CINXE.COM

<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0 plus SVG 1.1//EN" "http://www.w3.org/2002/04/xhtml-math-svg/xhtml-math-svg-flat.dtd" > <html xmlns="http://www.w3.org/1999/xhtml"> <head> <title> filtered object in nLab </title> <meta http-equiv="Content-Type" content="text/html; charset=UTF-8" /> <meta name="robots" content="index,follow" /> <meta name="viewport" content="width=device-width, initial-scale=1" /> <link href="/stylesheets/instiki.css?1676280126" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/mathematics.css?1660229990" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/syntax.css?1660229990" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/nlab.css?1676280126" media="all" rel="stylesheet" type="text/css" /> <link rel="stylesheet" type="text/css" href="https://cdn.jsdelivr.net/gh/dreampulse/computer-modern-web-font@master/fonts.css"/> <style type="text/css"> h1#pageName, div.info, .newWikiWord a, a.existingWikiWord, .newWikiWord a:hover, [actiontype="toggle"]:hover, #TextileHelp h3 { color: #226622; } a:visited.existingWikiWord { color: #164416; } </style> <style type="text/css"><![CDATA[/*></style> <script src="/javascripts/prototype.js?1660229990" type="text/javascript"></script> <script src="/javascripts/effects.js?1660229990" type="text/javascript"></script> <script src="/javascripts/dragdrop.js?1660229990" type="text/javascript"></script> <script src="/javascripts/controls.js?1660229990" type="text/javascript"></script> <script src="/javascripts/application.js?1660229990" type="text/javascript"></script> <script src="/javascripts/page_helper.js?1660229990" type="text/javascript"></script> <script src="/javascripts/thm_numbering.js?1660229990" type="text/javascript"></script> <script type="text/x-mathjax-config"> <![CDATA[//><!]]> </script> <script type="text/javascript"> <![CDATA[//><!]]> </script> <link href="https://ncatlab.org/nlab/atom_with_headlines" rel="alternate" title="Atom with headlines" type="application/atom+xml" /> <link href="https://ncatlab.org/nlab/atom_with_content" rel="alternate" title="Atom with full content" type="application/atom+xml" /> <script type="text/javascript"> document.observe("dom:loaded", function() { generateThmNumbers(); }); </script> </head> <body> <div id="Container"> <div id="Content"> <h1 id="pageName"> <span style="float: left; margin: 0.5em 0.25em -0.25em 0"> <svg xmlns="http://www.w3.org/2000/svg" width="1.872em" height="1.8em" viewBox="0 0 190 181"> <path fill="#226622" d="M72.8 145c-1.6 17.3-15.7 10-23.6 20.2-5.6 7.3 4.8 15 11.4 15 11.5-.2 19-13.4 26.4-20.3 3.3-3 8.2-4 11.2-7.2a14 14 0 0 0 2.9-11.1c-1.4-9.6-12.4-18.6-16.9-27.2-5-9.6-10.7-27.4-24.1-27.7-17.4-.3-.4 26 4.7 30.7 2.4 2.3 5.4 4.1 7.3 6.9 1.6 2.3 2.1 5.8-1 7.2-5.9 2.6-12.4-6.3-15.5-10-8.8-10.6-15.5-23-26.2-31.8-5.2-4.3-11.8-8-18-3.7-7.3 4.9-4.2 12.9.2 18.5a81 81 0 0 0 30.7 23c3.3 1.5 12.8 5.6 10 10.7-2.5 5.2-11.7 3-15.6 1.1-8.4-3.8-24.3-21.3-34.4-13.7-3.5 2.6-2.3 7.6-1.2 11.1 2.8 9 12.2 17.2 20.9 20.5 17.3 6.7 34.3-8 50.8-12.1z"/> <path fill="#a41e32" d="M145.9 121.3c-.2-7.5 0-19.6-4.5-26-5.4-7.5-12.9-1-14.1 5.8-1.4 7.8 2.7 14.1 4.8 21.3 3.4 12 5.8 29-.8 40.1-3.6-6.7-5.2-13-7-20.4-2.1-8.2-12.8-13.2-15.1-1.9-2 9.7 9 21.2 12 30.1 1.2 4 2 8.8 6.4 10.3 6.9 2.3 13.3-4.7 17.7-8.8 12.2-11.5 36.6-20.7 43.4-36.4 6.7-15.7-13.7-14-21.3-7.2-9.1 8-11.9 20.5-23.6 25.1 7.5-23.7 31.8-37.6 38.4-61.4 2-7.3-.8-29.6-13-19.8-14.5 11.6-6.6 37.6-23.3 49.2z"/> <path fill="#193c78" d="M86.3 47.5c0-13-10.2-27.6-5.8-40.4 2.8-8.4 14.1-10.1 17-1 3.8 11.6-.3 26.3-1.8 38 11.7-.7 10.5-16 14.8-24.3 2.1-4.2 5.7-9.1 11-6.7 6 2.7 7.4 9.2 6.6 15.1-2.2 14-12.2 18.8-22.4 27-3.4 2.7-8 6.6-5.9 11.6 2 4.4 7 4.5 10.7 2.8 7.4-3.3 13.4-16.5 21.7-16 14.6.7 12 21.9.9 26.2-5 1.9-10.2 2.3-15.2 3.9-5.8 1.8-9.4 8.7-15.7 8.9-6.1.1-9-6.9-14.3-9-14.4-6-33.3-2-44.7-14.7-3.7-4.2-9.6-12-4.9-17.4 9.3-10.7 28 7.2 35.7 12 2 1.1 11 6.9 11.4 1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> </span> <span class="webName">nLab</span> filtered object </h1> <div class="navigation"> <span class="skipNav"><a href='#navEnd'>Skip the Navigation Links</a> | </span> <span style="display:inline-block; width: 0.3em;"></span> <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/4408/#Item_19" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="category_theory">Category theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></strong></p> <h2 id="sidebar_concepts">Concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category">category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cat">Cat</a></p> </li> </ul> <h2 id="sidebar_universal_constructions">Universal constructions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+construction">universal construction</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/representable+functor">representable functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+functor">adjoint functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/limit">limit</a>/<a class="existingWikiWord" href="/nlab/show/colimit">colimit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/weighted+limit">weighted limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/end">end</a>/<a class="existingWikiWord" href="/nlab/show/coend">coend</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kan+extension">Kan extension</a></p> </li> </ul> </li> </ul> <h2 id="sidebar_theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+construction">Grothendieck construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+functor+theorem">adjoint functor theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monadicity+theorem">monadicity theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+lifting+theorem">adjoint lifting theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gabriel-Ulmer+duality">Gabriel-Ulmer duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/small+object+argument">small object argument</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freyd-Mitchell+embedding+theorem">Freyd-Mitchell embedding theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relation+between+type+theory+and+category+theory">relation between type theory and category theory</a></p> </li> </ul> <h2 id="sidebar_extensions">Extensions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/sheaf+and+topos+theory">sheaf and topos theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a></p> </li> </ul> <h2 id="sidebar_applications">Applications</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/applications+of+%28higher%29+category+theory">applications of (higher) category theory</a></li> </ul> <div> <p> <a href="/nlab/edit/category+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> <li><a href='#properties'>Properties</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> <div class="num_defn" id="Filtration"> <h6 id="definition_2">Definition</h6> <p>Given a <a class="existingWikiWord" href="/nlab/show/category">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>, then a <strong>filtered object</strong> is an <a class="existingWikiWord" href="/nlab/show/object">object</a> <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> 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">\mathcal{C}</annotation></semantics></math> equipped with a <em>filtration</em>:</p> <p>A <strong>descending filtration</strong> or <strong>decreasing filtrations</strong> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a sequence of <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> (often required to be <a class="existingWikiWord" href="/nlab/show/monomorphisms">monomorphisms</a>) of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>⋯</mi><mo>⟶</mo><msub><mi>X</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>⟶</mo><msub><mi>X</mi> <mi>n</mi></msub><mo>⟶</mo><msub><mi>X</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>⟶</mo><mi>⋯</mi><mo>⟶</mo><mi>X</mi></mrow><annotation encoding="application/x-tex"> \cdots \longrightarrow X_{n+1} \longrightarrow X_n \longrightarrow X_{n-1} \longrightarrow \cdots \longrightarrow X </annotation></semantics></math></div> <p>An <strong>ascending filtration</strong> or <strong>increasing filtration</strong> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>⟶</mo><mi>⋯</mi><mo>⟶</mo><msub><mi>X</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>⟶</mo><msub><mi>X</mi> <mi>n</mi></msub><mo>⟶</mo><msub><mi>X</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>⟶</mo><mi>⋯</mi></mrow><annotation encoding="application/x-tex"> X \longrightarrow \cdots \longrightarrow X_{n-1} \longrightarrow X_n \longrightarrow X_{n+1} \longrightarrow \cdots </annotation></semantics></math></div></div> <p>(In more generality, it is also possible to index using any <a class="existingWikiWord" href="/nlab/show/ordered+group">ordered abelian group</a>.)</p> <div class="num_defn" id="ExhaustiveHausdorffAndCompleteFiltrations"> <h6 id="definition_3">Definition</h6> <p>A decreasing filtration <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>X</mi> <mi>s</mi></msub><msub><mo stretchy="false">}</mo> <mi>s</mi></msub></mrow><annotation encoding="application/x-tex">\{X_s\}_s</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> (def. <a class="maruku-ref" href="#Filtration"></a>) is called</p> <ul> <li> <p><strong>exhaustive</strong> if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><munder><mi>lim</mi><mo>⟶</mo></munder> <mi>s</mi></msub><msub><mi>X</mi> <mi>s</mi></msub><mo>≃</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\underset{\longrightarrow}{\lim}_s X_s \simeq X</annotation></semantics></math> (<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> is the <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a> of the filter stages)</p> </li> <li> <p><strong>Hausdorff</strong> if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><munder><mi>lim</mi><mo>⟵</mo></munder> <mi>s</mi></msub><msub><mi>X</mi> <mi>s</mi></msub><mo>≃</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\underset{\longleftarrow}{\lim}_s X_s \simeq 0</annotation></semantics></math> (the <a class="existingWikiWord" href="/nlab/show/limit">limit</a> of the the filter stages is <a class="existingWikiWord" href="/nlab/show/initial+object">initial</a> (<a class="existingWikiWord" href="/nlab/show/zero+object">zero</a> in the case of an <a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a>))</p> </li> </ul> <p>and for a filtration of <a class="existingWikiWord" href="/nlab/show/abelian+groups">abelian groups</a>:</p> <ul> <li><strong>complete</strong> if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><munder><mi>lim</mi><mo>⟵</mo></munder> <mi>s</mi> <mn>1</mn></msubsup><msub><mi>X</mi> <mi>n</mi></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\underset{\longleftarrow}{\lim}^1_s X_n = 0</annotation></semantics></math> (also the first <a class="existingWikiWord" href="/nlab/show/derived+functor">derived</a> <a class="existingWikiWord" href="/nlab/show/limit">limit</a> (<a class="existingWikiWord" href="/nlab/show/lim%5E1">lim^1</a>) vanishes)</li> </ul> </div> <p>(<a href="#Boardman99">Boardman 99, def. 2.1</a>, see also <a href="#Rognes12">Rognes 12, section 2.1</a>)</p> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>Beware that for decreasing filtrations (def. <a class="maruku-ref" href="#Filtration"></a>) often exhaustiveness (def. <a class="maruku-ref" href="#ExhaustiveHausdorffAndCompleteFiltrations"></a>) is understood by default, and often it is even assumed by default that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>s</mi></msub><mo>=</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">X_s = X</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo>≤</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">s \leq 0</annotation></semantics></math>.</p> </div> <h2 id="properties">Properties</h2> <div class="num_prop" id="ReconstructingAbelianGroupFromExhaustiveCompleteHausdorffFiltering"> <h6 id="proposition">Proposition</h6> <p>If a decreasing filtration (def. <a class="maruku-ref" href="#Filtration"></a>) of <a class="existingWikiWord" href="/nlab/show/abelian+groups">abelian</a> <a class="existingWikiWord" href="/nlab/show/subgroups">subgroups</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>⋯</mi><mo>↪</mo><msub><mi>A</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>↪</mo><msub><mi>A</mi> <mi>n</mi></msub><mo>↪</mo><msub><mi>A</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>↪</mo><mi>⋯</mi><mo>↪</mo><mi>A</mi></mrow><annotation encoding="application/x-tex"> \cdots \hookrightarrow A_{n+1} \hookrightarrow A_n \hookrightarrow A_{n-1} \hookrightarrow \cdots \hookrightarrow A </annotation></semantics></math></div> <p>is exhaustive and complete Hausdorff (def. <a class="maruku-ref" href="#ExhaustiveHausdorffAndCompleteFiltrations"></a>) then <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> may be reobtained from the subquotients of the filtering as the <a class="existingWikiWord" href="/nlab/show/limit">limit</a>/<a class="existingWikiWord" href="/nlab/show/colimit">colimit</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mi>A</mi></mtd> <mtd><mo>≃</mo><msub><munder><mi>lim</mi><mo>⟵</mo></munder> <mi>s</mi></msub><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">/</mo><msub><mi>A</mi> <mi>s</mi></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><msub><munder><mi>lim</mi><mo>⟵</mo></munder> <mi>s</mi></msub><mo stretchy="false">(</mo><msub><munder><mi>lim</mi><mo>⟶</mo></munder> <mi>t</mi></msub><mo stretchy="false">(</mo><msub><mi>A</mi> <mi>t</mi></msub><mo stretchy="false">/</mo><msub><mi>A</mi> <mi>s</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} A & \simeq \underset{\longleftarrow}{\lim}_s (A/A_s) \\ & \simeq \underset{\longleftarrow}{\lim}_s (\underset{\longrightarrow}{\lim}_t ( A_t / A_s )) \end{aligned} \,. </annotation></semantics></math></div></div> <p>(<a href="#Boardman99">Boardman 99, prop. 2.5</a>)</p> <h2 id="related_concepts">Related concepts</h2> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/filtered+objects">filtered objects</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/filtered+topological+space">filtered topological space</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/spectral+sequence+of+a+tower+of+fibrations">spectral sequence of a tower of fibrations</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/filtered+vector+space">filtered vector space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/filtered+chain+complex">filtered chain complex</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+sequence+of+a+filtered+complex">spectral sequence of a filtered complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge+filtration">Hodge filtration</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/filtered+ring">filtered ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/filtered+object+in+an+%28%E2%88%9E%2C1%29-category">filtered object in an (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/filtered+stable+homotopy+type">filtered stable homotopy type</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/spectral+sequence+of+a+filtered+stable+homotopy+type">spectral sequence of a filtered stable homotopy type</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/filtered+probability+space">filtered probability space</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/associated+graded+objects">associated graded objects</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/associated+graded+vector+space">associated graded vector space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/associated+graded+ring">associated graded ring</a></p> </li> </ul> </div> <h2 id="references">References</h2> <ul> <li id="Boardman99"> <p><a class="existingWikiWord" href="/nlab/show/Michael+Boardman">Michael Boardman</a>, section I.2 of <em>Conditionally convergent spectral sequences</em>, 1999 (<a href="http://www.uio.no/studier/emner/matnat/math/MAT9580/v12/undervisningsmateriale/boardman-conditionally-1999.pdf">pdf</a>)</p> </li> <li id="Rognes12"> <p><a class="existingWikiWord" href="/nlab/show/John+Rognes">John Rognes</a>, <em>The Adams spectral sequence</em> (following <a href="Adams+spectral+sequence#Bruner09">Bruner 09</a>), 2012 (<a href="http://folk.uio.no/rognes/papers/notes.050612.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Paolo+Perrone">Paolo Perrone</a>, <em>Starting Category Theory</em>, World Scientific, 2024, Section 3.2.7. (<a href="https://www.worldscientific.com/worldscibooks/10.1142/13670">website</a>)</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on July 21, 2024 at 18:05:19. See the <a href="/nlab/history/filtered+object" 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/filtered+object" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/4408/#Item_19">Discuss</a><span class="backintime"><a href="/nlab/revision/filtered+object/16" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/filtered+object" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/filtered+object" accesskey="S" class="navlink" id="history" rel="nofollow">History (16 revisions)</a> <a href="/nlab/show/filtered+object/cite" style="color: black">Cite</a> <a href="/nlab/print/filtered+object" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/filtered+object" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>