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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> <h4 id="higher_linear_algebra">Higher linear algebra</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>, <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+theory">(∞,1)-category theory</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a></strong></p> <p>flavors: <a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable</a>, <a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+theory">equivariant</a>, <a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rational</a>, <a class="existingWikiWord" href="/nlab/show/p-adic+homotopy+theory">p-adic</a>, <a class="existingWikiWord" href="/nlab/show/proper+homotopy+theory">proper</a>, <a class="existingWikiWord" href="/nlab/show/geometric+homotopy+theory">geometric</a>, <a class="existingWikiWord" href="/nlab/show/cohesive+homotopy+theory">cohesive</a>, <a class="existingWikiWord" href="/nlab/show/directed+homotopy+theory">directed</a>…</p> <p>models: <a class="existingWikiWord" href="/nlab/show/topological+homotopy+theory">topological</a>, <a class="existingWikiWord" href="/nlab/show/simplicial+homotopy+theory">simplicial</a>, <a class="existingWikiWord" href="/nlab/show/localic+homotopy+theory">localic</a>, …</p> <p>see also <strong><a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a></strong></p> <p><strong>Introductions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Topology+--+2">Introduction to Basic Homotopy Theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Homotopy+Theory">Introduction to Abstract Homotopy Theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+homotopy+types">geometry of physics – homotopy types</a></p> </li> </ul> <p><strong>Definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a>, <a class="existingWikiWord" href="/nlab/show/higher+homotopy">higher homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pi-algebra">Pi-algebra</a>, <a class="existingWikiWord" href="/nlab/show/spherical+object+and+Pi%28A%29-algebra">spherical object and Pi(A)-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coherent+category+theory">homotopy coherent category theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopical+category">homotopical category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">category of fibrant objects</a>, <a class="existingWikiWord" href="/nlab/show/cofibration+category">cofibration category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Waldhausen+category">Waldhausen category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ho%28Top%29">Ho(Top)</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/homotopy+category+of+an+%28%E2%88%9E%2C1%29-category">homotopy category of an (∞,1)-category</a></li> </ul> </li> </ul> <p><strong>Paths and cylinders</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cylinder+object">cylinder object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/right+homotopy">right homotopy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/path+object">path object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cocone">mapping cocone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+universal+bundle">universal bundle</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interval+object">interval object</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+localization">homotopy localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infinitesimal+interval+object">infinitesimal interval object</a></p> </li> </ul> </li> </ul> <p><strong>Homotopy groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+group+of+a+topos">fundamental group of a topos</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brown-Grossman+homotopy+group">Brown-Grossman homotopy group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/categorical+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">categorical homotopy groups in an (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">geometric homotopy groups in an (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid">fundamental ∞-groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+groupoid">fundamental groupoid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/path+groupoid">path groupoid</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+in+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+of+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%28%E2%88%9E%2C1%29-category">fundamental (∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+category">fundamental category</a></li> </ul> </li> </ul> <p><strong>Basic facts</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+group+of+the+circle+is+the+integers">fundamental group of the circle is the integers</a></li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+theorem+of+covering+spaces">fundamental theorem of covering spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freudenthal+suspension+theorem">Freudenthal suspension theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Blakers-Massey+theorem">Blakers-Massey theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+homotopy+van+Kampen+theorem">higher homotopy van Kampen theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nerve+theorem">nerve theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitehead%27s+theorem">Whitehead's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hurewicz+theorem">Hurewicz theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Galois+theory">Galois theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis</a>-theorem</p> </li> </ul> </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='#general'>General</a></li> <li><a href='#history'>History</a></li> <li><a href='#as_day_convolution_spectra'>As Day convolution spectra</a></li> </ul> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#GradedCommutativity'>Graded commutativity</a></li> </ul> <li><a href='#definitions'>Definitions</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <h3 id="general">General</h3> <p>The <a class="existingWikiWord" href="/nlab/show/smash+product">smash product</a> on <a class="existingWikiWord" href="/nlab/show/pointed+topological+spaces">pointed topological spaces</a> induces a smash product on <a class="existingWikiWord" href="/nlab/show/spectra">spectra</a>.</p> <p>This is the canonical <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a> in the <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28infinity%2C1%29-category+of+spectra">symmetric monoidal (infinity,1)-category of spectra</a>. There are various <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> <a class="existingWikiWord" href="/nlab/show/presentable+%28infinity%2C1%29-category">presentations</a> which are symmetric <a class="existingWikiWord" href="/nlab/show/monoidal+model+categories">monoidal model categories</a> (such as the <a class="existingWikiWord" href="/nlab/show/highly+structured+spectra">highly structured spectra</a>: <a class="existingWikiWord" href="/nlab/show/S-modules">S-modules</a>, <a class="existingWikiWord" href="/nlab/show/symmetric+spectra">symmetric spectra</a> and <a class="existingWikiWord" href="/nlab/show/orthogonal+spectra">orthogonal spectra</a>, but also for instance <a class="existingWikiWord" href="/nlab/show/excisive+functors">excisive functors</a>). See at <em><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category+of+spectra">symmetric monoidal category of spectra</a></em> for more on this. Passing to the <a class="existingWikiWord" href="/nlab/show/stable+homotopy+category">stable homotopy category</a>, it become a <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a> under the smash product. Under mild assumptions, this is the essentially unique symmetric tensor product with <a class="existingWikiWord" href="/nlab/show/unit">unit</a> the <a class="existingWikiWord" href="/nlab/show/sphere+spectrum">sphere spectrum</a> (<a href="#Shipley01">Shipley 01</a>).</p> <h3 id="history">History</h3> <p>Historically the discussion proceeded in the opposite direction: available variants of the construction of smash products on <a class="existingWikiWord" href="/nlab/show/sequential+spectra">sequential spectra</a> (the “handicrafted” or “naive” smash products <a href="#Boardman65">Boardman 65</a>, <a href="#Adams74">Adams 74, part III, section 4</a>) were found to yield a <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a> structure <em>only</em> after passage to the <a class="existingWikiWord" href="/nlab/show/stable+homotopy+category">stable homotopy category</a>. Then models via non-sequential <a class="existingWikiWord" href="/nlab/show/highly+structured+spectra">highly structured spectra</a> were discovered which do admit a <a class="existingWikiWord" href="/nlab/show/symmetric+smash+product+of+spectra">symmetric smash product of spectra</a> in the sense of 1-category theory, as do <a class="existingWikiWord" href="/nlab/show/excisive+functors">excisive functors</a>, see at <em><a class="existingWikiWord" href="/nlab/show/model+structure+on+excisive+functors">model structure on excisive functors</a></em>.</p> <h3 id="as_day_convolution_spectra">As Day convolution spectra</h3> <p>Many versions of the smash product of spectra, <a class="existingWikiWord" href="/nlab/show/symmetric+smash+product+of+spectra">symmetric or not</a>, arise as <a class="existingWikiWord" href="/nlab/show/Day+convolution">Day convolution</a> products on a suitable <a class="existingWikiWord" href="/nlab/show/enriched+category">enriched</a> <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal</a> version of a <a class="existingWikiWord" href="/nlab/show/category">category</a> of “index spaces”.</p> <p>The <a class="existingWikiWord" href="/nlab/show/symmetric+smash+product+of+spectra">symmetric smash product of spectra</a> on, in particular, <a class="existingWikiWord" href="/nlab/show/symmetric+spectra">symmetric spectra</a> and <a class="existingWikiWord" href="/nlab/show/orthogonal+spectra">orthogonal spectra</a> is the Day convolution product for <a class="existingWikiWord" href="/nlab/show/Top">Top</a>-<a class="existingWikiWord" href="/nlab/show/enriched+functors">enriched functors</a> on monoidal categories of <a class="existingWikiWord" href="/nlab/show/symmetric+groups">symmetric groups</a> of <a class="existingWikiWord" href="/nlab/show/orthogonal+groups">orthogonal groups</a>, respectively (<a href="#MMSS00">MMSS 00, theorem 1.7 and section 21.</a>).</p> <p>Similarly the <a class="existingWikiWord" href="/nlab/show/symmetric+smash+product+of+spectra">symmetric smash product of spectra</a> on the <a class="existingWikiWord" href="/nlab/show/model+structure+for+excisive+functors">model structure for excisive functors</a> is Day convolution for <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a>-<a class="existingWikiWord" href="/nlab/show/enriched+functors">enriched functors</a> on the plain <a class="existingWikiWord" href="/nlab/show/smash+product">smash product</a> of finite pointed <a class="existingWikiWord" href="/nlab/show/simplicial+sets">simplicial sets</a> (<a href="#Lydakis98">Lydakis 98</a>).</p> <p>See also at <em><a class="existingWikiWord" href="/nlab/show/functor+with+smash+products">functor with smash products</a></em>.</p> <h2 id="properties">Properties</h2> <h3 id="GradedCommutativity">Graded commutativity</h3> <p>The smash product of spectra exhibits a certain graded commutativity akin to the graded commutativity in the <a class="existingWikiWord" href="/nlab/show/tensor+product+of+chain+complexes">tensor product of chain complexes</a> (in fact, under the <a class="existingWikiWord" href="/nlab/show/stable+Dold-Kan+correspondence">stable Dold-Kan correspondence</a> the latter maps to the former).</p> <p>This comes down to the following basic fact about the <a class="existingWikiWord" href="/nlab/show/smash+product">smash product</a> of <a class="existingWikiWord" href="/nlab/show/pointed+topological+spaces">pointed topological spaces</a>:</p> <div class="num_prop" id="GradedCommutativityOfSmashOfSpheres"> <h6 id="proposition">Proposition</h6> <p>There are <a class="existingWikiWord" href="/nlab/show/homeomorphisms">homeomorphisms</a> between <a class="existingWikiWord" href="/nlab/show/n-spheres">n-spheres</a> and their <a class="existingWikiWord" href="/nlab/show/smash+products">smash products</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>n</mi> <mn>2</mn></msub></mrow></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>S</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msup><mo>∧</mo><msup><mi>S</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup><mover><mo>⟶</mo><mo>≃</mo></mover><msup><mi>S</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup></mrow><annotation encoding="application/x-tex"> \phi_{n_1,n_2} \;\colon\; S^{n_1} \wedge S^{n_2} \stackrel{\simeq}{\longrightarrow} S^{n_1 + n_2} </annotation></semantics></math></div> <p>such that in <a class="existingWikiWord" href="/nlab/show/Ho%28Top%29">Ho(Top)</a> there are <a class="existingWikiWord" href="/nlab/show/commuting+diagrams">commuting diagrams</a> like so:</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><mo stretchy="false">(</mo><msup><mi>S</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msup><mo>∧</mo><msup><mi>S</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup><mo stretchy="false">)</mo><mo>∧</mo><msup><mi>S</mi> <mrow><msub><mi>n</mi> <mn>3</mn></msub></mrow></msup></mtd> <mtd></mtd> <mtd><mover><mo>⟶</mo><mo>≃</mo></mover></mtd> <mtd></mtd> <mtd><msup><mi>S</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msup><mo>∧</mo><mo stretchy="false">(</mo><msup><mi>S</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup><mo>∧</mo><msup><mi>S</mi> <mrow><msub><mi>n</mi> <mn>3</mn></msub></mrow></msup><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>ϕ</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>n</mi> <mn>2</mn></msub></mrow></msub><mo>∧</mo><mi>id</mi></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>id</mi><mo>∧</mo><msub><mi>ϕ</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>n</mi> <mn>3</mn></msub></mrow></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msup><mi>S</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup><mo>∧</mo><msup><mi>S</mi> <mrow><msub><mi>n</mi> <mn>3</mn></msub></mrow></msup></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mi>S</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msup><mo>∧</mo><msup><mi>S</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub><mo>+</mo><msub><mi>n</mi> <mn>3</mn></msub></mrow></msup></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>ϕ</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>n</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>n</mi> <mn>3</mn></msub></mrow></msub></mrow></mpadded></msub><mo>↘</mo></mtd> <mtd></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mrow><msub><mi>ϕ</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>n</mi> <mn>2</mn></msub><mo>+</mo><msub><mi>n</mi> <mn>3</mn></msub></mrow></msub></mrow></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mi>S</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>n</mi> <mn>2</mn></msub><mo>+</mo><msub><mi>n</mi> <mn>3</mn></msub></mrow></msup></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ (S^{n_1} \wedge S^{n_2}) \wedge S^{n_3} &&\stackrel{\simeq}{\longrightarrow}&& S^{n_1} \wedge (S^{n_2} \wedge S^{n_3}) \\ {}^{\mathllap{\phi_{n_1,n_2} \wedge id}}\downarrow && && \downarrow^{\mathrlap{id \wedge \phi_{n_2,n_3}}} \\ S^{n_1+n_2} \wedge S^{n_3} && && S^{n_1}\wedge S^{n_2 + n_3} \\ & {}_{\mathllap{\phi_{n_1+n_2, n_3}}}\searrow && \swarrow_{\mathrlap{\phi_{n_1,n_2+n_3}}} \\ && S^{n_1+n_2 + n_3} } \,. </annotation></semantics></math></div> <p>and</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><msup><mi>S</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msup><mo>∧</mo><msup><mi>S</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>b</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>n</mi> <mn>2</mn></msub></mrow></msub></mrow></mover></mtd> <mtd><msup><mi>S</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup><mo>∧</mo><msup><mi>S</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msup></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>ϕ</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>n</mi> <mn>2</mn></msub></mrow></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>ϕ</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>n</mi> <mn>1</mn></msub></mrow></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msup><mi>S</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup></mtd> <mtd><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><msub><mi>n</mi> <mn>1</mn></msub><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup></mrow></mover></mtd> <mtd><msup><mi>S</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ S^{n_1} \wedge S^{n_2} &\stackrel{b_{n_1,n_2}}{\longrightarrow}& S^{n_2} \wedge S^{n_1} \\ {}^{\mathllap{\phi_{n_1,n_2}}}\downarrow && \downarrow^{\mathrlap{\phi_{n_2,n_1}}} \\ S^{n_1 + n_2} &\stackrel{(-1)^{n_1 n_2}}{\longrightarrow}& S^{n_1 + n_2} } \,, </annotation></semantics></math></div> <p>where here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mi>n</mi></msup><mo lspace="verythinmathspace">:</mo><msup><mi>S</mi> <mi>n</mi></msup><mo>→</mo><msup><mi>S</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">(-1)^n \colon S^n \to S^n</annotation></semantics></math> denotes the homotopy class of a <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a> of <a class="existingWikiWord" href="/nlab/show/degree+of+a+continuous+function">degree</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mi>n</mi></msup><mo>∈</mo><mi>ℤ</mi><mo>≃</mo><mo stretchy="false">[</mo><msup><mi>S</mi> <mi>n</mi></msup><mo>,</mo><msup><mi>S</mi> <mi>n</mi></msup><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">(-1)^n \in \mathbb{Z} \simeq [S^n, S^n]</annotation></semantics></math>.</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>With the <a class="existingWikiWord" href="/nlab/show/n-sphere">n-sphere</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">S^n</annotation></semantics></math> realized as the <a class="existingWikiWord" href="/nlab/show/one-point+compactification">one-point compactification</a> of the <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>n</mi> <mn>2</mn></msub></mrow></msub></mrow><annotation encoding="application/x-tex">\phi_{n_1,n_2}</annotation></semantics></math> is given by the identity on <a class="existingWikiWord" href="/nlab/show/coordinates">coordinates</a> and the <a class="existingWikiWord" href="/nlab/show/braiding">braiding</a> homeomorphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>b</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>n</mi> <mn>2</mn></msub></mrow></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>S</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msup><mo>∧</mo><msup><mi>S</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup><mover><mo>⟶</mo><mi>σ</mi></mover><msup><mi>S</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup><mo>∧</mo><msup><mi>S</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msup></mrow><annotation encoding="application/x-tex"> b_{n_1,n_2} \;\colon\; S^{n_1} \wedge S^{n_2} \stackrel{\sigma}{\longrightarrow} S^{n_2} \wedge S^{n_1} </annotation></semantics></math></div> <p>is given by permuting the <a class="existingWikiWord" href="/nlab/show/coordinates">coordinates</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>x</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msub><mo>,</mo><msub><mi>y</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>y</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msub><mo stretchy="false">)</mo><mo>↦</mo><mo stretchy="false">(</mo><msub><mi>y</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>y</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msub><mo>,</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>x</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (x_1, \cdots, x_{n_1}, y_1, \cdots, y_{n_2}) \mapsto (y_1, \cdots, y_{n_2}, x_1, \cdots, x_{n_1}) \,. </annotation></semantics></math></div> <p>This has <a class="existingWikiWord" href="/nlab/show/degree+of+a+continuous+map">degree</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><msub><mi>n</mi> <mn>1</mn></msub><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup></mrow><annotation encoding="application/x-tex">(-1)^{n_1 n_2}</annotation></semantics></math> .</p> </div> <p>This statement passes to the <a class="existingWikiWord" href="/nlab/show/suspension+spectra">suspension spectra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Σ</mi> <mn>∞</mn></msup><msup><mi>S</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\Sigma^\infty S^n</annotation></semantics></math> of the spheres (<a href="#Adams74">Adams 74, part III, prop. 4.8</a>, <a href="#Schwede12">Schwede 12, chapter II.4, prop. 4.4</a>).</p> <div class="num_remark" id="WhySequentialSpectraHaveNoSymmetricSmashProduct"> <h6 id="remark">Remark</h6> <p>The phenomenon in prop. <a class="maruku-ref" href="#GradedCommutativityOfSmashOfSpheres"></a> is the reason why there is no <a class="existingWikiWord" href="/nlab/show/symmetric+smash+product+of+spectra">symmetric smash product of spectra</a> on plain <a class="existingWikiWord" href="/nlab/show/sequential+spectra">sequential spectra</a>, and in fact no appropriate functorial product operation at all.</p> <p>To see this, observe, by <a href="sequential+spectrum#QuillenEquivalenceBetweenSequentialBFAndExcisiveFunctors">this proposition</a>, that sequential spectra are Quillen equivalent to the model category of <a class="existingWikiWord" href="/nlab/show/excisive+%28infinity%2C1%29-functors">excisive (infinity,1)-functors</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Exc</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mn>∞</mn><msubsup><mi>Grpd</mi> <mi>fin</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup><mo>,</mo><mn>∞</mn><msup><mi>Grpd</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup><mo stretchy="false">)</mo><mo>↪</mo><mi>Func</mi><mo stretchy="false">(</mo><mn>∞</mn><msubsup><mi>Grpd</mi> <mi>fin</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup><mo>,</mo><mn>∞</mn><msup><mi>Grpd</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Exc^1(\infty Grpd^{\ast/}_{fin}, \infty Grpd^{\ast/}) \hookrightarrow Func(\infty Grpd^{\ast/}_{fin}, \infty Grpd^{\ast/}) </annotation></semantics></math></div> <p>under an equivalence given by restricting from the domain of all <a class="existingWikiWord" href="/nlab/show/pointed+homotopy+types">pointed</a> <a class="existingWikiWord" href="/nlab/show/finite+homotopy+types">finite homotopy types</a> <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> to its non-full subcategory <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>StdSpheres</mi></mrow><annotation encoding="application/x-tex">StdSpheres</annotation></semantics></math> of standard spheres with just the adjuncts of suspension maps between them.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ι</mi><mo lspace="verythinmathspace">:</mo><mi>StdSpheres</mi><mo>↪</mo><mn>∞</mn><msubsup><mi>Grpd</mi> <mi>fin</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \iota \colon StdSpheres \hookrightarrow \infty Grpd^{\ast/}_{fin} \,. </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Exc</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mn>∞</mn><msubsup><mi>Grpd</mi> <mi>fin</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup><mo>,</mo><mn>∞</mn><msup><mi>Grpd</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup><mo stretchy="false">)</mo><munderover><mo>⟶</mo><mo>≃</mo><mrow><msup><mi>ι</mi> <mo>*</mo></msup></mrow></munderover><mi>SeqSpectra</mi></mrow><annotation encoding="application/x-tex"> Exc^1(\infty Grpd^{\ast/}_{fin}, \infty Grpd^{\ast/}) \underoverset{\simeq}{\iota^\ast}{\longrightarrow} SeqSpectra </annotation></semantics></math></div> <p>Here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>StdSpheres</mi></mrow><annotation encoding="application/x-tex">StdSpheres</annotation></semantics></math> has objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>S</mi> <mi>std</mi> <mi>n</mi></msubsup><mo>≔</mo><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>1</mn></msup><msup><mo stretchy="false">)</mo> <mrow><mo>∧</mo><mi>n</mi></mrow></msup></mrow><annotation encoding="application/x-tex">S^n_{std} \coloneqq (S^1)^{\wedge n}</annotation></semantics></math> and as hom-spaces it has <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>StdSpheres</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mi>m</mi></msup><mo>,</mo><msup><mi>S</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo>=</mo><msup><mi>S</mi> <mrow><mi>max</mi><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mi>m</mi><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msup></mrow><annotation encoding="application/x-tex">StdSpheres(S^{m}, S^{n}) = S^{max(n-m,0)}</annotation></semantics></math>, identified as the <a class="existingWikiWord" href="/nlab/show/adjunct">adjunct</a> of the canonical isomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mi>m</mi></msup><mo>∧</mo><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mi>m</mi></mrow></msup><mo>→</mo><msup><mi>S</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">S^m \wedge S^{n-m} \to S^n</annotation></semantics></math>.</p> <p>Hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ι</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">\iota^\ast</annotation></semantics></math> identifies 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>th component space of a sequential spectrum with the value <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E(S^n)</annotation></semantics></math> of an excisive functor on 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>-sphere, and it identifies the structure map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup><mo>∧</mo><msub><mi>E</mi> <mi>n</mi></msub><mo>→</mo><msub><mi>E</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">S^1 \wedge E_n \to E_{n+1}</annotation></semantics></math> with part of the <a class="existingWikiWord" href="/nlab/show/enriched+functor">enriched functoriality</a> of the excisive functor.</p> <p>Now in the model category of excisive functors, the correct smash product of spectra is the <a class="existingWikiWord" href="/nlab/show/Day+convolution">Day convolution</a> over the symmetric monoidal category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><msubsup><mi>Grpd</mi> <mi>fin</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup><mo>,</mo><mo>∧</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty Grpd^{\ast/}_{fin},\wedge)</annotation></semantics></math>. The issue then is that the restricted hom-spaces of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>StdSpheres</mi></mrow><annotation encoding="application/x-tex">StdSpheres</annotation></semantics></math> do not see the non-trivial <a class="existingWikiWord" href="/nlab/show/braiding">braiding</a> of spheres in prop. <a class="maruku-ref" href="#GradedCommutativityOfSmashOfSpheres"></a> anymore.</p> <p>More concretely, the enriched category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>StdSpheres</mi></mrow><annotation encoding="application/x-tex">StdSpheres</annotation></semantics></math> does not inherit monoidal structure: defining the smash product on hom spaces requires permuting smash copies of spheres, which is not available. Thus there is no Day convolution product on sequential spectra at all.</p> <p>One could further restrict along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℕ</mi><mo>→</mo><mi>StdSpheres</mi></mrow><annotation encoding="application/x-tex">\mathbb{N} \to StdSpheres</annotation></semantics></math> and use the monoidal structure <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>ℕ</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathbb{N},+)</annotation></semantics></math> to define at least a smash product on sequences of pointed spaces by <a class="existingWikiWord" href="/nlab/show/Day+convolution">Day convolution</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>ℕ</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathbb{N},+)</annotation></semantics></math> as in (<a href="#MMSS00">MMSS 00, example 4.1</a>, <a href="#HoveyShipleySmith00">Hovey-Shipley-Smith 00, below prop. 2.3.4</a>). But then in addition to the above problem that this does not give a functorial smash product on spectra (it will not respect the structure maps), moreover <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>ℕ</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathbb{N},+)</annotation></semantics></math> is trivially <a class="existingWikiWord" href="/nlab/show/braided+monoidal+category">braided</a> and so, again, under restriction of excisive functors to <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">\mathbb{N}</annotation></semantics></math> there is no way to recover the information in the smash product of spectra that is encoded in the non-trivial braiding of the smash product of spheres.</p> </div> <h2 id="definitions">Definitions</h2> <ul> <li> <p><em><a href="sequential+spectrum#SmashProduct">smash product of sequential spectra</a></em></p> </li> <li> <p><em><a href="model+structure+for+excisive+functors#SmashProduct">smash product of excisive functors</a></em></p> </li> <li> <p><em><a href="symmetric+spectrum#SmashProduct">smash product of symmetric spectra</a></em></p> </li> <li> <p><em><a href="orthogonal+spectrum#SmashProduct">smash product of orthogonal spectra</a></em></p> </li> </ul> <h2 id="references">References</h2> <p>The original “handicrafted” constructions of the smash product on the <a class="existingWikiWord" href="/nlab/show/stable+homotopy+category">stable homotopy category</a> are due to</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 onwards</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Rainer+Vogt">Rainer Vogt</a>, <em>Boardman’s stable homotopy category</em>, lectures, spring 1969</p> </li> <li id="Puppe73"> <p><a class="existingWikiWord" href="/nlab/show/Dieter+Puppe">Dieter Puppe</a>, <em>On the stable homotopy category</em>, Topology and its application (1973) (<a class="existingWikiWord" href="/nlab/files/PuppeStableHomotopyCategory.pdf" title="pdf">pdf</a>)</p> </li> <li id="Adams74"> <p><a class="existingWikiWord" href="/nlab/show/Frank+Adams">Frank Adams</a>, part III, section 4 of <em><a class="existingWikiWord" href="/nlab/show/Stable+homotopy+and+generalised+homology">Stable homotopy and generalised homology</a></em>, 1974</p> </li> </ul> <p>The smash product on <a class="existingWikiWord" href="/nlab/show/connective+spectra">connective spectra</a> modeled as <a class="existingWikiWord" href="/nlab/show/Gamma+spaces">Gamma spaces</a> is discussed in</p> <ul> <li id="Lydakis98">Lydakis, <em>Smash products and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi></mrow><annotation encoding="application/x-tex">\Gamma</annotation></semantics></math>-spaces</em>, Math. Proc. Cam. Phil. Soc. 126 (1999), 311-328 (<a href="http://hopf.math.purdue.edu/Lydakis/smash_gamma.pdf">pdf</a>)</li> </ul> <p>The <a class="existingWikiWord" href="/nlab/show/symmetric+smash+product+on+spectra">symmetric smash product on spectra</a> (see there for more) on <a class="existingWikiWord" href="/nlab/show/highly+structured+ring+spectra">highly structured ring spectra</a> is discussed for instance in</p> <p>for <a class="existingWikiWord" href="/nlab/show/S-modules">S-modules</a>:</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>for <a class="existingWikiWord" href="/nlab/show/symmetric+spectra">symmetric spectra</a>:</p> <ul> <li id="HoveyShipleySmith00"> <p><a class="existingWikiWord" href="/nlab/show/Mark+Hovey">Mark Hovey</a>, <a class="existingWikiWord" href="/nlab/show/Brooke+Shipley">Brooke Shipley</a>, <a class="existingWikiWord" href="/nlab/show/Jeff+Smith">Jeff Smith</a>, <em>Symmetric spectra</em>, J. Amer. Math. Soc. 13 (2000), 149-208 (<a href="http://arxiv.org/abs/math/9801077">arXiv:math/9801077</a>)</p> </li> <li id="Schwede12"> <p><a class="existingWikiWord" href="/nlab/show/Stefan+Schwede">Stefan Schwede</a>, <em>Symmetric spectra</em>, 2012 (<a href="http://www.math.uni-bonn.de/people/schwede/SymSpec-v3.pdf">pdf</a>)</p> </li> </ul> <p>Discussion of the smash product as a suitable <a class="existingWikiWord" href="/nlab/show/Day+convolution">Day convolution</a> is, for <a class="existingWikiWord" href="/nlab/show/highly+structured+spectra">highly structured spectra</a>, in</p> <ul> <li id="MMSS00"><a class="existingWikiWord" href="/nlab/show/Michael+Mandell">Michael Mandell</a>, <a class="existingWikiWord" href="/nlab/show/Peter+May">Peter May</a>, <a class="existingWikiWord" href="/nlab/show/Stefan+Schwede">Stefan Schwede</a>, <a class="existingWikiWord" href="/nlab/show/Brooke+Shipley">Brooke Shipley</a>, <em><a class="existingWikiWord" href="/nlab/show/Model+categories+of+diagram+spectra">Model categories of diagram spectra</a></em>, Proceedings London Mathematical Society Volume 82, Issue 2, 2000 (<a href="http://www.math.uchicago.edu/~may/PAPERS/mmssLMSDec30.pdf">pdf</a>, <a href="http://plms.oxfordjournals.org/content/82/2/441.short?rss=1&ssource=mfc">publisher</a>)</li> </ul> <p>and for <a class="existingWikiWord" href="/nlab/show/excisive+functors">excisive functors</a> in</p> <ul> <li id="Lydakis98">Lydakis, <em>Simplicial functors and stable homotopy theory</em> Preprint, available via Hopf archive, 1998 (<a href="http://hopf.math.purdue.edu/Lydakis/s_functors.pdf">pdf</a>)</li> </ul> <p>The uniqueness of the smash product on spectra is discussed in</p> <ul> <li id="Shipley01"><a class="existingWikiWord" href="/nlab/show/Brooke+Shipley">Brooke Shipley</a>, <em>Monoidal uniqueness of Stable homotopy theory</em>, Advances in Mathematics Volume 160, Issue 2, 25 June 2001, Pages 217–240 (<a href="http://homepages.math.uic.edu/~bshipley/ideal.pdf">pdf</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on October 25, 2024 at 03:02:07. 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