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href="/nlab/show/Introduction+to+Stable+Homotopy+Theory">Introduction</a></em></p> <h1 id="ingredients">Ingredients</h1> <ul> <li><a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a></li> </ul> <h1 id="contents">Contents</h1> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/loop+space+object">loop space object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/suspension+object">suspension object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/looping+and+delooping">looping and delooping</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/stabilization">stabilization</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/spectrum+object">spectrum object</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+derivator">stable derivator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/triangulated+category">triangulated category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category+of+spectra">stable (∞,1)-category of spectra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/spectrum">spectrum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+homotopy+category">stable homotopy category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smash+product+of+spectra">smash product of spectra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+smash+product+of+spectra">symmetric smash product of spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Spanier-Whitehead+duality">Spanier-Whitehead duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+ring">A-∞ ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+ring">E-∞ ring</a></p> </li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/stable+homotopy+theory+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <ul> <li><a href='#via_spectrum_objects'>Via <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 objects</a></li> <li><a href='#ViaExcisiveFunctors'>Via excisive functors</a></li> <li><a href='#in_an_ordinary_category'>In an ordinary category</a></li> </ul> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#as_a_model_for_stabilization'>As a model for stabilization</a></li> <li><a href='#reflection_into_prespectrum_objects'>Reflection into pre-spectrum objects</a></li> </ul> <li><a href='#examples'>Examples</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>Every <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">\mathcal{C}</annotation></semantics></math> with <a class="existingWikiWord" href="/nlab/show/finite+%28%E2%88%9E%2C1%29-limits">finite (∞,1)-limits</a> has a <a class="existingWikiWord" href="/nlab/show/stabilization">stabilization</a> to 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>Stab</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Stab(\mathcal{C})</annotation></semantics></math>. This stabilization may be defined by abstract properties, but it may also be constructed explicitly as the category of <em>spectrum objects</em> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>.</p> <p>In the special case that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">\mathcal{C} = </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≃</mo><msub><mi>L</mi> <mi>whe</mi></msub></mrow><annotation encoding="application/x-tex">\simeq L_{whe}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Top">Top</a>, a spectrum object in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/spectrum">spectrum</a> in the traditional sense. There is an evident generalization of the traditional notion of <a class="existingWikiWord" href="/nlab/show/Omega-spectrum">Omega-spectrum</a> from <a class="existingWikiWord" href="/nlab/show/Top">Top</a> to any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> with <a class="existingWikiWord" href="/nlab/show/finite+%28%E2%88%9E%2C1%29-limits">finite (∞,1)-limits</a>: a spectrum object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">X_\bullet</annotation></semantics></math> is essentially a list of <a class="existingWikiWord" href="/nlab/show/pointed+objects">pointed objects</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">X_i</annotation></semantics></math> together with <a class="existingWikiWord" href="/nlab/show/equivalence+in+an+%28%E2%88%9E%2C1%29-category">equivalences</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>i</mi></msub><mo>→</mo><mi>Ω</mi><msub><mi>X</mi> <mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">X_i \to \Omega X_{i+1}</annotation></semantics></math>, from every object in the list to the <a class="existingWikiWord" href="/nlab/show/loop+space+object">loop space object</a> of its successor.</p> <h2 id="definition">Definition</h2> <h3 id="via_spectrum_objects">Via <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 objects</h3> <div class="num_defn"> <h6 id="definition_2">Definition</h6> <p>For <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> an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a>, a <strong>prespectrum object</strong> 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> is</p> <ul> <li> <p>a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>:</mo><mi>ℤ</mi><mo>×</mo><mi>ℤ</mi><mo>→</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">X : \mathbb{Z} \times \mathbb{Z} \to \mathcal{C}</annotation></semantics></math></p> </li> <li> <p>such that for all integers <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>≠</mo><mi>j</mi></mrow><annotation encoding="application/x-tex">i \neq j</annotation></semantics></math> we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">X(i,j) = 0</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/zero+object">zero object</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math></p> </li> </ul> <p>Notice that this definition is highly redundant. The point is that writing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>≔</mo><mi>X</mi><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X[n] \coloneqq X(n,n)</annotation></semantics></math> a spectrum object is for all <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> a (homotopy) <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commuting diagram</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mn>0</mn></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>X</mi><mo stretchy="false">[</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">]</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ X[n] &\to& 0 \\ \downarrow && \downarrow \\ 0 &\to& X[n+1] } \,. </annotation></semantics></math></div> <p>Recalling that in an <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-category">(infinity,1)-category</a> with <a class="existingWikiWord" href="/nlab/show/zero+object">zero object</a></p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><mi>X</mi><mo stretchy="false">[</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\Omega X[n+1]</annotation></semantics></math> denotes the pullback of such a diagram;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mi>X</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\Sigma X[n]</annotation></semantics></math> denotes the pushout of such a diagram</p> </li> </ul> <p>this induces maps</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>α</mi> <mi>n</mi></msub><mo>:</mo><mi>Σ</mi><mi>X</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>→</mo><mi>X</mi><mo stretchy="false">[</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> \alpha_n : \Sigma X[n] \to X[n+1] </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>β</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>:</mo><mi>X</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>→</mo><mi>Ω</mi><mi>X</mi><mo stretchy="false">[</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \beta_{n+1} : X[n] \to \Omega X[n+1] \,. </annotation></semantics></math></div> <p>A prespectrum object is</p> <ul> <li> <p>a <strong>spectrum object</strong> if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>β</mi> <mi>m</mi></msub></mrow><annotation encoding="application/x-tex">\beta_m</annotation></semantics></math> is an equivalence for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi><mo>∈</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">m \in \mathbb{Z}</annotation></semantics></math> (a <strong>spectrum below <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></strong>, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>β</mi> <mi>m</mi></msub></mrow><annotation encoding="application/x-tex">\beta_m</annotation></semantics></math> is an equivalence for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi><mo>≤</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">m \leq n</annotation></semantics></math>);</p> </li> <li> <p>a <strong>suspension spectrum</strong> if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>α</mi> <mi>m</mi></msub></mrow><annotation encoding="application/x-tex">\alpha_m</annotation></semantics></math> is an equivalence for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi><mo>∈</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">m \in \mathbb{Z}</annotation></semantics></math> (a <strong>suspension spectrum above <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></strong>, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>α</mi> <mi>m</mi></msub></mrow><annotation encoding="application/x-tex">\alpha_m</annotation></semantics></math> is an equivalence for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi><mo>≥</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">m \geq n</annotation></semantics></math>).</p> </li> </ul> </div> <p>(<a href="#StabCat">StabCat</a>)</p> <p>One writes</p> <ul> <li> <p><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(\mathcal{C})</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/full+sub-%28%E2%88%9E%2C1%29-category">full sub-(∞,1)-category</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Fun</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo>×</mo><mi>ℤ</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Fun(\mathbb{Z} \times \mathbb{Z},C)</annotation></semantics></math> on spectrum objects in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Stab</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo><mo>≔</mo><mi>Sp</mi><mo stretchy="false">(</mo><msub><mi>𝒞</mi> <mo>*</mo></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Stab(\mathcal{C}) \coloneqq Sp(\mathcal{C}_*)</annotation></semantics></math> – the <strong>stabilization of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math></strong> for the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-category of spectrum objects in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">C_*</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/pointed+object">pointed object</a>s of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>.</p> </li> </ul> <h3 id="ViaExcisiveFunctors">Via excisive functors</h3> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn><msub><mi>Grpd</mi> <mi>fin</mi></msub></mrow><annotation encoding="application/x-tex">\infty Grpd_{fin}</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> of <a class="existingWikiWord" href="/nlab/show/finite+homotopy+types">finite homotopy types</a>, hence those freely generated by <a class="existingWikiWord" href="/nlab/show/finite+%28%E2%88%9E%2C1%29-colimits">finite (∞,1)-colimits</a> from the point. Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn><msubsup><mi>Grpd</mi> <mi>fin</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">\infty Grpd_{fin}^{\ast/}</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/pointed+object">pointed</a> finite homotopy types.</p> <div class="num_defn"> <h6 id="definition_3">Definition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> be an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> with <a class="existingWikiWord" href="/nlab/show/finite+%28%E2%88%9E%2C1%29-limits">finite (∞,1)-limits</a>. Then a <strong>spectrum object</strong> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> is a reduced (i.e. <a class="existingWikiWord" href="/nlab/show/terminal+object+in+an+%28infinity%2C1%29-category">terminal object</a>-preserving) <a class="existingWikiWord" href="/nlab/show/excisive+%28%E2%88%9E%2C1%29-functor">excisive (∞,1)-functor</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><mn>∞</mn><msubsup><mi>Grpd</mi> <mi>fin</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup><mo>⟶</mo><mi>𝒞</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \infty Grpd_{fin}^{\ast/} \longrightarrow \mathcal{C} \,. </annotation></semantics></math></div></div> <p>(<a href="#HigherAlg">HigherAlg, def. 1.4.2.8 and around p. 823</a>).</p> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>This generalizes for instance to <a class="existingWikiWord" href="/nlab/show/G-spectra">G-spectra</a> (<a href="#Blumberg05">Blumberg 05</a>).</p> </div> <h3 id="in_an_ordinary_category">In an ordinary category</h3> <p>One can define <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Φ</mi></mrow><annotation encoding="application/x-tex">\Phi</annotation></semantics></math>-symmetric <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>-spectra in a 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>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Φ</mi></mrow><annotation encoding="application/x-tex">\Phi</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/graded+monoid">graded monoid</a> in the <a class="existingWikiWord" href="/nlab/show/category">category</a> of <a class="existingWikiWord" href="/nlab/show/groups">groups</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">F : C \to C</annotation></semantics></math> is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Φ</mi></mrow><annotation encoding="application/x-tex">\Phi</annotation></semantics></math>-symmetric <a class="existingWikiWord" href="/nlab/show/endofunctor">endofunctor</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>. Here we follow <a href="#Ayoub">Ayoub</a>.</p> <p>(One recovers the classical case described at <a class="existingWikiWord" href="/nlab/show/spectrum">spectrum</a> by taking <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> to be the <a class="existingWikiWord" href="/nlab/show/category">category</a> of <a class="existingWikiWord" href="/nlab/show/pointed+spaces">pointed spaces</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Φ</mi></mrow><annotation encoding="application/x-tex">\Phi</annotation></semantics></math> to be the trivial <a class="existingWikiWord" href="/nlab/show/graded+monoid">graded monoid</a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> to be the <a class="existingWikiWord" href="/nlab/show/suspension+functor">suspension functor</a>.)</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Φ</mi></mrow><annotation encoding="application/x-tex">\Phi</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/graded+monoid">graded monoid</a> in the category of <a class="existingWikiWord" href="/nlab/show/groups">groups</a>. Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Seq</mi><mo stretchy="false">(</mo><mi>Φ</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Seq(\Phi, C)</annotation></semantics></math> for the category of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Φ</mi></mrow><annotation encoding="application/x-tex">\Phi</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/symmetric+sequences">symmetric sequences</a>. Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">F : C \to C</annotation></semantics></math> be a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Φ</mi></mrow><annotation encoding="application/x-tex">\Phi</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/symmetric+endofunctor">symmetric endofunctor</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>. (Usually <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> will be the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><msub><mo>⊗</mo> <mi>C</mi></msub><mo>−</mo></mrow><annotation encoding="application/x-tex">T \otimes_C -</annotation></semantics></math> induced by <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a> with some object <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>.)</p> <p>A <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Φ</mi></mrow><annotation encoding="application/x-tex">\Phi</annotation></semantics></math>-symmetric <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>-spectrum</strong> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Φ</mi></mrow><annotation encoding="application/x-tex">\Phi</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/symmetric+sequence">symmetric sequence</a> <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>n</mi></msub><msub><mo stretchy="false">)</mo> <mrow><mi>n</mi><mo>∈</mo><mstyle mathvariant="bold"><mi>N</mi></mstyle></mrow></msub></mrow><annotation encoding="application/x-tex">(X_n)_{n \in \mathbf{N}}</annotation></semantics></math> together with <strong>assembly morphisms</strong></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>γ</mi> <mi>n</mi></msub><mo>:</mo><mi>F</mi><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>→</mo><msub><mi>X</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex"> \gamma_n : F(X_n) \to X_{n+1} </annotation></semantics></math></div> <p>such that the composite morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>F</mi> <mi>m</mi></msup><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>→</mo><msup><mi>F</mi> <mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><msub><mi>X</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mo>→</mo><mi>⋯</mi><mo>→</mo><msub><mi>X</mi> <mrow><mi>m</mi><mo>+</mo><mi>n</mi></mrow></msub></mrow><annotation encoding="application/x-tex"> F^m(X_n) \to F^{m-1}(X_{n+1}) \to \cdots \to X_{m+n} </annotation></semantics></math></div> <p>is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>Φ</mi> <mi>m</mi></msub><mo>×</mo><msub><mi>Φ</mi> <mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\Phi_m \times \Phi_n)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/equivariant">equivariant</a>. (Note that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Φ</mi> <mi>m</mi></msub></mrow><annotation encoding="application/x-tex">\Phi_m</annotation></semantics></math> acts on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>F</mi> <mi>m</mi></msup></mrow><annotation encoding="application/x-tex">F^m</annotation></semantics></math> by the definition of <a class="existingWikiWord" href="/nlab/show/symmetric+endofunctor">symmetric endofunctor</a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Φ</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">\Phi_n</annotation></semantics></math> acts on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">X_n</annotation></semantics></math> by the definition of <a class="existingWikiWord" href="/nlab/show/symmetric+sequence">symmetric sequence</a>.) A morphism of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Φ</mi></mrow><annotation encoding="application/x-tex">\Phi</annotation></semantics></math>-symmetric <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>-spectra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>n</mi></msub><msub><mo stretchy="false">)</mo> <mi>n</mi></msub><mo>→</mo><mi>Y</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mi>Y</mi> <mi>n</mi></msub><msub><mo stretchy="false">)</mo> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">X = (X_n)_n \to Y = (Y_n)_n</annotation></semantics></math> is a morphism of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Φ</mi></mrow><annotation encoding="application/x-tex">\Phi</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/symmetric+sequences">symmetric sequences</a> making the obvious <a class="existingWikiWord" href="/nlab/show/diagrams">diagrams</a> <a class="existingWikiWord" href="/nlab/show/commutative+diagram">commute</a>. We write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Spect</mi> <mi>F</mi> <mi>Φ</mi></msubsup><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spect^{\Phi}_F(C)</annotation></semantics></math> for the category of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Φ</mi></mrow><annotation encoding="application/x-tex">\Phi</annotation></semantics></math>-symmetric <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>-spectra in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>.</p> <p>When <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Φ</mi><mo>=</mo><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Phi = \Sigma</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/graded+monoid">graded monoid</a> of <a class="existingWikiWord" href="/nlab/show/symmetric+groups">symmetric groups</a>, <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>-symmetric <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>-spectra are called simply <strong>symmetric <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>-spectra</strong>. When <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Φ</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\Phi = 1</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math>-symmetric <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>-spectra are called simply <strong>nonsymmetric <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>-spectra</strong>. When the endofunctor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> is given by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo>⊗</mo><mo lspace="verythinmathspace" rspace="0em">−</mo></mrow><annotation encoding="application/x-tex">T \otimes -</annotation></semantics></math> for some object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">T \in C</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>-spectra are called <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>-spectra.</p> <p>When <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/symmetric+monoidal+category">symmetric monoidal category</a>, there is an induced <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+structure+on+spectrum+objects">symmetric monoidal structure on spectrum objects</a>.</p> <p>When <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 sufficiently nice <a class="existingWikiWord" href="/nlab/show/model+category">model category</a>, there are induced <span class="newWikiWord">model structures on spectrum objects<a href="/nlab/new/model+structures+on+spectrum+objects">?</a></span>.</p> <h2 id="properties">Properties</h2> <h3 id="as_a_model_for_stabilization">As a model for stabilization</h3> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is a pointed <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-category with finite <a class="existingWikiWord" href="/nlab/show/limit+in+quasi-categories">limits</a>, then <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> is a <a class="existingWikiWord" href="/nlab/show/stable+%28infinity%2C1%29-category">stable (infinity,1)-category</a>.</p> <h3 id="reflection_into_prespectrum_objects">Reflection into pre-spectrum objects</h3> <p>For <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> an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> with <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-pullbacks">(∞,1)-pullbacks</a> and <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-colimits">(∞,1)-colimits</a>, then the inclusion of spectrum objects into prespectum objects should be a <a class="existingWikiWord" href="/nlab/show/exact+%28%E2%88%9E%2C1%29-functor">left exact</a> <a class="existingWikiWord" href="/nlab/show/reflective+sub-%28%E2%88%9E%2C1%29-category">reflective sub-(∞,1)-category</a> inclusion (<a href="#Joyal08">Joyal 08, section 35</a>).</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Spec</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo><mover><mo>↪</mo><mover><mo>←</mo><mi>lex</mi></mover></mover><mi>PreSpec</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Spec(\mathcal{C}) \stackrel{\overset{lex}{\leftarrow}}{\hookrightarrow} PreSpec(\mathcal{C}) \,. </annotation></semantics></math></div> <p>This implies in particular that the <a class="existingWikiWord" href="/nlab/show/tangent+%28%E2%88%9E%2C1%29-category">tangent (∞,1)-category</a> of an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> is itself again an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> (<a href="#Joyal08">Joyal 08, section 35.5</a>), see at <em><a href="tangent+%28infinity%2C1%29-category#TangentTopos">tangent (∞,1)-category – Tangent (∞,1)-topos</a></em> .</p> <h2 id="examples">Examples</h2> <ul> <li> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>=</mo><mi>Top</mi></mrow><annotation encoding="application/x-tex">C = Top</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Stab</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Stab(C)</annotation></semantics></math> is the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-category version of the classical stable homotopy category of spaces: the <a class="existingWikiWord" href="/nlab/show/stable+%28infinity%2C1%29-category+of+spectra">stable (infinity,1)-category of spectra</a>.</p> </li> <li> <p>In the <a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+theory">equivariant homotopy theory</a> of <a class="existingWikiWord" href="/nlab/show/G-spaces">G-spaces</a> a spectrum object is a <em><a class="existingWikiWord" href="/nlab/show/spectrum+with+G-action">spectrum with G-action</a></em>.</p> </li> <li> <p>see also at <em><a class="existingWikiWord" href="/nlab/show/motivic+spectrum">motivic spectrum</a></em></p> </li> </ul> <h2 id="related_concepts">Related concepts</h2> <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> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+homotopy+type">stable homotopy type</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cospectrum">cospectrum</a></p> </li> </ul> <h2 id="references">References</h2> <p>Discussion in terms of <a class="existingWikiWord" href="/nlab/show/stable+%28infinity%2C1%29-categories">stable (infinity,1)-categories</a> is in</p> <ul> <li id="StabCat"> <p><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, section 8 of <em><a class="existingWikiWord" href="/nlab/show/Stable+Infinity-Categories">Stable Infinity-Categories</a></em></p> </li> <li id="HigherAlg"> <p><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, section 1.4.2 <em><a class="existingWikiWord" href="/nlab/show/Higher+Algebra">Higher Algebra</a></em></p> </li> <li id="Joyal08"> <p><a class="existingWikiWord" href="/nlab/show/Andr%C3%A9+Joyal">André Joyal</a>, section 35 <em>Notes on Logoi</em>, 2008 (<a href="http://www.math.uchicago.edu/~may/IMA/JOYAL/Joyal.pdf">pdf</a>)</p> </li> </ul> <p>Discussion of <a class="existingWikiWord" href="/nlab/show/model+structure+on+spectra">model structures for spectrum objects</a> includes</p> <ul> <li id="Hovey00"> <p><a class="existingWikiWord" href="/nlab/show/Mark+Hovey">Mark Hovey</a>, <em>Spectra and symmetric spectra in general model categories</em> (<a href="http://arxiv.org/abs/math/0004051">arXiv:0004051</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bj%C3%B8rn+Ian+Dundas">Bjørn Ian Dundas</a>, <a class="existingWikiWord" href="/nlab/show/Oliver+R%C3%B6ndigs">Oliver Röndigs</a>, <a class="existingWikiWord" href="/nlab/show/Paul+Arne+%C3%98stv%C3%A6r">Paul Arne Østvær</a>, <em>Enriched functors and stable homotopy theory</em>, Doc. Math., 8:409–488, 2003 (<a href="https://eudml.org/doc/123650">EuDML</a>)</p> </li> </ul> <p>A detailed treatment of the 1-categorical case is in the last chapter of</p> <ul> <li id="Ayoub"><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>)</li> </ul> <p>Generalization to <a class="existingWikiWord" href="/nlab/show/G-spectra">G-spectra</a> is in</p> <ul> <li id="Blumberg05"><a class="existingWikiWord" href="/nlab/show/Andrew+Blumberg">Andrew Blumberg</a>, <em>Continuous functors as a model for the equivariant stable homotopy category</em> (<a href="http://arxiv.org/abs/math.AT/0505512">arXiv:math.AT/0505512</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on January 16, 2024 at 09:26:16. See the <a href="/nlab/history/spectrum+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/spectrum+object" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/5657/#Item_7">Discuss</a><span class="backintime"><a href="/nlab/revision/spectrum+object/20" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/spectrum+object" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/spectrum+object" accesskey="S" class="navlink" id="history" rel="nofollow">History (20 revisions)</a> <a href="/nlab/show/spectrum+object/cite" style="color: black">Cite</a> <a href="/nlab/print/spectrum+object" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/spectrum+object" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>