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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='#properties'>Properties</a></li> <ul> <li><a href='#FiniteHomotopyLimitsOfSpectra'>Finite homotopy (co)limits of spectra</a></li> <li><a href='#monoidal_structure'>Monoidal structure</a></li> <li><a href='#prime_spectrum_and_morava_ktheory'>Prime spectrum and Morava K-theory</a></li> <li><a href='#model_category_presentation'>Model category presentation</a></li> </ul> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>The collection of <a class="existingWikiWord" href="/nlab/show/spectra">spectra</a> forms an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sp</mi><mo stretchy="false">(</mo><mn>∞</mn><mi>Grpd</mi><mo stretchy="false">)</mo><mo>=</mo></mrow><annotation encoding="application/x-tex">Sp(\infty Grpd) = </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Spectra">Spectra</a>, which is in fact a <a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a>. Indeed, it is the <em><a class="existingWikiWord" href="/nlab/show/universal+property">universal property</a></em> <a class="existingWikiWord" href="/nlab/show/stabilization">stabilization</a> of 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/%E2%88%9EGrpd">∞Grpd</a>, equivalently of the <a class="existingWikiWord" href="/nlab/show/simplicial+localization">simplicial localization</a> of the category <a class="existingWikiWord" href="/nlab/show/Top">Top</a> at the <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalences">weak homotopy equivalences</a>.</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sp</mi><mo stretchy="false">(</mo><mn>∞</mn><mi>Grpd</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sp(\infty Grpd)</annotation></semantics></math> plays a role in <a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable homotopy theory</a> analogous to the role played by the 1-<a class="existingWikiWord" href="/nlab/show/category">category</a> <a class="existingWikiWord" href="/nlab/show/Ab">Ab</a> of <a class="existingWikiWord" href="/nlab/show/abelian+groups">abelian groups</a> in <a class="existingWikiWord" href="/nlab/show/homological+algebra">homological algebra</a>, or rather of the <a class="existingWikiWord" href="/nlab/show/category+of+chain+complexes">category of chain complexes</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Ch</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>Ab</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ch_\bullet(Ab)</annotation></semantics></math> of abelian groups.</p> <h2 id="definition">Definition</h2> <p>In the context of <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-category">(∞,1)-categories</a> a <a class="existingWikiWord" href="/nlab/show/spectrum">spectrum</a> is a <a class="existingWikiWord" href="/nlab/show/spectrum+object">spectrum object</a> in the <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><msub><mi>L</mi> <mi>whe</mi></msub><msub><mi>Top</mi> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">L_{whe} Top_*</annotation></semantics></math> of pointed <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a>.</p> <p>Recall that <a class="existingWikiWord" href="/nlab/show/spectrum+objects">spectrum objects</a> in the <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-category">(infinity,1)-category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> form a <a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sp</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sp(C)</annotation></semantics></math>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a> of <a class="existingWikiWord" href="/nlab/show/spectrum+object">spectrum object</a>s in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mi>whe</mi></msub><msub><mi>Top</mi> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">L_{whe} Top_*</annotation></semantics></math> is the <strong>stable <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 of spectra</strong></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Stab</mi><mo stretchy="false">(</mo><msub><mi>L</mi> <mi>whe</mi></msub><mi>Top</mi><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><mi>Sp</mi><mo stretchy="false">(</mo><msub><mi>L</mi> <mi>whe</mi></msub><msub><mi>Top</mi> <mo>*</mo></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Stab(L_{whe}Top) := Sp(L_{whe}Top_*) \,. </annotation></semantics></math></div> <h2 id="properties">Properties</h2> <div> <h3 id="FiniteHomotopyLimitsOfSpectra">Finite homotopy (co)limits of spectra</h3> <p> <div class="num_prop" id="InSpectraHomotopyFiberSequencesAreHomotopyCofiberSequences"> <h6>Proposition</h6> <p></p> <p>A sequence of <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> of <a class="existingWikiWord" href="/nlab/show/spectra">spectra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>⟶</mo><mi>F</mi><mo>⟶</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">E \longrightarrow F \longrightarrow G</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/homotopy+fiber+sequence">homotopy fiber sequence</a> if and only if it is a <a class="existingWikiWord" 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stroke-linecap="round" stroke-linejoin="round" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M 53.512969 -5.978719 L 10.450469 -41.658406 " transform="matrix(1, 0, 0, -1, 7.03, 7.408)"></path> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#jfFjeZLfjE4cIxVOTaNQh2HFT6Y=-glyph-1-3" x="166.326" y="51.41"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#jfFjeZLfjE4cIxVOTaNQh2HFT6Y=-glyph-2-3" x="166.326" y="51.41"></use> </g> <path fill="none" stroke-width="0.478" stroke-linecap="round" stroke-linejoin="round" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M 203.619594 -5.979156 L 160.557094 -41.658844 " transform="matrix(1, 0, 0, -1, 8.314, 8.931)"></path> <path fill="none" stroke-width="0.478" stroke-linecap="round" stroke-linejoin="round" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M 203.618437 -5.978719 L 160.555937 -41.658406 " transform="matrix(1, 0, 0, -1, 7.03, 7.408)"></path> </svg> <p></p> </div> A <strong>proof</strong> is spelled out at <em><a class="existingWikiWord" href="/nlab/show/Introduction+to+Stable+homotopy+theory">Introduction to Stable homotopy theory</a></em> (<a href="Introduction+to+Stable+homotopy+theory+--+1-1#HomotopyCofiberSequencesAreHomotopyFiberSequencesInSpectra">this Prop.</a>, following <a href="equivariant+stable+homotopy+theory#LewisMaySteinberger86">Lewis-May-Steinberger 86, chapter III, theorem 2.4</a> )</p> <p>In fact:</p> <p> <div class="num_prop" id="HomotopyPullbacksOfSpectraAreHomotopyPushouts"> <h6>Proposition</h6> <p></p> <p>A <a class="existingWikiWord" href="/nlab/show/homotopy+coherent+diagram">homotopy</a>-<a class="existingWikiWord" href="/nlab/show/commuting+square">commuting square</a> in <a class="existingWikiWord" href="/nlab/show/Spectra">Spectra</a> is a <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a> if and only it is a <a class="existingWikiWord" 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transform="matrix(1, 0, 0, -1, 8.314, 8.931)"></path> <path fill="none" stroke-width="0.478" stroke-linecap="round" stroke-linejoin="round" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M 51.997344 -5.978719 L 10.309844 -41.634969 " transform="matrix(1, 0, 0, -1, 7.03, 7.408)"></path> <path fill="none" stroke-width="0.478" stroke-linecap="round" stroke-linejoin="round" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M 203.494594 -5.979156 L 203.486781 -5.979156 " transform="matrix(1, 0, 0, -1, 8.314, 8.931)"></path> <path fill="none" stroke-width="0.478" stroke-linecap="round" stroke-linejoin="round" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M 203.493437 -5.978719 L 203.485625 -5.978719 " transform="matrix(1, 0, 0, -1, 7.03, 7.408)"></path> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#Uf1DHcYmg2A5JpL3YORaeMDU9co=-glyph-4-1" x="190.941" y="40.25"></use> <use xlink:href="#Uf1DHcYmg2A5JpL3YORaeMDU9co=-glyph-4-2" x="193.819214" y="40.25"></use> <use xlink:href="#Uf1DHcYmg2A5JpL3YORaeMDU9co=-glyph-4-5" x="197.859474" y="40.25"></use> <use xlink:href="#Uf1DHcYmg2A5JpL3YORaeMDU9co=-glyph-4-4" x="201.899734" y="40.25"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#Uf1DHcYmg2A5JpL3YORaeMDU9co=-glyph-1-3" x="167.601" y="51.41"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#Uf1DHcYmg2A5JpL3YORaeMDU9co=-glyph-2-3" x="167.601" y="51.41"></use> </g> <path fill="none" stroke-width="0.478" stroke-linecap="round" stroke-linejoin="round" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M 203.494594 -5.979156 L 161.807094 -41.635406 " transform="matrix(1, 0, 0, -1, 8.314, 8.931)"></path> <path fill="none" stroke-width="0.478" stroke-linecap="round" stroke-linejoin="round" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M 203.493437 -5.978719 L 161.805937 -41.634969 " transform="matrix(1, 0, 0, -1, 7.03, 7.408)"></path> </svg> <p></p> </div> </p> <p>This follows from Prop. <a class="maruku-ref" href="#InSpectraHomotopyFiberSequencesAreHomotopyCofiberSequences"></a> by the fact that <a class="existingWikiWord" href="/nlab/show/Spectra">Spectra</a> is <a class="existingWikiWord" href="/nlab/show/additive+category">additive</a> (<a href="Introduction+to+Stable+homotopy+theory+--+1-1#TheStableHomotopyCategoryIsAdditive">this Prop.</a>).</p> <p>See also <a href="https://arxiv.org/abs/1906.04773">arXiv:1906.04773, Prop. 6.2.11</a>, <a href="https://mathoverflow.net/q/132347/381">MO:q/132347</a>.</p> <p> <div class="num_remark"> <h6>Remark</h6> <p>This property of Spectra (Prop. <a class="maruku-ref" href="#InSpectraHomotopyFiberSequencesAreHomotopyCofiberSequences"></a>, Prop. <a class="maruku-ref" href="#HomotopyPullbacksOfSpectraAreHomotopyPushouts"></a>) reflects one of the standard defining <a class="existingWikiWord" href="/nlab/show/axioms">axioms</a> on <a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-categories">stable (∞,1)-categories</a> (see <a href="stable+infinity-category#Definition">there</a>) and on <a class="existingWikiWord" href="/nlab/show/stable+derivators">stable derivators</a> (see <a href="stable+derivator#Definition">there</a>).</p> </div> </p> </div> <h3 id="monoidal_structure">Monoidal structure</h3> <ul> <li> <p>With the <a class="existingWikiWord" href="/nlab/show/smash+product+of+spectra">smash product of spectra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sp</mi><mo stretchy="false">(</mo><msub><mi>L</mi> <mi>whe</mi></msub><msub><mi>Top</mi> <mo>*</mo></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sp(L_{whe}Top_*)</annotation></semantics></math> becomes a <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28%E2%88%9E%2C1%29-category">symmetric monoidal (∞,1)-category</a>.</p> <ul> <li> <p>an <a class="existingWikiWord" href="/nlab/show/algebra+in+an+%28infinity%2C1%29-category">algebra object</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sp</mi><mo stretchy="false">(</mo><msub><mi>L</mi> <mi>whe</mi></msub><msub><mi>Top</mi> <mo>*</mo></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sp(L_{whe}Top_*)</annotation></semantics></math> with respect to this monoidal structure is an <a class="existingWikiWord" href="/nlab/show/associative+ring+spectrum">associative ring spectrum</a>;</p> </li> <li> <p>a <a class="existingWikiWord" href="/nlab/show/commutative+algebra+in+an+%28infinity%2C1%29-category">commutative algebra object</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sp</mi><mo stretchy="false">(</mo><msub><mi>L</mi> <mi>whe</mi></msub><msub><mi>Top</mi> <mo>*</mo></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sp(L_{whe}Top_*)</annotation></semantics></math> with respect to this monoidal structure is a <a class="existingWikiWord" href="/nlab/show/commutative+ring+spectrum">commutative ring spectrum</a>;</p> </li> </ul> </li> </ul> <h3 id="prime_spectrum_and_morava_ktheory">Prime spectrum and Morava K-theory</h3> <p>The <a class="existingWikiWord" href="/nlab/show/prime+spectrum+of+a+monoidal+stable+%28%E2%88%9E%2C1%29-category">prime spectrum of a monoidal stable (∞,1)-category</a> for <a class="existingWikiWord" href="/nlab/show/p-localization">p-local</a> and <a class="existingWikiWord" href="/nlab/show/finite+spectra">finite spectra</a> is labeled by the <a class="existingWikiWord" href="/nlab/show/Morava+K-theories">Morava K-theories</a>. This is the content of the <em><a class="existingWikiWord" href="/nlab/show/thick+subcategory+theorem">thick subcategory theorem</a></em>.</p> <h3 id="model_category_presentation">Model category presentation</h3> <p>There are several <a class="existingWikiWord" href="/nlab/show/presentable+%28infinity%2C1%29-category">presentations</a> of 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 of spectra by <a class="existingWikiWord" href="/nlab/show/model+categories+of+spectra">model categories of spectra</a>. In particular there are <a class="existingWikiWord" href="/nlab/show/symmetric+smash+product+of+spectra">symmetric</a> <a class="existingWikiWord" href="/nlab/show/monoidal+model+categories">monoidal model categories</a> where the <a class="existingWikiWord" href="/nlab/show/smash+product+of+spectra">smash product of spectra</a> is presented by an ordinary <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a>, so that <a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+rings">A-∞ rings</a>, <a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+rings">E-∞ rings</a> and <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-modules">∞-modules</a> are presented by 1-categorical <a class="existingWikiWord" href="/nlab/show/monoid+objects">monoid objects</a> and <a class="existingWikiWord" href="/nlab/show/module+objects">module objects</a>, respectively (“<a class="existingWikiWord" href="/nlab/show/brave+new+algebra">brave new algebra</a>”). See at:</p> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+spectra">model structure on spectra</a>, <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+smash+product+of+spectra">symmetric monoidal smash product of spectra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+spectrum">symmetric spectrum</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+on+symmetric+spectra">model structure on symmetric spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orthogonal+spectrum">orthogonal spectrum</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+on+orthogonal+spectra">model structure on orthogonal spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/S-module">S-module</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+on+S-modules">model structure on S-modules</a></p> </li> </ul> <h2 id="references">References</h2> <p>The stable <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 of spectra is described in chapter 1 of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, <a class="existingWikiWord" href="/nlab/show/Higher+Algebra">Higher Algebra</a></li> </ul> <p>or section 9 of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, <a class="existingWikiWord" href="/nlab/show/Stable+Infinity-Categories">Stable Infinity-Categories</a> .</li> </ul> <p>Its monoidal structure is described in section 4.2</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, <a class="existingWikiWord" href="/nlab/show/higher+algebra">Noncommutative algebra</a>.</li> </ul> <p>That this is a symmetric monoidal structure is described in section 6 of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, <a class="existingWikiWord" href="/nlab/show/higher+algebra">Commutative algebra</a>.</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on January 16, 2021 at 14:59:15. 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