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style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="superalgebra_and_supergeometry">Super-Algebra and Super-Geometry</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/superalgebra">superalgebra</a></strong> and (<a class="existingWikiWord" href="/nlab/show/synthetic+differential+supergeometry">synthetic</a> ) <strong><a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a></strong></p> <h2 id="background">Background</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra">algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry">geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/graded+object">graded object</a></p> </li> </ul> <h2 id="introductions">Introductions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+superalgebra">geometry of physics – superalgebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+supergeometry">geometry of physics – supergeometry</a></p> </li> </ul> <h2 id="superalgebra">Superalgebra</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/super+commutative+monoid">super commutative monoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+abelian+group">super abelian group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+ring">super ring</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/supercommutative+ring">supercommutative ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/exterior+ring">exterior ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Clifford+ring">Clifford ring</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+module">super module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+vector+space">super vector space</a>, <a class="existingWikiWord" href="/nlab/show/SVect">SVect</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+algebra">super algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/supercommutative+algebra">supercommutative algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/exterior+algebra">exterior algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Clifford+algebra">Clifford algebra</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/superdeterminant">superdeterminant</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Lie+algebra">super Lie algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/super+Poincare+Lie+algebra">super Poincare Lie algebra</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+complex+of+super+vector+spaces">chain complex of super vector spaces</a> (<a class="existingWikiWord" href="/nlab/show/model+structure+on+chain+complexes+of+super+vector+spaces">model structure</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+graded-commutative+superalgebra">differential graded-commutative superalgebra</a> (<a class="existingWikiWord" href="/nlab/show/model+structure+on+differential+graded-commutative+superalgebras">model structure</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+L-infinity+algebra">super L-infinity algebra</a></p> </li> </ul> <h2 id="supergeometry">Supergeometry</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/superpoint">superpoint</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Cartesian+space">super Cartesian space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supermanifold">supermanifold</a>, <a class="existingWikiWord" href="/nlab/show/SDiff">SDiff</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/NQ-supermanifold">NQ-supermanifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+vector+bundle">super vector bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complex+supermanifold">complex supermanifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Euclidean+supermanifold">Euclidean supermanifold</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+spacetime">super spacetime</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/super+Minkowski+spacetime">super Minkowski spacetime</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integration+over+supermanifolds">integration over supermanifolds</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Berezin+integral">Berezin integral</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Lie+group">super Lie group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Lie+group">super translation group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Euclidean+group">super Euclidean group</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+%E2%88%9E-groupoid">super ∞-groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+formal+smooth+%E2%88%9E-groupoid">super formal smooth ∞-groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+line+2-bundle">super line 2-bundle</a></p> </li> </ul> <h2 id="supersymmetry">Supersymmetry</h2> <p><a class="existingWikiWord" href="/nlab/show/supersymmetry">supersymmetry</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/division+algebra+and+supersymmetry">division algebra and supersymmetry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Poincare+Lie+algebra">super Poincare Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supermultiplet">supermultiplet</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/BPS+state">BPS state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/M-theory+super+Lie+algebra">M-theory super Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/type+II+super+Lie+algebra">type II super Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity+Lie+3-algebra">supergravity Lie 3-algebra</a>, <a class="existingWikiWord" href="/nlab/show/supergravity+Lie+6-algebra">supergravity Lie 6-algebra</a></p> </li> </ul> <h2 id="supersymmetric_field_theory">Supersymmetric field theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/superfield">superfield</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supersymmetric+quantum+mechanics">supersymmetric quantum mechanics</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/superparticle">superparticle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adinkra">adinkra</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Yang-Mills+theory">super Yang-Mills theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/gauged+supergravity">gauged supergravity</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/superstring+theory">superstring theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/type+II+string+theory">type II string theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/heterotic+string+theory">heterotic string theory</a></p> </li> </ul> </li> </ul> <h2 id="applications">Applications</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/geometric+model+for+elliptic+cohomology">geometric model for elliptic cohomology</a></li> </ul> <div> <p> <a href="/nlab/edit/supergeometry+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="higher_geometry">Higher geometry</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a></strong> / <strong><a class="existingWikiWord" href="/nlab/show/derived+geometry">derived geometry</a></strong></p> <p><strong>Ingredients</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+topos+theory">higher topos theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a></p> </li> </ul> <p><strong>Concepts</strong></p> <ul> <li> <p><strong>geometric <a class="existingWikiWord" href="/nlab/show/big+and+little+toposes">little</a> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a>es</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/structured+%28%E2%88%9E%2C1%29-topos">structured (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry+%28for+structured+%28%E2%88%9E%2C1%29-toposes%29">geometry (for structured (∞,1)-toposes)</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+scheme">generalized scheme</a></p> </li> </ul> </li> <li> <p><strong>geometric <a class="existingWikiWord" href="/nlab/show/big+and+little+toposes">big</a> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a>es</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/function+algebras+on+%E2%88%9E-stacks">function algebras on ∞-stacks</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/geometric+%E2%88%9E-stacks">geometric ∞-stacks</a></li> </ul> </li> </ul> <p><strong>Constructions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/loop+space+object">loop space object</a>, <a class="existingWikiWord" href="/nlab/show/free+loop+space+object">free loop space object</a></p> </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> / <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">of a locally ∞-connected (∞,1)-topos</a></p> </li> </ul> <p><strong>Examples</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+algebraic+geometry">derived algebraic geometry</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%C3%A9tale+%28%E2%88%9E%2C1%29-site">étale (∞,1)-site</a>, <a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Hochschild cohomology</a> of <a class="existingWikiWord" href="/nlab/show/dg-algebra">dg-algebra</a>s</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dg-geometry">dg-geometry</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/dg-scheme">dg-scheme</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/schematic+homotopy+type">schematic homotopy type</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+noncommutative+geometry">derived noncommutative geometry</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/noncommutative+geometry">noncommutative geometry</a></li> </ul> </li> <li> <p>derived smooth geometry</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a>, <a class="existingWikiWord" href="/nlab/show/differential+topology">differential topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+smooth+manifold">derived smooth manifold</a>, <a class="existingWikiWord" href="/nlab/show/dg-manifold">dg-manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth ∞-groupoid</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebroid">∞-Lie algebroid</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+symplectic+geometry">higher symplectic geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+Klein+geometry">higher Klein geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+Cartan+geometry">higher Cartan geometry</a></p> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Jones' theorem</a>, <a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Deligne-Kontsevich conjecture</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tannaka+duality+for+geometric+stacks">Tannaka duality for geometric stacks</a></p> </li> </ul> </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> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#SpectralSuperpoint'>Spectral superpoint</a></li> </ul> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>The concept of <em>spectral super-scheme</em> is supposed to be the refinement of the concept of <a class="existingWikiWord" href="/nlab/show/super-scheme">super-scheme</a> as one passes to <a class="existingWikiWord" href="/nlab/show/spectral+geometry">spectral geometry</a> in the sense of <a class="existingWikiWord" href="/nlab/show/derived+algebraic+geometry">derived algebraic geometry</a> over <a class="existingWikiWord" href="/nlab/show/E-infinity+rings">E-infinity rings</a> (<a class="existingWikiWord" href="/nlab/show/E-infinity+geometry">E-infinity geometry</a>).</p> <h2 id="Definition">Definition</h2> <p>The following is an argument for a good definition of spectral supergeometry. This was originally motivated from the observation in <a href="#Kapranov13">Kapranov 13</a> and uses results due to <a href="#Rezk09">Rezk 09</a> and <a href="#SagaveSchlichtkrull11">Sagave-Schlichtkrull 2011</a>.</p> <p>Observe that</p> <ol> <li id="EInfinityGeometryAlreadyZGraded"> <p><a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+geometry"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <msub><mi>E</mi> <mn>∞</mn></msub> </mrow> <annotation encoding="application/x-tex">E_\infty</annotation> </semantics> </math>-geometry</a> is <em>already in itself</em> a <a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometric</a> version of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math>-graded supergeometry (in the sense discussed at <em><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+superalgebra">geometry of physics – superalgebra</a></em>).</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></p> <p>At the level of <a class="existingWikiWord" href="/nlab/show/homotopy+groups">homotopy groups</a> this is the following basic fact:</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/homotopy+commutative+ring+spectrum">homotopy commutative ring spectrum</a>, its <a class="existingWikiWord" href="/nlab/show/stable+homotopy+groups">stable homotopy groups</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_\bullet(E)</annotation></semantics></math> inherit the structure of a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math>-graded <a class="existingWikiWord" href="/nlab/show/super-commutative+ring">super-commutative ring</a> (according to <a href="geometry+of+physics+--+superalgebra#SupercommutativeSuperalgebraZGraded">this</a>). See <a href="Introduction+to+Stable+homotopy+theory+--+1-2#HomotopyGroupsOfHomotopyCommutativeRingSpectrum">this proposition</a> in the section <em><a href="Introduction+to+Stable+homotopy+theory+--+1-2#HomotopyRingSpectra">1-2 Homotopy commutative ring spectra</a></em> of <em><a class="existingWikiWord" href="/nlab/show/Introduction+to+Stable+homotopy+theory">Introduction to Stable homotopy theory</a></em>.</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></p> <p>But more is true: the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">E_\infty</annotation></semantics></math>-analog of the <a class="existingWikiWord" href="/nlab/show/integers">integers</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math>, is the <a class="existingWikiWord" href="/nlab/show/sphere+spectrum">sphere spectrum</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝕊</mi><mspace width="thinmathspace"></mspace><mo>≃</mo><mspace width="thinmathspace"></mspace><msup><mi>Σ</mi> <mn>∞</mn></msup><msup><mi>S</mi> <mn>0</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{S} \,\simeq\, \Sigma^\infty S^0</annotation></semantics></math>, and every <a class="existingWikiWord" href="/nlab/show/E-infinity+ring"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <msub><mi>E</mi> <mn>∞</mn></msub> </mrow> <annotation encoding="application/x-tex">E_\infty</annotation> </semantics> </math>-ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>E</mi><mo>,</mo><mo>⋅</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(E, \cdot)</annotation></semantics></math> is canonically <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝕊</mi></mrow><annotation encoding="application/x-tex">\mathbb{S}</annotation></semantics></math>-graded, in that (<a href="#SagaveSchlichtkrull11">Sagave-Schlichtkrull 2011, theorem 1.7-1.8</a>):</p> <p>on <a href="Introduction+to+Stable+homotopy+theory+--+1-1#SigmaInfinityOmegaInfinity">underlying</a> <a class="existingWikiWord" href="/nlab/show/E-infinity+spaces"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <msub><mi>E</mi> <mn>∞</mn></msub> </mrow> <annotation encoding="application/x-tex">E_\infty</annotation> </semantics> </math>-spaces</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>0</mn></msub><mspace width="thinmathspace"></mspace><mo>≔</mo><mspace width="thinmathspace"></mspace><msup><mi>Ω</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E_0 \,\coloneqq\, \Omega^\infty(E)</annotation></semantics></math>, at least, realized as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mi>𝒥</mi></msup><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Omega^{\mathcal{J}}(E)</annotation></semantics></math> in <a href="#SagaveSchlichtkrull11">SaSc11 (4.4)</a>, they are canonically equipped with an <a class="existingWikiWord" href="/nlab/show/commutative+monoid+in+a+symmetric+monoidal+%28infinity%2C1%29-category"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <msub><mi>E</mi> <mn>∞</mn></msub> </mrow> <annotation encoding="application/x-tex">E_\infty</annotation> </semantics> </math>-monoid</a> <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</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>E</mi> <mn>0</mn></msub><mo>,</mo><mo>⋅</mo><mo stretchy="false">)</mo><mover><mo>→</mo><mspace width="thickmathspace"></mspace></mover><mo stretchy="false">(</mo><msub><mi>𝕊</mi> <mn>0</mn></msub><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (E_0, \cdot) \xrightarrow{\;} (\mathbb{S}_0, +) </annotation></semantics></math></div> <p>to the <em>additive</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">E_\infty</annotation></semantics></math>-space underlying the <a class="existingWikiWord" href="/nlab/show/sphere+spectrum">sphere spectrum</a> (<a href="Bousfield-Friedlander+model+structure#Spectrification">traditionally</a> denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi><msup><mi>S</mi> <mn>0</mn></msup></mrow><annotation encoding="application/x-tex">Q S^0 </annotation></semantics></math>, which is notation for a construction that yields <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mn>∞</mn></msup><msup><mi>Σ</mi> <mn>∞</mn></msup><msup><mi>S</mi> <mn>0</mn></msup></mrow><annotation encoding="application/x-tex">\Omega^\infty \Sigma^\infty S^0</annotation></semantics></math>).</p> <p>The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">E_\infty</annotation></semantics></math>-monoid homomorphisms of this form are the evident homotopy-theoretic generalization of morphisms of <a class="existingWikiWord" href="/nlab/show/commutative+monoids">commutative monoids</a> to the additive group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>ℤ</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathbb{Z}, +)</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/integers">integers</a>, and these are evidently equivalent to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/graded+monoid">gradings</a> on the <a class="existingWikiWord" href="/nlab/show/domain">domain</a> monoid.</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></p> <p>So <a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+geometry"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <msub><mi>E</mi> <mn>∞</mn></msub> </mrow> <annotation encoding="application/x-tex">E_\infty</annotation> </semantics> </math>-geometry</a> in itself is already a <a class="existingWikiWord" href="/nlab/show/categorification">categorified</a>/homotopified version of <a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a>, but of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math>-graded supergeometry, not of the proper <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">\mathbb{Z}/2</annotation></semantics></math>-graded supergeometry.</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></p> <p>(That grading over the <a class="existingWikiWord" href="/nlab/show/sphere+spectrum">sphere spectrum</a> is closely related to <a class="existingWikiWord" href="/nlab/show/superalgebra">superalgebra</a> had been highlighted in <a href="#Kapranov13">Kapranov 2013</a>, but the issue of the difference between homotopified <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math>-grading compared to homotopified <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">\mathbb{Z}/2</annotation></semantics></math>-grading had been left open.)</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></p> </li> <li id="SuperGeometryAsPeriodifiedZGradedGeometry"> <p>But ordinary <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">\mathbb{Z}/2</annotation></semantics></math>-graded <a class="existingWikiWord" href="/nlab/show/supercommutative+superalgebra">supercommutative superalgebra</a> is equivalently <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math>-graded <a class="existingWikiWord" href="/nlab/show/supercommutative+superalgebra">supercommutative superalgebra</a> over the free even periodic <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math>-graded supercommutative superalgebra (<a href="geometry+of+physics+--+superalgebra#ModulesOverRbeta">this prop.</a>).</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></p> </li> <li id="PassageToAlgebraOverPeriodicRingSpectra"> <p>In view of the <a href="#EInfinityGeometryAlreadyZGraded">first point</a>, the <a href="#SuperGeometryAsPeriodifiedZGradedGeometry">second point</a> has an evident analog in <a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+geometry">E-∞ geometry</a>:</p> <p>The higher/derived analog of an even periodic <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math>-graded commutative algebra is an <a class="existingWikiWord" href="/nlab/show/E-infinity+algebra">E-infinity algebra</a> over an <a class="existingWikiWord" href="/nlab/show/even+cohomology+theory">even</a> <a class="existingWikiWord" href="/nlab/show/periodic+ring+spectrum">periodic ring spectrum</a>.</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></p> <p>That <a class="existingWikiWord" href="/nlab/show/E-infinity+algebras">E-infinity algebras</a> over <a class="existingWikiWord" href="/nlab/show/even+cohomology+theory">even</a> <a class="existingWikiWord" href="/nlab/show/periodic+ring+spectra">periodic ring spectra</a> are usefully regarded from the point of view of <a class="existingWikiWord" href="/nlab/show/supercommutative+superalgebra">supercommutative superalgebra</a> was highlighted in <a href="#Rezk09">Rezk 09, section 2</a>.</p> </li> </ol> <p>Hence it makes sense to say:</p> <p><strong>Definition.</strong> <em>Spectral/<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">E_\infty</annotation></semantics></math> super-geometry</em> is simply the <a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+geometry"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <msub><mi>E</mi> <mn>∞</mn></msub> </mrow> <annotation encoding="application/x-tex">E_\infty</annotation> </semantics> </math>-geometry</a> over <a class="existingWikiWord" href="/nlab/show/even+cohomology+theory">even</a> <a class="existingWikiWord" href="/nlab/show/periodic+ring+spectra">periodic ring spectra</a>.</p> <h2 id="examples">Examples</h2> <h3 id="SpectralSuperpoint">Spectral superpoint</h3> <p>The ordinary <a class="existingWikiWord" href="/nlab/show/superpoint">superpoint</a> over some <a class="existingWikiWord" href="/nlab/show/field">field</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/spectrum+of+a+commutative+ring">spectrum of a commutative ring</a> of the graded <a class="existingWikiWord" href="/nlab/show/symmetric+algebra">symmetric algebra</a> on a single odd generator (“graded <a class="existingWikiWord" href="/nlab/show/ring+of+dual+numbers">ring of dual numbers</a>”)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>𝔸</mi> <mi>k</mi> <mrow><mn>0</mn><mo stretchy="false">|</mo><mn>1</mn></mrow></msubsup><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mi>Spec</mi><mo stretchy="false">(</mo><mspace width="thinmathspace"></mspace><msub><mi>Sym</mi> <mi>k</mi></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathbb{A}_k^{0 \vert 1} \;\simeq\; Spec( \,Sym_k (k[1])\, ) </annotation></semantics></math></div> <p>Accordingly, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> an <a class="existingWikiWord" href="/nlab/show/even+cohomology+theory">even</a> <a class="existingWikiWord" href="/nlab/show/periodic+ring+spectrum">periodic ring spectrum</a>, then the <em>spectral superpoint</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>R</mi> <mrow><mn>0</mn><mo stretchy="false">|</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">R^{0 \vert 1}</annotation></semantics></math> should be the <a class="existingWikiWord" href="/nlab/show/spectral+scheme">spectral scheme</a> given by the <a class="existingWikiWord" href="/nlab/show/spectral+symmetric+algebra">spectral symmetric algebra</a> on the <a class="existingWikiWord" href="/nlab/show/suspension+spectrum">suspension spectrum</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msup><mi>R</mi> <mrow><mn>0</mn><mo stretchy="false">|</mo><mn>1</mn></mrow></msup></mtd> <mtd><mo>≔</mo><mi>Spec</mi><mrow><mo>(</mo><msub><mi>Sym</mi> <mi>R</mi></msub><mo stretchy="false">(</mo><mi>Σ</mi><mi>R</mi><mo stretchy="false">)</mo><mo>)</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><mi>Spec</mi><mrow><mo>(</mo><mi>R</mi><mo>∧</mo><msub><mrow><mo>(</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow></munder><mi>B</mi><mi>Σ</mi><mo stretchy="false">(</mo><mi>n</mi><msup><mo stretchy="false">)</mo> <mrow><msub><mi>τ</mi> <mi>n</mi></msub></mrow></msup><mo>)</mo></mrow> <mo>+</mo></msub><mo>)</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><mi>Spec</mi><mrow><mo>(</mo><mi>R</mi><mo>∧</mo><msub><mi>Sym</mi> <mi>𝕊</mi></msub><mo stretchy="false">(</mo><mi>Σ</mi><mi>𝕊</mi><mo stretchy="false">)</mo><mo>)</mo></mrow></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} R^{0 \vert 1} &\coloneqq Spec \left( Sym_R (\Sigma R) \right) \\ & \simeq Spec\left( R \wedge \left( \underset{n \in \mathbb{N}}{\coprod} B \Sigma(n)^{\tau_n} \right)_+ \right) \\ & \simeq Spec\left( R \wedge Sym_{\mathbb{S}}(\Sigma \mathbb{S}) \right) \end{aligned} \,. </annotation></semantics></math></div> <p>where on the right we have the <a class="existingWikiWord" href="/nlab/show/Thom+space">Thom space</a> of the <a class="existingWikiWord" href="/nlab/show/vector+bundle">vector bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">\tau_n</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/associated+bundle">associated</a> to the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Sigma(n)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/universal+principal+bundle">universal principal bundle</a> via the canonical <a class="existingWikiWord" href="/nlab/show/action">action</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Sigma(n)</annotation></semantics></math> on <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> (see also at <a href="permutation#ClassifyingSpaceAndThomSpace">symmetric group – Classifying space and Thom space</a>).</p> <h2 id="references">References</h2> <p>The proposal for spectral super-geometry <a href="#Definition">above</a> invokes observations from</p> <ul> <li id="Rezk09"> <p><a class="existingWikiWord" href="/nlab/show/Charles+Rezk">Charles Rezk</a>, section 2 of <em>The congruence criterion for power operations in Morava E-theory</em>, Homology, Homotopy and Applications, Vol. 11 (2009), No. 2, pp.327-379 (<a href="https://arxiv.org/abs/0902.2499">arXiv:0902.2499</a>)</p> </li> <li id="SagaveSchlichtkrull11"> <p><a class="existingWikiWord" href="/nlab/show/Steffen+Sagave">Steffen Sagave</a>, <a class="existingWikiWord" href="/nlab/show/Christian+Schlichtkrull">Christian Schlichtkrull</a>, <em>Diagram spaces and symmetric spectra</em>, Advances in Mathematics, Volume 231, Issues 3–4, October–November 2012, Pages 2116–2193 (<a href="https://arxiv.org/abs/1103.2764">arXiv:1103.2764</a>)</p> </li> </ul> <p>The proposal <a href="#Definition">above</a> was originally motivated from the discussion of the <a class="existingWikiWord" href="/nlab/show/sphere+spectrum">sphere spectrum</a> in relation to <a class="existingWikiWord" href="/nlab/show/super+algebra">super algebra</a> highlighted in</p> <ul> <li id="Kapranov13"> <p><a class="existingWikiWord" href="/nlab/show/Mikhail+Kapranov">Mikhail Kapranov</a>, <em>Categorification of supersymmetry and stable homotopy groups of spheres</em>, talk at <em><a href="https://web.archive.org/web/20130617191515/http://mathserver.neu.edu/~bwebster/ACRT/">Algebra, Combinatorics and Representation Theory: in memory of Andrei Zelevinsky (1953-2013)</a></em> (April 2013) [<a class="existingWikiWord" href="/nlab/files/Webster-ACRTAbstracts2013.pdf" title="pdf">pdf</a>, video:<a href="https://www.youtube.com/watch?v=WBsZrcYOrxI">YT</a>]</p> <blockquote> <p><strong>Abstract:</strong>. The “minimal sign skeleton” necessary to formulate the <a class="existingWikiWord" href="/nlab/show/signs+in+supergeometry">Koszul sign rule</a> is a certain <a class="existingWikiWord" href="/nlab/show/Picard+category">Picard category</a>, a <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a> with all <a class="existingWikiWord" href="/nlab/show/invertible+object">objects</a> and <a class="existingWikiWord" href="/nlab/show/invertible+morphism">morphisms invertible</a>. It can be seen as the free Picard category generated by one object and corresponds, by <a class="existingWikiWord" href="/nlab/show/Grothendieck">Grothendieck</a>‘s dictionary, to the <a class="existingWikiWord" href="/nlab/show/n-truncation">truncation</a> of the <a class="existingWikiWord" href="/nlab/show/sphere+spectrum">spherical spectrum</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> in degrees 0 and 1, so that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mo>±</mo><mn>1</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{\pm 1\}</annotation></semantics></math> appears as the <a class="existingWikiWord" href="/nlab/show/first+stable+homotopy+group+of+spheres">first stable homotopy group of spheres</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><msup><mi>S</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_{n+1}(S^n)</annotation></semantics></math>. This suggest a “<a class="existingWikiWord" href="/nlab/show/higher+structures">higher</a>” or <a class="existingWikiWord" href="/nlab/show/categorification">categorified</a> versions of <a class="existingWikiWord" href="/nlab/show/super-algebra">super-mathematics</a> which utilize deeper structure of <a class="existingWikiWord" href="/nlab/show/sphere+spectrum"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>S</mi> </mrow> <annotation encoding="application/x-tex">S</annotation> </semantics> </math></a>. The first concept on this path is that of a supersymmetric monoidal category which is <a class="existingWikiWord" href="/nlab/show/categorification">categorified</a> version of the concept of a <a class="existingWikiWord" href="/nlab/show/supercommutative+algebra">supercommutative algebra</a>.</p> </blockquote> </li> <li id="Kapranov15"> <p><a class="existingWikiWord" href="/nlab/show/Mikhail+Kapranov">Mikhail Kapranov</a>, <em>Supergeometry in mathematics and physics</em>, in <a class="existingWikiWord" href="/nlab/show/Gabriel+Catren">Gabriel Catren</a>, <a class="existingWikiWord" href="/nlab/show/Mathieu+Anel">Mathieu Anel</a>, (eds.) <em><a class="existingWikiWord" href="/nlab/show/New+Spaces+for+Mathematics+and+Physics">New Spaces for Mathematics and Physics</a></em> (<a href="http://arxiv.org/abs/1512.07042">arXiv:1512.07042</a>)</p> </li> <li id="Kapranov15b"> <p><a class="existingWikiWord" href="/nlab/show/Mikhail+Kapranov">Mikhail Kapranov</a>, <em>Super-geometry</em>, talk at <em><a href="http://ercpqg-espace.sciencesconf.org/program">New Spaces for Mathematics & Physics</a></em>, IHP Paris (Oct-Sept 2015) [<a href="https://www.youtube.com/watch?v=bjsNwKYT8JE">video recording</a>]</p> </li> </ul> <p>A closely related suggestion later appears in:</p> <ul> <li>MathOverflow comment <a href="https://mathoverflow.net/a/403274/381">MO:a/40327</a>, Sep 2021</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on October 25, 2023 at 11:24:41. 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