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It does not yet have a dedicated thread; feel free to create one, giving it the same name as the title of this page" 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> <p>This entry is about the notion of spectrum in <a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable homotopy theory</a>. For other uses of the term ‘’spectrum’‘ see <a class="existingWikiWord" href="/nlab/show/spectrum+-+disambiguation">spectrum - disambiguation</a>.</p> <hr /> <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='#idea'>Idea</a></li> <ul> <li><a href='#connective_and_nonconnective_spectra_infinite_loop_spaces'>Connective and non-connective spectra; infinite loop spaces</a></li> </ul> <li><a href='#definition'>Definition</a></li> <ul> <li><a href='#SequentialPreSpectra'>Sequential pre-spectra</a></li> <li><a href='#OmegaSpectrum'><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi></mrow><annotation encoding="application/x-tex">\Omega</annotation></semantics></math>-spectra</a></li> <li><a href='#coordinatefree_spectra'>Coordinate-free spectra</a></li> <li><a href='#combinatorial_spectra'>Combinatorial spectra</a></li> <li><a href='#general_context'>General context</a></li> </ul> <li><a href='#examples'>Examples</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#stabilization'>Stabilization</a></li> <li><a href='#symmetric_monoidal_structure'>Symmetric monoidal structure</a></li> <li><a href='#closed_structure'>Closed structure</a></li> <li><a href='#model_category_structure'>Model category structure</a></li> <li><a href='#relation_to_symmetric_monoidal_groupoids'>Relation to symmetric monoidal ∞-groupoids</a></li> </ul> <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 <a class="existingWikiWord" href="/nlab/show/topological+space">topological</a> spectrum is an <a class="existingWikiWord" href="/nlab/show/object">object</a> in the universal <a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sp</mi><mo stretchy="false">(</mo><mi>Top</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>Sp</mi><mo stretchy="false">(</mo><mn>∞</mn><mi>Grpd</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sp(Top) \simeq Sp(\infty Grpd)</annotation></semantics></math> that stabilizes the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> <a class="existingWikiWord" href="/nlab/show/Top">Top</a> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≃</mo></mrow><annotation encoding="application/x-tex">\simeq</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Grpd">∞-Grpd</a> of <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a> or <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoids">∞-groupoids</a> under the operations of forming <a class="existingWikiWord" href="/nlab/show/loop+space+objects">loop space objects</a> and <a class="existingWikiWord" href="/nlab/show/reduced+suspensions">reduced suspensions</a>: the <a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category+of+spectra">stable (∞,1)-category of spectra</a>.</p> <p>More generally, one may consider <a class="existingWikiWord" href="/nlab/show/spectrum+objects">spectrum objects</a> in any <a class="existingWikiWord" href="/nlab/show/presentable+%28%E2%88%9E%2C1%29-category">presentable (∞,1)-category</a>.</p> <h3 id="connective_and_nonconnective_spectra_infinite_loop_spaces">Connective and non-connective spectra; infinite loop spaces</h3> <p>As opposed to the <a class="existingWikiWord" href="/nlab/show/homotopy+groups">homotopy groups</a> of a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>, the <a class="existingWikiWord" href="/nlab/show/homotopy+groups+of+a+spectrum">homotopy groups of a spectrum</a> are defined and may be non-trivial in <em>negative</em> integer degree. This follows from the fact that the <a class="existingWikiWord" href="/nlab/show/loop+space">loop space</a> operation is by construction invertible on spectra, which implies that for every spectrum <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> these and all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>, the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-fold looping <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\Omega^n</annotation></semantics></math> has <a class="existingWikiWord" href="/nlab/show/stable+homotopy+groups">stable homotopy groups</a> given by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mrow><mi>k</mi><mo>−</mo><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><msup><mi>Ω</mi> <mi>n</mi></msup><mi>E</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>π</mi> <mi>k</mi></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_{k-n}(\Omega^n E) \simeq \pi_k(E)</annotation></semantics></math>.</p> <p>Those spectra for which the <a class="existingWikiWord" href="/nlab/show/homotopy+groups+of+spectra">homotopy groups of spectra</a> in negative degree happen to vanish are called <em><a class="existingWikiWord" href="/nlab/show/connective+spectra">connective spectra</a></em>. They are equivalent to <a class="existingWikiWord" href="/nlab/show/infinite+loop+spaces">infinite loop spaces</a>, i.e. grouplike <a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+spaces">E-∞ spaces</a>.</p> <p>Connective spectra in the image of the <a class="existingWikiWord" href="/nlab/show/nerve">nerve</a> operation of the classical <a class="existingWikiWord" href="/nlab/show/Dold-Kan+correspondence">Dold-Kan correspondence</a>: this identifies <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoids">∞-groupoids</a> that are not only connective spectra but even have a <em>strict</em> symmetric monoidal <a class="existingWikiWord" href="/nlab/show/group">group</a> structure with non-negatively graded <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complexes</a> of abelian groups.</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><msub><mi>Ch</mi> <mo>+</mo></msub></mtd> <mtd><mover><mo>→</mo><mrow><mi>Dold</mi><mo>−</mo><mi>Kan</mi><mspace width="thickmathspace"></mspace><mi>nerve</mi></mrow></mover></mtd> <mtd><mi>ConnectSp</mi><mo stretchy="false">(</mo><mn>∞</mn><mi>Grp</mi><mo stretchy="false">)</mo><mo>⊂</mo><mn>∞</mn><mi>Grpd</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">(</mo><mi>⋯</mi><msub><mi>A</mi> <mn>2</mn></msub><mover><mo>→</mo><mo>∂</mo></mover><msub><mi>A</mi> <mn>1</mn></msub><mover><mo>→</mo><mo>∂</mo></mover><msub><mi>A</mi> <mn>0</mn></msub><mo>→</mo><mn>0</mn><mo>→</mo><mn>0</mn><mo>→</mo><mi>⋯</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>↦</mo><mrow></mrow></mover></mtd> <mtd><mi>N</mi><mo stretchy="false">(</mo><msub><mi>A</mi> <mo>•</mo></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ Ch_+ &\stackrel{Dold-Kan \; nerve}{\to}& ConnectSp(\infty Grp) \subset \infty Grpd \\ (\cdots A_2 \stackrel{\partial}{\to} A_1 \stackrel{\partial}{\to} A_0 \to 0 \to 0 \to \cdots) &\stackrel{}{\mapsto}& N(A_\bullet) } </annotation></semantics></math></div> <p>The <a class="existingWikiWord" href="/nlab/show/homology">homology</a> groups of the chain complex correspond precisely to the <a class="existingWikiWord" href="/nlab/show/homotopy+groups">homotopy groups</a> of the corresponding <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> or <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a>.</p> <p>The free stabilization of the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> of non-negatively graded chain complexes is simply the <a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a> of arbitrary chain complexes. There is a <a class="existingWikiWord" href="/nlab/show/stable+Dold-Kan+correspondence">stable Dold-Kan correspondence</a> (see at <em><a class="existingWikiWord" href="/nlab/show/module+spectrum">module spectrum</a></em> the section <em><a href="module+spectrum#StableDoldKanCorrespondence">stable Dold-Kan correspondence</a></em> ) that identifies these with special objects in <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>.</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>Ch</mi></mtd> <mtd><mover><mo>→</mo><mrow><mi>Dold</mi><mo>−</mo><mi>Kan</mi><mspace width="thickmathspace"></mspace><mi>nerve</mi></mrow></mover></mtd> <mtd><mi>Sp</mi><mo stretchy="false">(</mo><mn>∞</mn><mi>Grp</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">(</mo><mi>⋯</mi><msub><mi>A</mi> <mn>2</mn></msub><mover><mo>→</mo><mo>∂</mo></mover><msub><mi>A</mi> <mn>1</mn></msub><mover><mo>→</mo><mo>∂</mo></mover><msub><mi>A</mi> <mn>0</mn></msub><mover><mo>→</mo><mo>∂</mo></mover><msub><mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msub><mover><mo>→</mo><mo>∂</mo></mover><msub><mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>2</mn></mrow></msub><mo>→</mo><mi>⋯</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>↦</mo><mrow></mrow></mover></mtd> <mtd><mi>N</mi><mo stretchy="false">(</mo><msub><mi>A</mi> <mo>•</mo></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ Ch &\stackrel{Dold-Kan \; nerve}{\to}& Sp(\infty Grp) \\ (\cdots A_2 \stackrel{\partial}{\to} A_1 \stackrel{\partial}{\to} A_0 \stackrel{\partial}{\to} A_{-1} \stackrel{\partial}{\to} A_{-2} \to \cdots) &\stackrel{}{\mapsto}& N(A_\bullet) } </annotation></semantics></math></div> <p>So it is the homologically nontrivial parts of the chain complexes in negative degree that corresponds to the non-connectiveness of a spectrum.</p> <h2 id="definition">Definition</h2> <p>There are many “models” for spectra, all of which present the same (<a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable</a>) <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> (and in fact, nearly all of them are <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalent</a> <a class="existingWikiWord" href="/nlab/show/model+category">model categories</a>). For more details see at <em><a class="existingWikiWord" href="/nlab/show/Introduction+to+Stable+Homotopy+Theory">Introduction to Stable Homotopy Theory</a></em>.</p> <h3 id="SequentialPreSpectra">Sequential pre-spectra</h3> <p>A simple first definition is to define a <a class="existingWikiWord" href="/nlab/show/pre-spectrum">pre-spectrum</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>E</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{E}</annotation></semantics></math> to be a sequence of pointed spaces <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>E</mi> <mi>n</mi></msub><msub><mo stretchy="false">)</mo> <mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow></msub></mrow><annotation encoding="application/x-tex">(E_n)_{n\in\mathbb{N}}</annotation></semantics></math> together with structure maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mrow></mrow><msub><mi>E</mi> <mi>n</mi></msub><mo>→</mo><mrow></mrow><msub><mi>E</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">\Sigma{}E_n\to{}E_{n+1}</annotation></semantics></math> (where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> denotes the <a class="existingWikiWord" href="/nlab/show/reduced+suspension">reduced suspension</a>). See at <em><a class="existingWikiWord" href="/nlab/show/model+structure+on+sequential+spectra">model structure on sequential spectra</a></em>.</p> <p>There are various conditions that can be put on the spaces <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">E_n</annotation></semantics></math> and the structure maps, for example if the spaces are CW-complexes and the structure maps are inclusions of subcomplexes, the spectrum is called a <strong><a class="existingWikiWord" href="/nlab/show/CW-spectrum">CW-spectrum</a></strong>.</p> <p>Without any condition, this is just called a <strong>spectrum</strong>, or sometimes a <strong><a class="existingWikiWord" href="/nlab/show/pre-spectrum">pre-spectrum</a></strong>. In order to distinguish from various other richer definitions (such as <a class="existingWikiWord" href="/nlab/show/coordinate-free+spectra">coordinate-free spectra</a>, one also speaks of <em><a class="existingWikiWord" href="/nlab/show/sequential+spectra">sequential spectra</a></em>).</p> <p>For details see <em><a class="existingWikiWord" href="/nlab/show/Introduction+to+Stable+homotopy+theory+--+1-1">Introduction to stable homotopy theory – 1.1 Sequential Spectra</a></em>.</p> <h3 id="OmegaSpectrum"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi></mrow><annotation encoding="application/x-tex">\Omega</annotation></semantics></math>-spectra</h3> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi></mrow><annotation encoding="application/x-tex">\Omega</annotation></semantics></math> denotes the <a class="existingWikiWord" href="/nlab/show/loop+space">loop space</a> functor on the category of pointed spaces, we know that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> is left adjoint to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi></mrow><annotation encoding="application/x-tex">\Omega</annotation></semantics></math>. In particular, given a spectrum <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>E</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{E}</annotation></semantics></math>, the structure maps can be transformed into maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mi>n</mi></msub><mo>→</mo><mi>Ω</mi><mrow></mrow><msub><mi>E</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">E_n\to\Omega{}E_{n+1}</annotation></semantics></math>. If these maps are isomorphisms (depending on the situation it can be weak equivalences or homeomorphisms), then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>E</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{E}</annotation></semantics></math> is called an <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi></mrow><annotation encoding="application/x-tex">\Omega</annotation></semantics></math>-spectrum</strong>.</p> <p>The idea is that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">E_0</annotation></semantics></math> contains the information of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>E</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{E}</annotation></semantics></math> in dimensions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>≥</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">k\ge 0</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">E_1</annotation></semantics></math> contains the information of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>E</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{E}</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>≥</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">k\ge -1</annotation></semantics></math> (but shifted up by one, so that it is modeled by the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≥</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\ge 0</annotation></semantics></math> information in the space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">E_1</annotation></semantics></math>), and so on.</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi></mrow><annotation encoding="application/x-tex">\Omega</annotation></semantics></math>-spectra are special cases of <a class="existingWikiWord" href="/nlab/show/sequential+spectrum">sequential</a> pre-spectra as <a href="#SequentialPreSpectra">above</a>, and are in fact the <a class="existingWikiWord" href="/nlab/show/fibrant+objects">fibrant objects</a> for some <a class="existingWikiWord" href="/nlab/show/model+structure+on+spectra">model structure on spectra</a>.</p> <p>Given any sequential pre-spectrum <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>E</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{E}</annotation></semantics></math>, it induces an equivalent <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi></mrow><annotation encoding="application/x-tex">\Omega</annotation></semantics></math>-spectrum <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>F</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{F}</annotation></semantics></math> (a fibrant replacement of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>E</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{E}</annotation></semantics></math>, its <em><a class="existingWikiWord" href="/nlab/show/spectrification">spectrification</a></em>) given by (<a href="#LewisMaySteinberger86">Lewis-May-Steinberger 86, p. 3</a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mi>n</mi></msub><mo>≔</mo><munder><mi>lim</mi> <mrow><mi>m</mi><mo>→</mo><mn>∞</mn></mrow></munder><msup><mi>Ω</mi> <mi>m</mi></msup><msub><mi>E</mi> <mrow><mi>n</mi><mo>+</mo><mi>m</mi></mrow></msub></mrow><annotation encoding="application/x-tex"> F_n \coloneqq \lim_{m\to\infty}\Omega^m E_{n+m} </annotation></semantics></math></div> <p>(using that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi></mrow><annotation encoding="application/x-tex">\Omega</annotation></semantics></math> commutes with the <a class="existingWikiWord" href="/nlab/show/filtered+colimits">filtered colimits</a>).</p> <h3 id="coordinatefree_spectra">Coordinate-free spectra</h3> <p>A definition of spectrum consisting of spaces indexed by index sets less “coordinatized” than the integers is a</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/coordinate-free+spectrum">coordinate-free spectrum</a>.</li> </ul> <p>See there for details.</p> <h3 id="combinatorial_spectra">Combinatorial spectra</h3> <p>There might be a type of categorical structure related to a spectrum in the same way that <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 related to <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. In other words, it would contain <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>-cells for all integers <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>, not necessarily invertible. Some people have called this conjectural object a <strong><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">Z</annotation></semantics></math>-category</strong>. “Connective” <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">Z</annotation></semantics></math>-categories could perhaps then be identified with stably monoidal <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> <p>One realization of this kind of idea is the notion of <a class="existingWikiWord" href="/nlab/show/combinatorial+spectrum">combinatorial spectrum</a>.</p> <h3 id="general_context">General context</h3> <p>See <a class="existingWikiWord" href="/nlab/show/spectrum+object">spectrum object</a>.</p> <h2 id="examples">Examples</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/suspension+spectrum">suspension spectrum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sphere+spectrum">sphere spectrum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Eilenberg-MacLane+spectrum">Eilenberg-MacLane spectrum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K-theory+spectrum">K-theory spectrum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/elliptic+spectrum">elliptic spectrum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tmf">tmf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Thom+spectrum">Thom spectrum</a>, <a class="existingWikiWord" href="/nlab/show/complex+cobordism+spectrum">complex cobordism spectrum</a></p> </li> </ul> <h2 id="properties">Properties</h2> <h3 id="stabilization">Stabilization</h3> <p>In direct analogy to how topological spaces form the archetypical example, <a class="existingWikiWord" href="/nlab/show/Top">Top</a>, of an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a>, spectra form the archetypical example <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> of a <a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a>. In fact, there is a general procedure for turning any <a class="existingWikiWord" href="/nlab/show/pointed+category">pointed</a> <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> into 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>, and doing this to the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">Top_*</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/pointed+object">pointed</a> spaces yields <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>.</p> <h3 id="symmetric_monoidal_structure">Symmetric monoidal structure</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/smash+product+of+spectra">smash product of spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+smash+product+of+spectra">symmetric monoidal smash product of spectra</a></p> </li> </ul> <h3 id="closed_structure">Closed structure</h3> <ul> <li><a class="existingWikiWord" href="/nlab/show/mapping+spectrum">mapping spectrum</a></li> </ul> <h3 id="model_category_structure">Model category structure</h3> <ul> <li><a class="existingWikiWord" href="/nlab/show/model+structure+on+spectra">model structure on spectra</a></li> </ul> <h3 id="relation_to_symmetric_monoidal_groupoids">Relation to symmetric monoidal ∞-groupoids</h3> <ul> <li><a class="existingWikiWord" href="/nlab/show/stable+homotopy+hypothesis">stable homotopy hypothesis</a></li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+group+of+a+spectrum">homotopy group of a spectrum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sheaf+of+spectra">sheaf of spectra</a>, <a class="existingWikiWord" href="/nlab/show/parametrized+spectrum">parametrized spectrum</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/smooth+spectrum">smooth spectrum</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectrum+with+G-action">spectrum with G-action</a>, <a class="existingWikiWord" href="/nlab/show/G-spectrum">G-spectrum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ring+spectrum">ring spectrum</a>, <a class="existingWikiWord" href="/nlab/show/algebra+spectrum">algebra spectrum</a>, <a class="existingWikiWord" href="/nlab/show/module+spectrum">module spectrum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/zero+spectrum">zero spectrum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/finite+spectrum">finite spectrum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/p-local+spectrum">p-local spectrum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bousfield+localization+of+spectra">Bousfield localization of spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/motivic+spectrum">motivic spectrum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brown+representability+theorem">Brown representability theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+homotopy+hypothesis">stable homotopy hypothesis</a></p> </li> </ul> <div> <table><thead><tr><th></th><th><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-operad">(∞,1)-operad</a></th><th><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-algebra+over+an+%28%E2%88%9E%2C1%29-operad">∞-algebra</a></th><th>grouplike version</th><th>in <a class="existingWikiWord" href="/nlab/show/Top">Top</a></th><th>generally</th></tr></thead><tbody><tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+operad">A-∞ operad</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+algebra">A-∞ algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+space">A-∞ space</a>, e.g. <a class="existingWikiWord" href="/nlab/show/loop+space">loop space</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/loop+space+object">loop space object</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/E-k+operad">E-k operad</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/E-k+algebra">E-k algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/k-monoidal+%E2%88%9E-group">k-monoidal ∞-group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/iterated+loop+space">iterated loop space</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/iterated+loop+space+object">iterated loop space object</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+operad">E-∞ operad</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+algebra">E-∞ algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/abelian+%E2%88%9E-group">abelian ∞-group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+space">E-∞ space</a>, if grouplike: <a class="existingWikiWord" href="/nlab/show/infinite+loop+space">infinite loop space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≃</mo></mrow><annotation encoding="application/x-tex">\simeq</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-space">∞-space</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/infinite+loop+space+object">infinite loop space object</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≃</mo></mrow><annotation encoding="application/x-tex">\simeq</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/connective+spectrum">connective spectrum</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≃</mo></mrow><annotation encoding="application/x-tex">\simeq</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/connective+spectrum+object">connective spectrum object</a></td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/stabilization">stabilization</a></strong></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/spectrum">spectrum</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/spectrum+object">spectrum object</a></td></tr> </tbody></table> <ul> <li><a class="existingWikiWord" href="/nlab/show/looping+and+delooping">looping and delooping</a>, <a class="existingWikiWord" href="/nlab/show/stabilization+hypothesis">stabilization hypothesis</a></li> </ul> </div> <h2 id="references">References</h2> <p>According to (<a href="#Adams74">Adams 74, p. 131</a>) the notion of spectrum is due to</p> <ul> <li id="Lima58">E. L. Lima, <em>Duality and Postnikov invariants</em>, Thesis, University of Chicago, Chicago 1958</li> </ul> <blockquote> <p>It is generally supposed that <a class="existingWikiWord" href="/nlab/show/George+Whitehead">G. W. Whitehead</a> also had something to do with it, but the latter takes a modest attitude about that. (<a href="#Adams74">Adams 74, p. 131</a>)</p> </blockquote> <p>Early notes include</p> <ul> <li id="Boardman65"> <p><a class="existingWikiWord" href="/nlab/show/Michael+Boardman">Michael Boardman</a>, <em>Stable homotopy theory</em>, mimeographed notes, University of Warwick, 1965 onward</p> </li> <li id="Adams74"> <p><a class="existingWikiWord" href="/nlab/show/Frank+Adams">Frank Adams</a>, Part III, section 2 <em><a class="existingWikiWord" href="/nlab/show/Stable+homotopy+and+generalised+homology">Stable homotopy and generalised homology</a></em>, 1974</p> </li> </ul> <p>See the references at <em><a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable homotopy theory</a></em>.</p> <p>Lecture notes include</p> <ul> <li><em><a class="existingWikiWord" href="/nlab/show/Introduction+to+Stable+homotopy+theory+--+1-1">Introduction to stable homotopy theory – 1.1 Sequential Spectra</a></em></li> </ul> <p>More modern developments are due to</p> <ul> <li id="LewisMaySteinberger86"><a class="existingWikiWord" href="/nlab/show/L.+Gaunce+Lewis">L. Gaunce Lewis</a>, <a class="existingWikiWord" href="/nlab/show/Peter+May">Peter May</a>, M. Steinberger, <em>Equivariant stable homotopy theory</em>, Springer Lecture Notes in Mathematics, 1986 (<a href="http://www.math.uchicago.edu/~may/BOOKS/equi.pdf">pdf</a>)</li> </ul> <p>The quick idea is surveyed for instance in</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Cary+Malkiewich">Cary Malkiewich</a>, <em>The stable homotopy category</em>, 2014 (<a href="http://math.stanford.edu/~carym/stable.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Aaron+Mazel-Gee">Aaron Mazel-Gee</a>, <em>An introduction to spectra</em> (<a href="https://math.berkeley.edu/~aaron/writing/an-introduction-to-spectra.pdf">pdf</a>)</p> </li> </ul> <p>The first review of stable homotopy theory with <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+smash+product+of+spectra">symmetric monoidal smash product of spectra</a> is (in terms of <a class="existingWikiWord" href="/nlab/show/S-modules">S-modules</a>) in</p> <ul> <li id="ElmendorfKrizMay95"><a class="existingWikiWord" href="/nlab/show/Anthony+Elmendorf">Anthony Elmendorf</a>, <a class="existingWikiWord" href="/nlab/show/Igor+Kriz">Igor Kriz</a>, <a class="existingWikiWord" href="/nlab/show/Peter+May">Peter May</a>, <em><a class="existingWikiWord" href="/nlab/show/Modern+foundations+for+stable+homotopy+theory">Modern foundations for stable homotopy theory</a></em>, in <a class="existingWikiWord" href="/nlab/show/Ioan+Mackenzie+James">Ioan Mackenzie James</a>, <em><a class="existingWikiWord" href="/nlab/show/Handbook+of+Algebraic+Topology">Handbook of Algebraic Topology</a></em>, Amsterdam: North-Holland, (1995) pp. 213–253, (<a href="http://hopf.math.purdue.edu/Elmendorf-Kriz-May/modern_foundations.pdf">pdf</a>)</li> </ul> <p>A comprehensive account of the symmetric model in terms of <a class="existingWikiWord" href="/nlab/show/symmetric+spectra">symmetric spectra</a> is in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Stefan+Schwede">Stefan Schwede</a>, <em>Symmetric spectra</em> (<a href="http://www.math.uni-bonn.de/~schwede/SymSpec-v3.pdf">pdf</a>)</li> </ul> <p>and in terms of <a class="existingWikiWord" href="/nlab/show/orthogonal+spectra">orthogonal spectra</a> in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Stefan+Schwede">Stefan Schwede</a>, <em><a class="existingWikiWord" href="/nlab/show/Global+homotopy+theory">Global homotopy theory</a></em> (take <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><mn>1</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\mathcal{F} = \{1\}</annotation></semantics></math>, on p. 4, to be the trivial collection of groups, in order to specialize from <a class="existingWikiWord" href="/nlab/show/global+equivariant+stable+homotopy+theory">global equivariant stable homotopy theory</a> to plain stable homotopy theory).</li> </ul> <p>See also</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Robert+Thomason">Robert Thomason</a>, <em>Symmetric Monoidal Categories Model All Connective Spectra</em> (<a href="http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.37.8193">web</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Frank+Adams">Frank Adams</a>, <em>Infinite loop spaces</em>, Princeton University Press, 1978</p> </li> <li id="Ayoub"> <p><a class="existingWikiWord" href="/nlab/show/Joseph+Ayoub">Joseph Ayoub</a>, <em>Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique, I</em>. Astérisque, Vol. 314 (2008). Société Mathématique de France. (<a href="http://user.math.uzh.ch/ayoub/PDF-Files/THESE.PDF">pdf</a>)</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on November 4, 2024 at 01:12:16. See the <a href="/nlab/history/spectrum" 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/spectrum" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussions/?CategoryID=0">Discuss</a><span class="backintime"><a href="/nlab/revision/spectrum/69" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/spectrum" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/spectrum" accesskey="S" class="navlink" id="history" rel="nofollow">History (69 revisions)</a> <a href="/nlab/show/spectrum/cite" style="color: black">Cite</a> <a href="/nlab/print/spectrum" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/spectrum" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>