWord" href="/nlab/show/J.+Daniel+Christensen">J. Daniel Christensen</a>, <a class="existingWikiWord" href="/nlab/show/Jarl+G.+Taxer%C3%A5s+Flaten">Jarl G. Taxerås Flaten</a>, <a class="existingWikiWord" href="/nlab/show/Egbert+Rijke">Egbert Rijke</a>, <em>Central H-spaces and banded types</em> &lbrack;<a href="https://arxiv.org/abs/2301.02636">arXiv:2301.02636</a>&rbrack;</p> </li> <li id="W&#xE4;rn23"> <p><a class="existingWikiWord" href="/nlab/show/David+W%C3%A4rn">David Wärn</a>, <em>Eilenberg-MacLane spaces and stabilisation in homotopy type theory</em> &lbrack;<a href="https://arxiv.org/abs/2301.03685">arXiv:2301.03685</a>&rbrack;</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on September 19, 2023 at 05:52:43. See the <a href="/nlab/history/delooping" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/delooping" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/15872/#Item_2">Discuss</a><span class="backintime"><a href="/nlab/revision/delooping/39" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/delooping" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/delooping" accesskey="S" class="navlink" id="history" rel="nofollow">History (39 revisions)</a> <a href="/nlab/show/delooping/cite" style="color: black">Cite</a> <a href="/nlab/print/delooping" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/delooping" id="view_source" rel="nofollow">Source</a> </div> </div> <!-- Content --> </div> <!-- Container --> </body> </html>

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