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href="/nlab/show/%28%E2%88%9E%2C1%29-category+theory">(∞,1)-category theory</a></strong></p> <p><strong>Background</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28n%2Cr%29-category">(n,r)-category</a></p> </li> </ul> <p><strong>Basic concepts</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hom-object+in+a+quasi-category">hom-objects</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivalence+in+a+quasi-category">equivalences in</a>/<a class="existingWikiWord" href="/nlab/show/equivalence+of+quasi-categories">of</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-categories</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sub-quasi-category">sub-(∞,1)-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/reflective+sub-%28%E2%88%9E%2C1%29-category">reflective sub-(∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/localization+of+an+%28%E2%88%9E%2C1%29-category">reflective localization</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/opposite+quasi-category">opposite (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/over+quasi-category">over (∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/join+of+quasi-categories">join of quasi-categories</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/exact+%28%E2%88%9E%2C1%29-functor">exact (∞,1)-functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-functors">(∞,1)-category of (∞,1)-functors</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-presheaves">(∞,1)-category of (∞,1)-presheaves</a></p> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/fibrations+of+quasi-categories">fibrations</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/inner+fibration">inner fibration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/left+fibration">left/right fibration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cartesian+fibration">Cartesian fibration</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Cartesian+morphism">Cartesian morphism</a></li> </ul> </li> </ul> </li> </ul> <p><strong>Universal constructions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/limit+in+quasi-categories">limit</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/terminal+object+in+a+quasi-category">terminal object</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+%28%E2%88%9E%2C1%29-functor">adjoint functors</a></p> </li> </ul> <p><strong>Local presentation</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+presentable+%28%E2%88%9E%2C1%29-category">locally presentable</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/essentially+small+%28%E2%88%9E%2C1%29-category">essentially small</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+small+%28%E2%88%9E%2C1%29-category">locally small</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/accessible+%28%E2%88%9E%2C1%29-category">accessible</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/idempotent-complete+%28%E2%88%9E%2C1%29-category">idempotent-complete</a></p> </li> </ul> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-Yoneda+lemma">(∞,1)-Yoneda lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-Grothendieck+construction">(∞,1)-Grothendieck construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+%28%E2%88%9E%2C1%29-functor+theorem">adjoint (∞,1)-functor theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monadicity+theorem">(∞,1)-monadicity theorem</a></p> </li> </ul> <p><strong>Extra stuff, structure, properties</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a></p> </li> </ul> <p><strong>Models</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category+with+weak+equivalences">category with weak equivalences</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derivator">derivator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quasi-category">quasi-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+quasi-categories">model structure for quasi-categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+Cartesian+fibrations">model structure for Cartesian fibrations</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relation+between+quasi-categories+and+simplicial+categories">relation to simplicial categories</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coherent+nerve">homotopy coherent nerve</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/simplicial+model+category">simplicial model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/presentable+%28%E2%88%9E%2C1%29-category">presentable quasi-category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+sets">model structure for Kan complexes</a></li> </ul> </li> </ul> </div></div> <h4 id="stable_homotopy_theory">Stable Homotopy theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable homotopy theory</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/homological+algebra">homological algebra</a>, <a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a></li> </ul> <p><em><a class="existingWikiWord" href="/nlab/show/Introduction+to+Stable+Homotopy+Theory">Introduction</a></em></p> <h1 id="ingredients">Ingredients</h1> <ul> <li><a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a></li> </ul> <h1 id="contents">Contents</h1> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/loop+space+object">loop space object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/suspension+object">suspension object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/looping+and+delooping">looping and delooping</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/stabilization">stabilization</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/spectrum+object">spectrum object</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+derivator">stable derivator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/triangulated+category">triangulated category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category+of+spectra">stable (∞,1)-category of spectra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/spectrum">spectrum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+homotopy+category">stable homotopy category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smash+product+of+spectra">smash product of spectra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+smash+product+of+spectra">symmetric smash product of spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Spanier-Whitehead+duality">Spanier-Whitehead duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+ring">A-∞ ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+ring">E-∞ ring</a></p> </li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/stable+homotopy+theory+-+contents">Edit this sidebar</a> </p> </div></div></div> </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='#constructions_in_stable_categories'>Constructions in stable <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-categories</a></li> <ul> <li><a href='#looping_and_delooping'>Looping and delooping</a></li> <li><a href='#stabilization'>Stabilization</a></li> </ul> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#enrichment_over_spectra'>Enrichment over spectra</a></li> <li><a href='#TheTriangulatedHomotopyCategory'>The homotopy category: triangulated categories</a></li> <li><a href='#alternativemodels'>Models</a></li> <li><a href='#StabGiraud'>Stabilization and localization of presheaf <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</a></li> <li><a href='#AsCategoriesOfModules'>As categories of modules over <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>-ring(oid)s</a></li> <li><a href='#doldkan_correspondence'>Dold-Kan correspondence</a></li> </ul> <li><a href='#examples'>Examples</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>A stable <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>, is a <a class="existingWikiWord" href="/nlab/show/pointed+%28%E2%88%9E%2C1%29-category">pointed (∞,1)-category</a> with <a class="existingWikiWord" href="/nlab/show/finite+%28%E2%88%9E%2C1%29-limit">finite (∞,1)-limit</a> which is <em>stable</em> under forming <a class="existingWikiWord" href="/nlab/show/loop+space+objects">loop space objects</a>:</p> <p><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> has a <a class="existingWikiWord" href="/nlab/show/zero+object">zero object</a> and the corresponding loop <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex"> \Omega : C \to C </annotation></semantics></math></div> <p>is an <em>equivalence</em> with inverse the <a class="existingWikiWord" href="/nlab/show/suspension+object">suspension object</a> functor</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><mi>C</mi><mo>:</mo><mi>Σ</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> C \leftarrow C : \Sigma \,. </annotation></semantics></math></div> <p>This means that the objects of a 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 are stable in the sense of <a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable homotopy theory</a>: they behave as if they were <a class="existingWikiWord" href="/nlab/show/spectrum">spectra</a>.</p> <p>Indeed, every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-category with finite <a class="existingWikiWord" href="/nlab/show/limit">limit</a>s has a free <a class="existingWikiWord" href="/nlab/show/stabilization">stabilization</a> to a 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 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Stab</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Stab(C)</annotation></semantics></math>, and the objects of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Stab</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Stab(C)</annotation></semantics></math> are the <a class="existingWikiWord" href="/nlab/show/spectrum+object">spectrum object</a>s of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/homotopy+category+of+an+%28%E2%88%9E%2C1%29-category">homotopy category of an (∞,1)-category</a> of a stable <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-category is a <a class="existingWikiWord" href="/nlab/show/triangulated+category">triangulated category</a>.</p> <p>Notice that the definition of triangulated categories is involved and their behaviour is bad, whereas the definition of stable <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-category is simple and natural. The complexity and bad behavior of triangulated categories comes from them being the <a class="existingWikiWord" href="/nlab/show/decategorification">decategorification</a> of a structure that is natural in <a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a>.</p> <h2 id="Definition">Definition</h2> <p>As with ordinary categories, an object in a <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-category">(infinity,1)-category</a> is a <a class="existingWikiWord" href="/nlab/show/zero+object">zero object</a> if it is both <a class="existingWikiWord" href="/nlab/show/initial+object">initial object</a> and a <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a>. An <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-category with a zero object is a <strong>pointed</strong> <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.</p> <div class="num_defn"> <h6 id="definition_2">Definition</h6> <p>In a pointed <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> with <a class="existingWikiWord" href="/nlab/show/zero+object">zero object</a> <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>, the <strong>kernel</strong> of a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>:</mo><mi>Y</mi><mo>→</mo><mi>Z</mi></mrow><annotation encoding="application/x-tex">g : Y \to Z</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-pullback">(∞,1)-pullback</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>ker</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mi>g</mi></msup></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>Z</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ ker(g) &\to& Y \\ \downarrow && \downarrow^g \\ 0 &\to& Z } </annotation></semantics></math></div> <p>(so that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ker</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Y</mi><mover><mo>→</mo><mi>g</mi></mover><mi>Z</mi></mrow><annotation encoding="application/x-tex">ker(g) \to Y \stackrel{g}{\to} Z</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/fibration+sequence">fibration sequence</a>)</p> <p>and the <strong>cokernel</strong> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f:X\to Y</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-pushout">(∞,1)-pushout</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mi>f</mi></mover></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>coker</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ X &\stackrel{f}{\to}& Y \\ \downarrow && \downarrow \\ 0 &\to& coker(f) } \,. </annotation></semantics></math></div> <p>An arbitrary commuting square in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mi>f</mi></mover></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mi>g</mi></msup></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>Z</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ X &\stackrel{f}{\to}& Y \\ \downarrow && \downarrow^g \\ 0 &\to& Z } </annotation></semantics></math></div> <p>is a <strong>triangle</strong> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>. A pullback triangle is called an <strong>exact triangle</strong> and a pushout triangle a <strong>coexact triangle</strong>. By the universal property of pullback and pushout, to any triangle are associated canonical morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mi>ker</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X\to\ker(g)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>coker</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Z</mi></mrow><annotation encoding="application/x-tex">coker(f)\to Z</annotation></semantics></math>. In particular, for every exact triangle there is a canonical morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>coker</mi><mo stretchy="false">(</mo><mi>ker</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Z</mi></mrow><annotation encoding="application/x-tex">coker(ker(g)\to Y)\to Z</annotation></semantics></math> and for every coexact triangle there is a canonical morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mi>ker</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>→</mo><mi>coker</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X\to ker(Y\to coker(f))</annotation></semantics></math>.</p> </div> <div class="num_defn"> <h6 id="definition_3">Definition</h6> <p>A <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</strong> is a pointed <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 such that</p> <ul> <li> <p>for every morphism in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> kernel and cokernel exist;</p> </li> <li> <p>every exact triangle is coexact and vice versa, i.e. every morphism is the cokernel of its kernel and the kernel of its cokernel.</p> </li> </ul> </div> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>The notion of stable <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-category should not be confused with that of a <a class="existingWikiWord" href="/nlab/show/k-tuply+monoidal+n-category">stably monoidal</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-category. A connection between the terms is that the <a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category+of+spectra">stable (∞,1)-category of spectra</a> is the prototypical stable <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-category, while <em>connective</em> spectra (not all spectra) can be identified with <a class="existingWikiWord" href="/nlab/show/k-tuply+groupal+n-groupoid">stably groupal</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoids, aka <em>infinite loop spaces</em> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">E_\infty</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/E-infinity+space">spaces</a>.</p> </div> <h2 id="constructions_in_stable_categories">Constructions in stable <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-categories</h2> <h3 id="looping_and_delooping">Looping and delooping</h3> <p>The relevance of the axioms of a 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 is that they imply that not only does every object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> have a <a class="existingWikiWord" href="/nlab/show/loop+space+object">loop space object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">\Omega X</annotation></semantics></math> defined by the exact triangle</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><mn>0</mn></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>X</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \Omega X &\to& 0 \\ \downarrow && \downarrow \\ 0 &\to& X } </annotation></semantics></math></div> <p>but also that, conversely, every object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> has a <a class="existingWikiWord" href="/nlab/show/suspension+object">suspension object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">\Sigma X</annotation></semantics></math> defined by the coexact triangle</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>X</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mn>0</mn></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>Σ</mi><mi>X</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ X &\to& 0 \\ \downarrow && \downarrow \\ 0 &\to& \Sigma X } \,. </annotation></semantics></math></div> <p>These arrange into <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>-endofunctors</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex"> \Omega : C \to 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>Σ</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex"> \Sigma : C \to C </annotation></semantics></math></div> <p>which are <a class="existingWikiWord" href="/nlab/show/autoequivalences">autoequivalences</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> that are inverses of each other.</p> <h3 id="stabilization">Stabilization</h3> <p>For every pointed <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 with finite <a class="existingWikiWord" href="/nlab/show/limit">limit</a>s which is not yet stable there is its free <a class="existingWikiWord" href="/nlab/show/stabilization">stabilization</a> (see there for more details):</p> <p>a 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 <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> that can be defined as the <a class="existingWikiWord" href="/nlab/show/limit+in+a+quasi-category">limit</a> in the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-categories">(∞,1)-category of (∞,1)-categories</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Sp</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><mi>holim</mi><mo stretchy="false">(</mo><mi>⋯</mi><mo>→</mo><mi>C</mi><mover><mo>→</mo><mi>Ω</mi></mover><mi>C</mi><mover><mo>→</mo><mi>Ω</mi></mover><mi>C</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Sp(C) := holim( \cdots \to C \stackrel{\Omega}{\to} C \stackrel{\Omega}{\to} C ) \,. </annotation></semantics></math></div> <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 <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 topological spaces, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sp</mi><mo stretchy="false">(</mo><mi>Top</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sp(Top)</annotation></semantics></math> is the familiar <a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category+of+spectra">stable (∞,1)-category of spectra</a> (whose <a class="existingWikiWord" href="/nlab/show/homotopy+category+of+an+%28%E2%88%9E%2C1%29-category">homotopy category</a> is the <a class="existingWikiWord" href="/nlab/show/stable+homotopy+category">stable homotopy category</a>) used in <a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable homotopy theory</a> (which gives 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>-categories their name).</p> <p>Moreover, every <a class="existingWikiWord" href="/nlab/show/derived+category">derived category</a> of an <a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a> is the triangulated homotopy category of a 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.</p> <p>Hence stable homotopy theory and homological algebra are both special cases of the theory of 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>-categories.</p> <h2 id="properties">Properties</h2> <h3 id="enrichment_over_spectra">Enrichment over spectra</h3> <p>Stable <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-categories are naturally <a class="existingWikiWord" href="/nlab/show/enriched+%28%E2%88%9E%2C1%29-categories">enriched (∞,1)-categories</a> over the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+spectra">(∞,1)-category of spectra</a> (<a href="#GepnerHaugseng13">Gepner-Haugseng 13</a>).</p> <h3 id="TheTriangulatedHomotopyCategory">The homotopy category: triangulated categories</h3> <p>The <a class="existingWikiWord" href="/nlab/show/homotopy+category+of+an+%28%E2%88%9E%2C1%29-category">homotopy category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(C)</annotation></semantics></math> of a 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 <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> – its <a class="existingWikiWord" href="/nlab/show/decategorification">decategorification</a> to an ordinary <a class="existingWikiWord" href="/nlab/show/category">category</a> – is less well behaved than the original 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, but remembers a shadow of some of its structure: this shadow is the structure of a <a class="existingWikiWord" href="/nlab/show/triangulated+category">triangulated category</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(C)</annotation></semantics></math></p> <ul> <li> <p>the <strong>translation functor</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo>:</mo><mi>Ho</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Ho</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">T : Ho(C) \to Ho(C)</annotation></semantics></math> comes from the <a class="existingWikiWord" href="/nlab/show/suspension+object">suspension</a> functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">\Sigma : C \to C</annotation></semantics></math>;</p> </li> <li> <p>the <strong>distinguished triangles</strong> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(C)</annotation></semantics></math> are pieces of the <a class="existingWikiWord" href="/nlab/show/fibration+sequence">fibration sequence</a>s in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>.</p> </li> </ul> <p>For details see <a href="http://arxiv.org/PS_cache/math/pdf/0608/0608228v5.pdf#page=6">StabCat, section 3</a>.</p> <p>Alternately, one can first pass to a <a class="existingWikiWord" href="/nlab/show/stable+derivator">stable derivator</a>, and thence to a triangulated category. Any suitably complete and cocomplete <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 has an underlying <a class="existingWikiWord" href="/nlab/show/derivator">derivator</a>, and the underlying derivator of a 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 is always stable—while the underlying category of any stable derivator is triangulated. But the derivator retains more useful information about the original 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 than does its triangulated homotopy category.</p> <h3 id="alternativemodels">Models</h3> <p>In direct analogy to how a general <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> may be <a class="existingWikiWord" href="/nlab/show/presentable+%28%E2%88%9E%2C1%29-category">presented</a> by <a class="existingWikiWord" href="/nlab/show/model+category">model category</a>, a 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>-categories may be presented by any of the following models.</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+model+category">stable model category</a>;</p> </li> <li> <p>stable <a class="existingWikiWord" href="/nlab/show/quasicategory">quasicategory</a> (as devloped by <a class="existingWikiWord" href="/nlab/show/Lurie">Lurie</a> in Chapter 1 of <a class="existingWikiWord" href="/nlab/show/Higher+Algebra">Higher Algebra</a>);</p> </li> <li> <p>pre-triangulated <a class="existingWikiWord" href="/nlab/show/spectral+category">spectral category</a>.</p> </li> </ul> <p>There are further variants and special cases of these models. The following three concepts are equivalent to each other and special cases of the above models, or equivalent in characteristic 0.</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/pretriangulated+dg-category">pretriangulated dg-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Frobenius+category">Frobenius category</a></p> </li> </ul> <p>(e.g. <a href="#Cohn13">Cohn 13</a>, see also <a href="#Schwede">Schwede</a>)</p> <p>A <a class="existingWikiWord" href="/nlab/show/triangulated+category">triangulated category</a> linear over a field <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> can canonically be refined to</p> <ul> <li> <p>an <a class="existingWikiWord" href="/nlab/show/enhanced+triangulated+category">enhanced triangulated category</a>:</p> <ul> <li>a <a class="existingWikiWord" href="/nlab/show/pretriangulated+dg-category">pretriangulated</a> <a class="existingWikiWord" href="/nlab/show/dg-category">dg-category</a>;</li> </ul> </li> <li> <p>an <a class="existingWikiWord" href="/nlab/show/A-infinity-category">A-infinity-category</a>;</p> </li> <li> <p>a 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">(\infinity,1)</annotation></semantics></math>-category.</p> </li> </ul> <p>If <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> has characteristic 0, then all these three concepts become equivalent.</p> <h3 id="StabGiraud">Stabilization and localization of presheaf <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</h3> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> be <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-category">(∞,1)-categories</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Func</mi><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mi>D</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Func(C,D)</annotation></semantics></math> the <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> between them.</p> <p>Its <a class="existingWikiWord" href="/nlab/show/stabilization">stabilization</a> is equivalent to the functor category into the stabilization of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>:</p> <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><mi>Func</mi><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mi>D</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≃</mo><mi>Func</mi><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mi>Stab</mi><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Stab(Func(C,D)) \simeq Func(C,Stab(D)) \,. </annotation></semantics></math></div> <p>In particular, consider the case <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">D = </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a> where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Stab</mi><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo><mo>=</mo><mi>Stab</mi><mo stretchy="false">(</mo><mn>∞</mn><mi>Grpd</mi><mo stretchy="false">)</mo><mo>=</mo><mi>Sp</mi></mrow><annotation encoding="application/x-tex">Stab(D) = Stab(\infty Grpd) = Sp</annotation></semantics></math> (= the <a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category+of+spectra">stable (∞,1)-category of spectra</a>). One has <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Func</mi><mo stretchy="false">(</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>D</mi><mo stretchy="false">)</mo><mo>=</mo><mi>Func</mi><mo stretchy="false">(</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mn>∞</mn><mi>Grpd</mi><mo stretchy="false">)</mo><mo>=</mo><mo>:</mo><msub><mi>PSh</mi> <mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Func(C^{op}, D) = Func(C^{op}, \infty Grpd) =: PSh_{(\infty,1)}(C)</annotation></semantics></math> is the <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>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Func</mi><mo stretchy="false">(</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>Sp</mi><mo stretchy="false">)</mo><mo>=</mo><mo>:</mo><msubsup><mi>PSh</mi> <mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow> <mi>Sp</mi></msubsup><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Func(C^{op},Sp) =: PSh_{(\infty,1)}^{Sp}(C)</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> of <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-presheaves">(∞,1)-presheaves</a> of <a class="existingWikiWord" href="/nlab/show/spectra">spectra</a>, we get</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>PSh</mi> <mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≃</mo><msubsup><mi>PSh</mi> <mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow> <mi>Sp</mi></msubsup><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Stab(PSh_{(\infty,1)}(C)) \simeq PSh_{(\infty,1)}^{Sp}(C) \,. </annotation></semantics></math></div></div> <p>This is <a href="#StabCat">StabCat, example 10.13</a> .</p> <div class="num_prop" id="StableGiraud"> <h6 id="proposition_2">Proposition</h6> <p><strong>(“stable Giraud theorem”)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> be an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a>. Then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is stable and <a class="existingWikiWord" href="/nlab/show/presentable+%28%E2%88%9E%2C1%29-category">presentable (∞,1)-category</a> if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is equivalent to an <a class="existingWikiWord" href="/nlab/show/accessible+%28infinity%2C1%29-category">accessible</a> left-exact <a class="existingWikiWord" href="/nlab/show/localization+of+an+%28%E2%88%9E%2C1%29-category">localization</a> of the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> of presheaves of spectra on some small <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>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>, so that there is an adjunction</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>C</mi><mover><mo>↪</mo><mover><mo>←</mo><mi>lex</mi></mover></mover><msup><mi>PSh</mi> <mi>Sp</mi></msup><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>Stab</mi><mo stretchy="false">(</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> C \stackrel{\stackrel{lex}{\leftarrow}}{\hookrightarrow} PSh^{Sp}(E) \simeq Stab(PSh(E)) \,. </annotation></semantics></math></div></div> <p>This is <a href="#HigherAlgebra">Higher Algebra, Proposition 1.4.4.9</a>.</p> <p>This is the stable analog of the statement that every <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-sheaves">(∞,1)-category of (∞,1)-sheaves</a> is a left exact localization of an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-category of presheaves.</p> <p>A more intrinsic <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a>-theoretic version of this statement (not mentioning a choice of <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-site">(∞,1)-site</a>) is the following:</p> <div class="num_prop" id="StabilizationBySheavesOfSpectra"> <h6 id="proposition_3">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> be an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> and write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Sh</mi> <mn>∞</mn></msub><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>,</mo><mi>Spectra</mi><mo stretchy="false">)</mo><mo>≔</mo><msub><mi>Func</mi> <mi>lex</mi></msub><mo stretchy="false">(</mo><msup><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>op</mi></msup><mo>,</mo><mi>Spectra</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Sh_\infty(\mathbf{H}, Spectra) \coloneqq Func_{lex}(\mathbf{H}^{op}, Spectra) </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> of <em><a class="existingWikiWord" href="/nlab/show/sheaves+of+spectra">sheaves of spectra</a></em> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> (with respect to its <a class="existingWikiWord" href="/nlab/show/canonical+topology">canonical topology</a>), hence the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> of <a class="existingWikiWord" href="/nlab/show/left+exact+%28%E2%88%9E%2C1%29-functors">left exact (∞,1)-functors</a> from the <a class="existingWikiWord" href="/nlab/show/opposite+%28%E2%88%9E%2C1%29-category">opposite (∞,1)-category</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> to the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+spectra">(∞,1)-category of spectra</a>.</p> <p>This exhibits the <a class="existingWikiWord" href="/nlab/show/stabilization">stabilization</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>:</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><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>Sh</mi> <mn>∞</mn></msub><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>,</mo><mi>Spectra</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Stab(\mathbf{H}) \simeq Sh_\infty(\mathbf{H}, Spectra) \,. </annotation></semantics></math></div></div> <p>This is (<a class="existingWikiWord" href="/nlab/show/Spectral+Schemes">Lurie "Spectral Schemes", remark 1.2</a>).</p> <p>See at <em><a class="existingWikiWord" href="/nlab/show/sheaf+of+spectra">sheaf of spectra</a></em> and <em><a class="existingWikiWord" href="/nlab/show/model+structure+on+presheaves+of+spectra">model structure on presheaves of spectra</a></em> for more.</p> <h3 id="AsCategoriesOfModules">As categories of modules over <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>-ring(oid)s</h3> <p>In terms of (<a class="existingWikiWord" href="/nlab/show/stable+model+category">stable</a>) <a class="existingWikiWord" href="/nlab/show/model+category">model categories</a>, something like an analog of this statement is (<a href="#SchwedeShipley">Schwede-Shipley, theorem 3.3.3</a>):</p> <div class="num_prop" id="ModuleProp"> <h6 id="proposition_4">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> be a <em><a class="existingWikiWord" href="/nlab/show/stable+model+category">stable model category</a></em> that is in addition</p> <ul> <li> <p>a <a class="existingWikiWord" href="/nlab/show/simplicial+model+category">simplicial model category</a>;</p> </li> <li> <p>a <a class="existingWikiWord" href="/nlab/show/cofibrantly+generated+model+category">cofibrantly generated model category</a>;</p> </li> <li> <p>a <a class="existingWikiWord" href="/nlab/show/proper+model+category">proper model category</a>;</p> </li> <li> <p>with a <a class="existingWikiWord" href="/nlab/show/set">set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/compact+object">compact</a> generators;</p> </li> </ul> <p>then there is a chain of <a class="existingWikiWord" href="/nlab/show/simplicial+Quillen+adjunction">sSet-enriched</a> <a class="existingWikiWord" href="/nlab/show/Quillen+equivalences">Quillen equivalences</a> linking <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">\mathcal{C}</annotation></semantics></math> to the the <a class="existingWikiWord" href="/nlab/show/spectrum">spectrum</a>-<a class="existingWikiWord" href="/nlab/show/enriched+functor+category">enriched functor category</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><msub><mo>≃</mo> <mi>Q</mi></msub><msub><mi>A</mi> <mi>S</mi></msub><mi>Mod</mi><mo>≔</mo><mi>Sp</mi><mi>Cat</mi><mo stretchy="false">(</mo><msub><mi>A</mi> <mi>S</mi></msub><mo>,</mo><mi>Sp</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathcal{C} \simeq_Q A_S Mod \coloneqq Sp Cat(A_S, Sp) </annotation></semantics></math></div> <p>equipped with the <a class="existingWikiWord" href="/nlab/show/global+model+structure+on+functors">global model structure on functors</a>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mi>S</mi></msub></mrow><annotation encoding="application/x-tex">A_S</annotation></semantics></math> is the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sp</mi></mrow><annotation encoding="application/x-tex">Sp</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/enriched+category">enriched category</a> whose set of objects is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math></p> </div> <p>This is in (<a href="#SchwedeShipley">Schwede-Shipley, theorem 3.3.3</a>)</p> <div class="num_remark"> <h6 id="remark_2">Remark</h6> <p>An <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sp</mi></mrow><annotation encoding="application/x-tex">Sp</annotation></semantics></math>-enriched category is a homotopy-theoretic analog of an <a class="existingWikiWord" href="/nlab/show/Ab-enriched+category">Ab-enriched category</a>, which may be thought of as a many-object version of a <a class="existingWikiWord" href="/nlab/show/ring">ring</a>, a “<a class="existingWikiWord" href="/nlab/show/ringoid">ringoid</a>”. Accordingly, an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sp</mi></mrow><annotation encoding="application/x-tex">Sp</annotation></semantics></math>-enriched category is 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>-ringoid. It is has a single object then (as a pointed category) it is an <a class="existingWikiWord" href="/nlab/show/A-infinity+algebra">A-infinity algebra</a>.</p> </div> <p>Hence:</p> <div class="num_cor"> <h6 id="corollary">Corollary</h6> <p>If in prop. <a class="maruku-ref" href="#ModuleProp"></a> there is just one compact generator <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">P \in \mathcal{C}</annotation></semantics></math>, then there is a one-object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sp</mi></mrow><annotation encoding="application/x-tex">Sp</annotation></semantics></math>-enriched category, hence an <a class="existingWikiWord" href="/nlab/show/A-infinity+algebra">A-infinity algebra</a> <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>, which is the <a class="existingWikiWord" href="/nlab/show/endomorphisms">endomorphisms</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>≃</mo><msub><mi>End</mi> <mi>𝒞</mi></msub><mo stretchy="false">(</mo><mi>P</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A \simeq End_{\mathcal{C}}(P)</annotation></semantics></math>, and the <a class="existingWikiWord" href="/nlab/show/stable+model+category">stable model category</a> is its <a class="existingWikiWord" href="/nlab/show/category+of+modules">category of modules</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><msub><mo>≃</mo> <mi>Q</mi></msub><mi>A</mi><mi>Mod</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathcal{C} \simeq_Q A Mod \,. </annotation></semantics></math></div></div> <p>This is in (<a href="#SchwedeShipley">Schwede-Shipley, theorem 3.1.1</a>)</p> <p>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 an <a class="existingWikiWord" href="/nlab/show/Eilenberg-MacLane+spectrum">Eilenberg-MacLane spectrum</a>, then this identifies the corresponding stable model categories with the <a class="existingWikiWord" href="/nlab/show/model+structure+on+unbounded+chain+complexes">model structure on unbounded chain complexes</a>.</p> <p>This is (<a href="#SchwedeShipley03">Schwede-Shipley 03, theorem 5.1.6</a>).</p> <div class="num_remark"> <h6 id="remark_3">Remark</h6> <p>This may be thought of as a homotopy-theoretic analog of the <a class="existingWikiWord" href="/nlab/show/Freyd-Mitchell+embedding+theorem">Freyd-Mitchell embedding theorem</a> for <a class="existingWikiWord" href="/nlab/show/abelian+categories">abelian categories</a>.</p> </div> <div class="num_remark"> <h6 id="remark_4">Remark</h6> <p>One way to read this is that <a class="existingWikiWord" href="/nlab/show/formal+duals">formal duals</a> of presentable <a class="existingWikiWord" href="/nlab/show/stable+infinity-categories">stable infinity-categories</a> are a model for <a class="existingWikiWord" href="/nlab/show/spaces">spaces</a> in (“<a class="existingWikiWord" href="/nlab/show/derived+noncommutative+geometry">derived</a>”) <a class="existingWikiWord" href="/nlab/show/noncommutative+algebraic+geometry">noncommutative algebraic geometry</a>.</p> </div> <h3 id="doldkan_correspondence">Dold-Kan correspondence</h3> <ul> <li><a class="existingWikiWord" href="/nlab/show/infinity-Dold-Kan+correspondence">infinity-Dold-Kan correspondence</a></li> </ul> <h2 id="examples">Examples</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+chain+complexes">(∞,1)-category of chain complexes</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-modules">(∞,1)-category of (∞,1)-modules</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fukaya+category">Fukaya category</a>, <a class="existingWikiWord" href="/nlab/show/Lagrangian+cobordism">Lagrangian cobordism</a></p> </li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/additive+%28%E2%88%9E%2C1%29-category">additive (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Stab%28%E2%88%9E%2C1%29Cat">Stab(∞,1)Cat</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable homotopy theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+model+category">stable model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+category">spectral category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dg-category">dg-category</a>, <a class="existingWikiWord" href="/nlab/show/dg-nerve">dg-nerve</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+category">A-∞ category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/triangulated+category">triangulated category</a>, <a class="existingWikiWord" href="/nlab/show/enhanced+triangulated+category">enhanced triangulated category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/t-structure+on+a+stable+%28%E2%88%9E%2C1%29-category">t-structure on a stable (∞,1)-category</a>, <a class="existingWikiWord" href="/nlab/show/heart+of+a+stable+%28%E2%88%9E%2C1%29-category">heart of a stable (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K-theory+of+a+stable+%28%E2%88%9E%2C1%29-category">K-theory of a stable (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+sequence+of+a+filtered+stable+homotopy+type">spectral sequence of a filtered stable homotopy type</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/prime+spectrum+of+a+symmetric+monoidal+stable+%28%E2%88%9E%2C1%29-category">prime spectrum of a symmetric monoidal stable (∞,1)-category</a></p> </li> </ul> <h2 id="references">References</h2> <p>The abstract <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+theory">(∞,1)-category theoretical</a> notion was introduced and studied in</p> <ul> <li id="StabCat"><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, <em><a class="existingWikiWord" href="/nlab/show/Stable+Infinity-Categories">Stable Infinity-Categories</a></em> [<a href="https://arxiv.org/abs/math/0608228">arXiv:math/0608228</a></li> </ul> <p>This appears in a more comprehensive context of <a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a></p> <ul> <li id="HigherAlgebra"><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, as section 1 of: <em><a class="existingWikiWord" href="/nlab/show/Higher+Algebra">Higher Algebra</a></em></li> </ul> <p>A brief introduction is in</p> <ul> <li id="Harpaz2013"><a class="existingWikiWord" href="/nlab/show/Yonatan+Harpaz">Yonatan Harpaz</a>, <em>Introduction to stable <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-categories</em>, October 2013 (<a class="existingWikiWord" href="/nlab/files/HarpazStableInfinityCategory2013.pdf" title="pdf">pdf</a>)</li> </ul> <p>Discussion of how <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>-linear <a class="existingWikiWord" href="/nlab/show/dg-categories">dg-categories</a>/<a class="existingWikiWord" href="/nlab/show/A-infinity+categories">A-infinity categories</a> present <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>-linear 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>-categories is in</p> <ul> <li id="Cohn13"> <p><a class="existingWikiWord" href="/nlab/show/Lee+Cohn">Lee Cohn</a>, <em>Differential Graded Categories are k-linear Stable Infinity Categories</em> (<a href="http://arxiv.org/abs/1308.2587">arXiv:1308.2587</a>)</p> </li> <li id="Faonte13"> <p><a class="existingWikiWord" href="/nlab/show/Giovanni+Faonte">Giovanni Faonte</a>, <em>Simplicial nerve of an A-infinity category</em> (<a href="http://arxiv.org/abs/1312.2127">arXiv:1312.2127</a>)</p> </li> </ul> <p>A diagram of the interrelation of all models for 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>-categories (in the guise of <a class="existingWikiWord" href="/nlab/show/enhanced+triangulated+categories">enhanced triangulated categories</a>) with a list of further literature:</p> <ul> <li id="Schwede"><a class="existingWikiWord" href="/nlab/show/Stefan+Schwede">Stefan Schwede</a>, <a class="existingWikiWord" href="/nlab/show/Markus+Hausmann">Markus Hausmann</a>, <em>Enhancements of triangulated categories</em>, Graduate Seminar Topology, Bonn 2019-2020 (<a class="existingWikiWord" href="/nlab/files/Schwede_EnhancedSeminar.pdf" title="pdf">pdf</a>)</li> </ul> <p>For discussion of the <a class="existingWikiWord" href="/nlab/show/stable+model+category">stable model category</a> models of stable <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-categories see</p> <ul> <li id="SchwedeShipley"><a class="existingWikiWord" href="/nlab/show/Stefan+Schwede">Stefan Schwede</a>, <a class="existingWikiWord" href="/nlab/show/Brooke+Shipley">Brooke Shipley</a>, <em>Classification of stable model categories</em> (<a href="http://hopf.math.purdue.edu/Schwede-Shipley/class.final.pdf">Hopf pdf</a>) and (<a href="http://arxiv.org/abs/math/0108143">arXiv:math/0108143</a>)</li> </ul> <p>The enrichment over spectra is made precise in</p> <ul> <li id="GepnerHaugseng13"><a class="existingWikiWord" href="/nlab/show/David+Gepner">David Gepner</a>, <a class="existingWikiWord" href="/nlab/show/Rune+Haugseng">Rune Haugseng</a>, <em>Enriched ∞-categories via non-symmetric ∞-operads</em> (<a href="http://arxiv.org/abs/1312.3178">arXiv:1312.3178</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on November 16, 2023 at 10:02:12. See the <a href="/nlab/history/stable+%28infinity%2C1%29-category" 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/stable+%28infinity%2C1%29-category" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/758/#Item_9">Discuss</a><span class="backintime"><a href="/nlab/revision/stable+%28infinity%2C1%29-category/64" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/stable+%28infinity%2C1%29-category" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/stable+%28infinity%2C1%29-category" accesskey="S" class="navlink" id="history" rel="nofollow">History (64 revisions)</a> <a href="/nlab/show/stable+%28infinity%2C1%29-category/cite" style="color: black">Cite</a> <a href="/nlab/print/stable+%28infinity%2C1%29-category" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/stable+%28infinity%2C1%29-category" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>