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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="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='#Definition'>Definition</a></li> <ul> <li><a href='#OnTriangulatedCategories'>On triangulated categories</a></li> <li><a href='#InStableInfinityCategories'>On 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='#properties'>Properties</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#relation_to_spectral_sequences'>Relation to spectral sequences</a></li> <li><a href='#RelationToNormalTorsionTheories'>Relation to normal torsion theories</a></li> <li><a href='#towers'>Towers</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="Definition">Definition</h2> <p>Originally, t-structures were defined</p> <ul> <li><a href="#OnTriangulatedCategories">on triangulated categories</a></li> </ul> <p>These typically arise as <a class="existingWikiWord" href="/nlab/show/homotopy+category+of+an+%28infinity%2C1%29-category">homotopy categories</a> of t-structures</p> <ul> <li><a href="#InStableInfinityCategories">on 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> <h3 id="OnTriangulatedCategories">On triangulated categories</h3> <p> <div class='num_defn' id='tStructure'> <h6>Definition</h6> <p><strong>(t-structure on a triangulated category)</strong> <br /> 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 a <a class="existingWikiWord" href="/nlab/show/triangulated+category">triangulated category</a>. A <em>t-structure</em> on <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/pair">pair</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔱</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mi>C</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msub><mo>,</mo><msub><mi>C</mi> <mrow><mo>≤</mo><mn>0</mn></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathfrak{t}=(C_{\ge 0}, C_{\le 0})</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/strictly+full+subcategories">strictly full subcategories</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msub><mo>,</mo><msub><mi>C</mi> <mrow><mo>≤</mo><mn>0</mn></mrow></msub><mo>↪</mo><mi>C</mi></mrow><annotation encoding="application/x-tex"> C_{\geq 0}, C_{\leq 0} \hookrightarrow C </annotation></semantics></math></div> <p>such that</p> <ol> <li> <p>for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><msub><mi>C</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">X \in C_{\geq 0}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>∈</mo><msub><mi>C</mi> <mrow><mo>≤</mo><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">Y \in C_{\leq 0}</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/hom+object">hom object</a> is the <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><msub><mi>Hom</mi> <mi>C</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">Hom_{C}(X, Y[-1]) = 0</annotation></semantics></math>;</p> </li> <li> <p>the subcategories are closed under <a class="existingWikiWord" href="/nlab/show/suspension">suspension</a>/desuspension: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msub><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo>⊂</mo><msub><mi>C</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">C_{\geq 0}[1] \subset C_{\geq 0}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mrow><mo>≤</mo><mn>0</mn></mrow></msub><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><mo stretchy="false">]</mo><mo>⊂</mo><msub><mi>C</mi> <mrow><mo>≤</mo><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">C_{\leq 0}[-1] \subset C_{\leq 0}</annotation></semantics></math>.</p> </li> <li> <p>For all <a class="existingWikiWord" href="/nlab/show/objects">objects</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">X \in C</annotation></semantics></math> there is a <a class="existingWikiWord" href="/nlab/show/fiber+sequence">fiber sequence</a> (i.e. an <a class="existingWikiWord" href="/nlab/show/exact+triangle">exact triangle</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>→</mo><mi>X</mi><mo>→</mo><mi>Z</mi></mrow><annotation encoding="application/x-tex">Y \to X \to Z</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>∈</mo><msub><mi>C</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">Y \in C_{\geq 0}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Z</mi><mo>∈</mo><msub><mi>C</mi> <mrow><mo>≤</mo><mn>0</mn></mrow></msub><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">Z \in C_{\leq 0}[-1]</annotation></semantics></math>.</p> </li> </ol> <p></p> </div> </p> <p> <div class='num_defn' id='HeartOfATStructure'> <h6>Definition</h6> <p>Given a t-structure (Def. <a class="maruku-ref" href="#tStructure"></a>), its <em>heart</em> is the intersection</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msub><mo>∩</mo><msub><mi>C</mi> <mrow><mo>≤</mo><mn>0</mn></mrow></msub><mo>↪</mo><mi>C</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> C_{\geq 0} \cap C_{\leq 0} \hookrightarrow C \,. </annotation></semantics></math></div> <p></p> </div> </p> <h3 id="InStableInfinityCategories">On 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</h3> <p> <div class='num_defn' id='tStructureOnStableInfinityCategory'> <h6>Definition</h6> <p><strong>(t-structure in 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)</strong> <br /> A <em>t-structure</em> on 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>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> is a t-structure in the above sense (Def. <a class="maruku-ref" href="#tStructure"></a>) on its <a class="existingWikiWord" href="/nlab/show/underlying">underlying</a> <a class="existingWikiWord" href="/nlab/show/homotopy+category+of+an+%28%E2%88%9E%2C1%29-category">homotopy category</a> (which is <a class="existingWikiWord" href="/nlab/show/triangulated+category">triangulated</a>, see <a href="stable+infinity-category#TheTriangulatedHomotopyCategory">there</a>).</p> </div> (<a href="#LurieHA">Lurie, Higher Algebra, Def. 1.2.1.4</a>)</p> <p>Therefore, a t-structure on 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 <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> is a system of <a class="existingWikiWord" href="/nlab/show/full+sub-%28%E2%88%9E%2C1%29-categories">full sub-(∞,1)-categories</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mrow><mo>≥</mo><mi>n</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\mathcal{C}_{\geq n}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mrow><mo>≤</mo><mi>n</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\mathcal{C}_{\leq n}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{Z}</annotation></semantics></math>.</p> <p> <div class='num_prop'> <h6>Proposition</h6> <p>In this situation (Def. <a class="maruku-ref" href="#tStructureOnStableInfinityCategory"></a>)</p> <ol> <li> <p>the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mrow><mo>≤</mo><mi>n</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\mathcal{C}_{\leq n} </annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/reflective+sub-%28%E2%88%9E%2C1%29-categories">reflective sub-(∞,1)-categories</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mrow><mo>≤</mo><mi>n</mi></mrow></msub><munderover><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>⊥</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow><munder><mo>↪</mo><mrow></mrow></munder><mover><mo>⟵</mo><mrow><msub><mi>τ</mi> <mrow><mo>≤</mo><mi>n</mi></mrow></msub></mrow></mover></munderover><mi>𝒞</mi></mrow><annotation encoding="application/x-tex"> \mathcal{C}_{\leq n} \underoverset {\underset{}{\hookrightarrow}} {\overset{\tau_{\leq n}}{\longleftarrow}} {\;\; \bot \;\;} \mathcal{C} </annotation></semantics></math></div></li> <li> <p>the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mrow><mo>≥</mo><mi>n</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\mathcal{C}_{\geq n}</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/coreflective+sub-%28%E2%88%9E%2C1%29-categories">coreflective sub-(∞,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>𝒞</mi><munderover><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>⊥</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow><munder><mo>⟶</mo><mrow><msub><mi>τ</mi> <mrow><mo>≥</mo><mi>n</mi></mrow></msub></mrow></munder><mo>↩</mo></munderover><msub><mi>𝒞</mi> <mrow><mo>≤</mo><mi>n</mi></mrow></msub></mrow><annotation encoding="application/x-tex"> \mathcal{C} \underoverset {\underset{\tau_{\geq n}}{\longrightarrow}} {\hookleftarrow} {\;\; \bot \;\;} \mathcal{C}_{\leq n} </annotation></semantics></math></div></li> </ol> <p></p> </div> (<a href="#LurieHA">Lurie, Higher Algebra, Def. 1.2.1.5, Cor. 1.2.1.6, Ntn. 1.2.1.7</a>)</p> <center> <img src="https://ncatlab.org/nlab/files/HeartOfTStructureAsCoModals-230408.jpg" width="600" /> </center> <h2 id="properties">Properties</h2> <h3 id="general">General</h3> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>The heart 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 an <a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a>.</p> </div> <p>(<a href="#BeilinsonBernsteinDeligne82">BBD 82</a>, <a class="existingWikiWord" href="/nlab/show/Higher+Algebra">Higher Algebra, remark 1.2.1.12</a>, <a href="#FL16">FL16, Ex. 4.1</a> and <a href="#FLM19">FLM19, §3.1</a>)</p> <h3 id="relation_to_spectral_sequences">Relation to spectral sequences</h3> <p>If the heart (Def. <a class="maruku-ref" href="#HeartOfATStructure"></a>) of a t-structure on a <a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a> with <a class="existingWikiWord" href="/nlab/show/sequential+limits">sequential limits</a> is an <a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a>, then the <a class="existingWikiWord" href="/nlab/show/spectral+sequence+of+a+filtered+stable+homotopy+type">spectral sequence of a filtered stable homotopy type</a> <a href="spectral+sequence#ConvergenceOfSpectralSequences">converges</a> (see there).</p> <h3 id="RelationToNormalTorsionTheories">Relation to normal torsion theories</h3> <p>In the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+theory">(∞,1)-category theory</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math>-structures arise as <a class="existingWikiWord" href="/nlab/show/torsion+theory">torsion/torsionfree</a> classes associated with suitable <a class="existingWikiWord" href="/nlab/show/orthogonal+factorization+system">factorization systems</a> on <a class="existingWikiWord" href="/nlab/show/stable+%E2%88%9E-categories">stable ∞-categories</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>.</p> <ul> <li> <p>In <a class="existingWikiWord" href="/nlab/show/stable+%E2%88%9E-category">stable ∞-category</a>-theory, the relevant sub-<a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-categories">(∞,1)-categories</a> are closed under de/suspension simply because they are (co-)<a class="existingWikiWord" href="/nlab/show/reflective+sub-%28%E2%88%9E%2C1%29-categories">reflective</a>, arising from co/<a class="existingWikiWord" href="/nlab/show/reflective+factorization+systems">reflective factorization systems</a> on <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> <li> <p>A <em>bireflective</em> factorization system on 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 <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> consists of a <a class="existingWikiWord" href="/nlab/show/orthogonal+factorization+system">factorization system</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔽</mi><mo>=</mo><mo stretchy="false">(</mo><mi>E</mi><mo>,</mo><mi>M</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{F}=(E,M)</annotation></semantics></math> where both classes satisfy the <a class="existingWikiWord" href="/nlab/show/two-out-of-three">two-out-of-three</a> property.</p> </li> <li> <p>A bireflective factorization system <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>E</mi><mo>,</mo><mi>M</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(E,M)</annotation></semantics></math> on 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 <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 called <em>normal</em> if the diagram <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mi>x</mi><mo>→</mo><mi>x</mi><mo>→</mo><mi>R</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">S x\to x\to R x</annotation></semantics></math> obtained from the reflection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo lspace="verythinmathspace">:</mo><mi>C</mi><mo>→</mo><mi>M</mi><mo stretchy="false">/</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">R\colon C\to M/0</annotation></semantics></math> and the coreflection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo lspace="verythinmathspace">:</mo><mi>C</mi><mo>→</mo><mo>*</mo><mspace width="negativethinmathspace"></mspace><mo stretchy="false">/</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">S\colon C\to *\!/E</annotation></semantics></math> (where the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mo stretchy="false">/</mo><mspace width="negativethinmathspace"></mspace><mo>*</mo><mo>=</mo><mo stretchy="false">{</mo><mi>A</mi><mo lspace="mediummathspace" rspace="mediummathspace">∣</mo><mo stretchy="false">(</mo><mn>0</mn><mo>→</mo><mi>A</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>M</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">M/\!* =\{A\mid (0\to A)\in M\}</annotation></semantics></math> is obtained as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ψ</mi><mo stretchy="false">(</mo><mi>E</mi><mo>,</mo><mi>M</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Psi(E,M)</annotation></semantics></math> under the adjunction <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Φ</mi><mo>⊣</mo><mi>Ψ</mi></mrow><annotation encoding="application/x-tex">\Phi \dashv \Psi</annotation></semantics></math> described at <em><a class="existingWikiWord" href="/nlab/show/reflective+factorization+system">reflective factorization system</a></em> and in <a href="#CHK85">CHK85</a>; see also <a href="#FL16">FL16, §1.1</a>) is <em>exact</em>, meaning that the square in</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mtable displaystyle="false" rowspacing="0.5ex" columnalign="center center center center center center"><mtr><mtd><mn>0</mn></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>S</mi><mi>X</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mn>0</mn></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>R</mi><mi>X</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mn>0</mn></mtd></mtr></mtable></mrow><annotation encoding="application/x-tex"> \begin{array}{cccccc} 0 &\to& S X &\to& X\\ && \downarrow&&\downarrow\\ && 0 &\to& R X\\ && && \downarrow\\ && && 0 \end{array} </annotation></semantics></math></div> <p>is a fiber sequence for any object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>; see <a href="#FL16">FL16, Def 3.5 and Prop. 3.10</a> for equivalent conditions for normality.</p> </li> </ul> <p> <div class='num_remark'> <h6>Remark</h6> <p><a href="#CHK85">CHK85</a> established a hierarchy between the three notions of simple, semi-exact and normal factorization system: in the setting 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 the three notions turn out to be equivalent: see <a href="#FL16">FL16, Thm 3.11</a>.</p> </div> </p> <p> <div class='num_prop' id='CorrespondenceWithNormalTorsionTheories'> <h6>Proposition</h6> <p>There is a bijective correspondence between the class <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>TS</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">TS( C )</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math>-structures and the class of normal torsion theories on 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 <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>, induced by the following correspondence:</p> <ul> <li> <p>On the one side, given a normal, bireflective factorization system <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>E</mi><mo>,</mo><mi>M</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(E,M)</annotation></semantics></math> on <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> we define the two classes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>C</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msub><mo stretchy="false">(</mo><mi>𝔽</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>C</mi> <mrow><mo><</mo><mn>0</mn></mrow></msub><mo stretchy="false">(</mo><mi>𝔽</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(C_{\ge0}(\mathbb{F}), C_{\lt 0}(\mathbb{F}))</annotation></semantics></math> of a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math>-structure <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔱</mi><mo stretchy="false">(</mo><mi>𝔽</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathfrak{t}(\mathbb{F})</annotation></semantics></math> to be the torsion and torsionfree classes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo>*</mo><mspace width="negativethinmathspace"></mspace><mo stretchy="false">/</mo><mi>E</mi><mo>,</mo><mi>M</mi><mo stretchy="false">/</mo><mspace width="negativethinmathspace"></mspace><mo>*</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(*\!/E, M/\!*)</annotation></semantics></math> associated to the factorization <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>E</mi><mo>,</mo><mi>M</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(E,M)</annotation></semantics></math>.</p> </li> <li> <p>On the other side, given a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math>-structure on <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> we set</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">{</mo><mi>f</mi><mo>∈</mo><msup><mi>C</mi> <mrow><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mrow></msup><mo lspace="mediummathspace" rspace="mediummathspace">∣</mo><msub><mi>τ</mi> <mrow><mo><</mo><mn>0</mn></mrow></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mtext> is an equivalence</mtext><mo stretchy="false">}</mo><mo>;</mo></mrow><annotation encoding="application/x-tex">E(t)=\{f\in C^{\Delta[1]} \mid \tau_{\lt 0}(f) \;\text{ is an equivalence}\};</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>M</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">{</mo><mi>f</mi><mo>∈</mo><msup><mi>C</mi> <mrow><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mrow></msup><mo lspace="mediummathspace" rspace="mediummathspace">∣</mo><msub><mi>τ</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mtext> is an equivalence</mtext><mo stretchy="false">}</mo><mo>.</mo></mrow><annotation encoding="application/x-tex"> M(t)=\{f\in C^{\Delta[1]} \mid \tau_{\geq0}(f) \;\text{ is an equivalence}\}. </annotation></semantics></math></div></li> </ul> <p></p> </div> </p> <p>This is <a href="#FL16">FL16, Theorem 3.13</a></p> <p> <div class='num_prop'> <h6>Proposition</h6> <p>There is a natural monotone action of the group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math> of integers on the class <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>TS</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">TS( C )</annotation></semantics></math> (now confused with the class <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>FS</mi> <mi>ν</mi></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">FS_\nu( C )</annotation></semantics></math> of normal torsion theories on <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>) given by the suspension functor: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔽</mi><mo>=</mo><mo stretchy="false">(</mo><mi>E</mi><mo>,</mo><mi>M</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{F}=(E,M)</annotation></semantics></math> goes to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔽</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo>=</mo><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo>,</mo><mi>M</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{F}[1] = (E[1], M[1])</annotation></semantics></math>.</p> </div> </p> <p>This correspondence leads to study <em>families</em> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math>-structures <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>𝔽</mi> <mi>i</mi></msub><msub><mo stretchy="false">}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\{\mathbb{F}_i\}_{i\in I}</annotation></semantics></math>; more precisely, we are led to study <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math>-equivariant</em> <a class="existingWikiWord" href="/nlab/show/k-ary+factorization+system">multiple factorization systems</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi><mo>→</mo><mi>TS</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">J\to TS( C )</annotation></semantics></math>.</p> <p> <div class='num_prop' id='StableTStructure'> <h6>Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔱</mi><mo>∈</mo><mi>TS</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathfrak{t} \in TS(C)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔽</mi><mo>=</mo><mo stretchy="false">(</mo><mi>E</mi><mo>,</mo><mi>M</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{F}=(E,M)</annotation></semantics></math> correspond each other under the above bijection (Prop. <a class="maruku-ref" href="#CorrespondenceWithNormalTorsionTheories"></a>); then the following conditions are equivalent:</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔱</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo>=</mo><mi>𝔱</mi></mrow><annotation encoding="application/x-tex">\mathfrak{t}[1]=\mathfrak{t}</annotation></semantics></math>, i.e. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mrow><mo>≥</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>C</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">C_{\geq 1}= C_{\geq 0}</annotation></semantics></math>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msub><mo>=</mo><mo>*</mo><mspace width="negativethinmathspace"></mspace><mo stretchy="false">/</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">C_{\geq 0}=*\!/E</annotation></semantics></math> is 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;</p> </li> <li> <p>the class <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> is closed under pullback.</p> </li> </ol> <p></p> </div> </p> <p>In each of these cases, we say that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔱</mi></mrow><annotation encoding="application/x-tex">\mathfrak{t}</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>E</mi><mo>,</mo><mi>M</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(E,M)</annotation></semantics></math> is <em>stable</em>.</p> <p>This is <a href="#FLM19">FLM19, Theorem 6.3</a></p> <p>This results allows us to recognize <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math>-structures with stable classes</em> precisely as those which are fixed in the natural <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math>-action on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>TS</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">TS( C )</annotation></semantics></math>.</p> <p>Two “extremal” choices of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math>-chains of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math>-structures draw a connection between two apparently separated constructions in the theory of derived categories: <em>Harder-Narashiman filtrations</em> and <em>semiorthogonal decompositions</em> on triangulated categories: we adopt the shorthand <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔱</mi> <mrow><mn>1</mn><mo>,</mo><mi>…</mi><mo>,</mo><mi>n</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\mathfrak{t}_{1,\dots, n}</annotation></semantics></math> to denote the tuple <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔱</mi> <mn>1</mn></msub><mo>⪯</mo><msub><mi>𝔱</mi> <mn>2</mn></msub><mo>⪯</mo><mi>⋯</mi><mo>⪯</mo><msub><mi>𝔱</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">\mathfrak{t}_1\preceq \mathfrak{t}_2\preceq\cdots\preceq \mathfrak{t}_n</annotation></semantics></math>, each of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔱</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\mathfrak{t}_i</annotation></semantics></math> being a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math>-structure <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msub><mi>C</mi> <mi>i</mi></msub><msub><mo stretchy="false">)</mo> <mrow><mo>≥</mo><mn>0</mn></mrow></msub><mo>,</mo><mo stretchy="false">(</mo><msub><mi>C</mi> <mi>i</mi></msub><msub><mo stretchy="false">)</mo> <mrow><mo><</mo><mn>0</mn></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">((C_i)_{\ge 0}, (C_i)_{\lt 0})</annotation></semantics></math> on <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 we denote similarly <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔱</mi> <mi>ω</mi></msub></mrow><annotation encoding="application/x-tex">\mathfrak{t}_\omega</annotation></semantics></math>. Then</p> <ul> <li>In the <em>stable case</em> the tuple <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>t</mi> <mrow><mn>1</mn><mo>,</mo><mi>…</mi><mo>,</mo><mi>n</mi></mrow></msub></mrow><annotation encoding="application/x-tex">t_{1,\dots, n}</annotation></semantics></math> is endowed with a (monotone) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math>-action, and the map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mn>0</mn><mo><</mo><mn>1</mn><mi>⋯</mi><mo><</mo><mi>n</mi><mo stretchy="false">}</mo><mo>→</mo><mi>TS</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\{0\lt 1\cdots\lt n\}\to TS( C )</annotation></semantics></math> is equivariant with respect to this action; the absence of nontrivial <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math>-actions on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mn>0</mn><mo><</mo><mn>1</mn><mi>⋯</mi><mo><</mo><mi>n</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{0\lt 1\cdots\lt n\}</annotation></semantics></math> forces each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>t</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">t_i</annotation></semantics></math> to be stable.</li> <li>In the <em>orbit case</em> we consider an <em>infinite</em> family <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>t</mi> <mi>ω</mi></msub></mrow><annotation encoding="application/x-tex">t_\omega</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math>-structures on <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>, obtained as the orbit of a fixed <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>E</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>M</mi> <mn>0</mn></msub><mo stretchy="false">)</mo><mo>∈</mo><mi>TS</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(E_0, M_0)\in TS( C )</annotation></semantics></math> with respect to the natural <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math>-action.</li> </ul> <h3 id="towers">Towers</h3> <p>The HN-filtration induced by a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math>-structure and the factorization induced by a <a class="existingWikiWord" href="/nlab/show/semiorthogonal+decomposition">semiorthogonal decomposition</a> on <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> both are the byproduct of the <em>tower</em> associated to a tuple <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔱</mi> <mrow><mn>1</mn><mo>,</mo><mi>…</mi><mo>,</mo><mi>n</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\mathfrak{t}_{1,\dots, n}</annotation></semantics></math>:</p> <h2 id="Examples">Examples</h2> <p>The archetypical and historically motivating example (cf. <a href="#GelfandManin96">Gelfand & Manin (1996), IV.4 §1</a>) is the following:</p> <p> <div class='num_remark' id='TStructureOnDerivedCategoryOfAbelianCategory'> <h6>Example</h6> <p><strong>(canonical t-structure on the <a class="existingWikiWord" href="/nlab/show/derived+category">derived category</a> of an <a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a>)</strong> <br /> For <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">\mathcal{A}</annotation></semantics></math> an <a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a>, its unbounded <a class="existingWikiWord" href="/nlab/show/derived+category">derived category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒟</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>𝒜</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{D}_\bullet(\mathcal{A})</annotation></semantics></math></p> <ol> <li> <p>carries a t-structure (Def. <a class="maruku-ref" href="#tStructure"></a>) for which <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒟</mi><mo stretchy="false">(</mo><mi>𝒜</mi><msub><mo stretchy="false">)</mo> <mrow><mo>≥</mo><mi>n</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\mathcal{D}(\mathcal{A})_{\geq n}</annotation></semantics></math> (rep. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒟</mi><mo stretchy="false">(</mo><mi>𝒜</mi><msub><mo stretchy="false">)</mo> <mrow><mo>≤</mo><mi>n</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\mathcal{D}(\mathcal{A})_{\leq n}</annotation></semantics></math>) is the <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> of <a class="existingWikiWord" href="/nlab/show/objects">objects</a> presented by <a class="existingWikiWord" href="/nlab/show/chain+complexes">chain complexes</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">V_\bullet</annotation></semantics></math> whose <a class="existingWikiWord" href="/nlab/show/chain+homology">chain homology</a>-<a class="existingWikiWord" href="/nlab/show/homology+groups">groups</a> are <a class="existingWikiWord" href="/nlab/show/trivial+group">trivial</a> in degrees <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo><</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">\lt n</annotation></semantics></math> (resp. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>></mo><mi>n</mi></mrow><annotation encoding="application/x-tex">\gt n</annotation></semantics></math>);</p> </li> <li> <p>whose heart (Def. <a class="maruku-ref" href="#HeartOfATStructure"></a>) is <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalent</a> to <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">\mathcal{A}</annotation></semantics></math> (embedded as the <a class="existingWikiWord" href="/nlab/show/chain+complexes">chain complexes</a> which are concentrated in degree 0).</p> </li> </ol> <p></p> </div> </p> <p>(eg. <a href="#GelfandManin96">Gelfand & Manin (1996), IV.4 §3</a>)</p> <p> <div class='num_remark' id='CanonicalTStructureOnSpectra'> <h6>Example</h6> <p><strong>(canonical t-structure on spectra)</strong> <br /> The <a class="existingWikiWord" href="/nlab/show/stable+%28infinity%2C1%29-category+of+spectra">stable (infinity,1)-category of spectra</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spectra</mi></mrow><annotation encoding="application/x-tex">Spectra</annotation></semantics></math>, carries a canonical t-structure for which</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Spectra</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">Spectra_{\geq 0}</annotation></semantics></math> is the sub-category of <a class="existingWikiWord" href="/nlab/show/connective+spectra">connective spectra</a>, with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msub><mo lspace="verythinmathspace">:</mo><mi>Spectra</mi><mo>→</mo><msub><mi>Spectra</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">\tau_{\geq 0} \colon Spectra \to Spectra_{\geq 0}</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/connective+cover">connective cover</a>-construction.</p> </li> <li> <p>…</p> </li> </ul> <p></p> </div> </p> <p>(e.g. <a href="#LurieHA">Lurie, Higher Algebr, pp. 150</a>)</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Bridgeland+stability">Bridgeland stability</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/n-connective+object">n-connective object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connective+cover">connective cover</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/theorem+of+the+heart">theorem of the heart</a></p> </li> </ul> <h2 id="references">References</h2> <p>For <a class="existingWikiWord" href="/nlab/show/triangulated+categories">triangulated categories</a>:</p> <p>the notion is due to</p> <ul> <li id="BeilinsonBernsteinDeligne82"> <p><a class="existingWikiWord" href="/nlab/show/Alexander+Beilinson">Alexander Beilinson</a>, <a class="existingWikiWord" href="/nlab/show/Joseph+Bernstein">Joseph Bernstein</a>, <a class="existingWikiWord" href="/nlab/show/Pierre+Deligne">Pierre Deligne</a>, <em>Faisceaux pervers</em>, Astérisque <strong>100</strong> (1982) [<a href="https://smf.emath.fr/publications/faisceaux-pervers">ISBN:978-2-85629-878-7</a>, <a href="https://publications.ias.edu/sites/default/files/Faisceaux%20pervers.pdf">pdf</a>, <a href="http://www.ams.org/mathscinet-getitem?mr=751966">MR86g:32015</a>]</p> <blockquote> <p>(otherwise introducing <a class="existingWikiWord" href="/nlab/show/perverse+sheaves">perverse sheaves</a>)</p> </blockquote> </li> </ul> <p>Further development:</p> <ul> <li id="GelfandManin96"> <p><a class="existingWikiWord" href="/nlab/show/Sergei+Gelfand">Sergei Gelfand</a>, <a class="existingWikiWord" href="/nlab/show/Yuri+Manin">Yuri Manin</a>, Section IV.4 of: <em><a class="existingWikiWord" href="/nlab/show/Methods+of+homological+algebra">Methods of homological algebra</a></em>, transl. from the 1988 Russian (Nauka Publ.) original, Springer (1996, 2002) [<a href="https://doi.org/10.1007/978-3-662-12492-5">doi:10.1007/978-3-662-12492-5</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Donu+Arapura">Donu Arapura</a>, <em>Triangulated categories and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math>-structures</em> [<a href="http://www.math.purdue.edu/~dvb/preprints/perv2.pdf">pdf</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dan+Abramovich">Dan Abramovich</a>, <a class="existingWikiWord" href="/nlab/show/Alexander+Polishchuk">Alexander Polishchuk</a>, <em>Sheaves of t-structures and valuative criteria for stable complexes</em>, J. reine angew. Math. <strong>590</strong> (2006) 89-130 [<a href="https://arxiv.org/abs/math/0309435">arXiv:math/0309435</a>, <a href="https://doi.org/10.1515/CRELLE.2006.005">doi:10.1515/CRELLE.2006.005</a>]</p> </li> <li> <p>A. L. Gorodentsev, S. A. Kuleshov, A. N. Rudakov, <em>t-stabilities and t-structures on triangulated categories</em>, Izv. Ross. Akad. Nauk Ser. Mat. <strong>68</strong> (2004), no. 4, 117-150</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Alexander+Polishchuk">Alexander Polishchuk</a>, <em>Constant families of t-structures on derived categories of coherent sheaves</em>, Moscow Math. J. <strong>7</strong> (2007) 109-134 [<a href="https://arxiv.org/abs/math/0606013">arXiv:math/0606013</a>]</p> </li> <li> <p>John Collins, <a class="existingWikiWord" href="/nlab/show/Alexander+Polishchuk">Alexander Polishchuk</a>, <em>Gluing stability conditions</em> [<a href="http://arxiv.org/abs/0902.0323">arxiv/0902.0323</a>]</p> </li> </ul> <p>For <a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-categories">stable (∞,1)-categories</a>:</p> <ul> <li id="LurieHA"><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, Section 1.2.1 in: <em><a class="existingWikiWord" href="/nlab/show/Higher+Algebra">Higher Algebra</a></em></li> </ul> <p>On <a class="existingWikiWord" href="/nlab/show/reflective+factorization+systems">reflective factorization systems</a>:</p> <ul> <li id="CHK85"> <p>C. Cassidy, M. Hébert, <a class="existingWikiWord" href="/nlab/show/Max+Kelly">Max Kelly</a>, <em>Reflective subcategories, localizations, and factorization systems</em>, J. Austral. Math Soc. (Series A) <strong>38</strong> (1985) 287-329 [<a href="https://doi.org/10.1017/S1446788700023624">doi:10.1017/S1446788700023624</a>]</p> </li> <li id="RosickyTholen08"> <p><a class="existingWikiWord" href="/nlab/show/Jiri+Rosicky">Jiri Rosicky</a>, <a class="existingWikiWord" href="/nlab/show/Walter+Tholen">Walter Tholen</a>, <em>Factorization, Fibration and Torsion</em>, Journal of Homotopy and Related Structures, Vol. 2(2007), No. 2, pp. 295-314 [<a href="http://arxiv.org/abs/0801.0063">arXiv:0801.0063</a>, <a href="http://www.emis.de/journals/JHRS/volumes/2007/n2a14/">publisher</a>]</p> </li> </ul> <p>and on normal torsion theories 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:</p> <ul> <li id="FL16"> <p><a class="existingWikiWord" href="/nlab/show/Domenico+Fiorenza">Domenico Fiorenza</a>, <a class="existingWikiWord" href="/nlab/show/Fosco+Loregian">Fosco Loregian</a>, <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math>-Structures are normal torsion theories</em>, Appl Categor Struct <strong>24</strong> (2016) 181–208 [<a href="http://arxiv.org/abs/1408.7003">arxiv:1408.7003</a>, <a href="https://doi.org/10.1007/s10485-015-9393-z">doi:10.1007/s10485-015-9393-z</a>]</p> </li> <li id="FLM19"> <p><a class="existingWikiWord" href="/nlab/show/Domenico+Fiorenza">Domenico Fiorenza</a>, <a class="existingWikiWord" href="/nlab/show/Fosco+Loregian">Fosco Loregian</a>, <a class="existingWikiWord" href="/nlab/show/Giovanni+Marchetti">Giovanni Marchetti</a>, <em>Hearts and towers 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</em>, J. Homotopy Relat. Struct. <strong>14</strong> (2019) 993–1042 [<a href="https://arxiv.org/abs/1501.04658">arXiv:1501.04658</a>, <a href="https://doi.org/10.1007/s40062-019-00237-0">doi:10.1007/s40062-019-00237-0</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fosco+Loregian">Fosco Loregian</a>, <a class="existingWikiWord" href="/nlab/show/Simone+Virili">Simone Virili</a> <em>Triangulated factorization systems and t-structures</em>, Journal of Algebra <strong>550</strong> (2020) 219-241 [<a href="https://doi.org/10.1016/j.jalgebra.2019.12.021">doi:10.1016/j.jalgebra.2019.12.021</a>]</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on April 20, 2023 at 06:33:40. 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