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width: 0.3em;"></span> <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/6623/#Item_49" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Supergeometry</title></head> <body> <blockquote> <p>this entry is one section of “<a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+supergeometry+and+superphysics">geometry of physics – supergeometry and superphysics</a>”</p> <p>which is one chapter of “<a class="existingWikiWord" href="/nlab/show/geometry+of+physics">geometry of physics</a>”</p> <p>previous sections: <em><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+superalgebra">superalgebra</a></em>, <em><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+categories+and+toposes">categories and toposes</a></em>, <em><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+smooth+sets">smooth sets</a></em></p> <p>following sections: <em><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+supersymmetry">geometry of physics – supersymmetry</a></em>, <em><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+smooth+homotopy+types">geometry of physics – smooth homotopy types</a></em></p> </blockquote> <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><a class="existingWikiWord" href="/nlab/show/supergeometry">Supergeometry</a> is the generalization of <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a> (or <a class="existingWikiWord" href="/nlab/show/algebraic+geometry">algebraic geometry</a>) to the situation where <a class="existingWikiWord" href="/nlab/show/algebras+of+functions">algebras of functions</a> are generalized from <a class="existingWikiWord" href="/nlab/show/commutative+algebras">commutative algebras</a> to <a class="existingWikiWord" href="/nlab/show/supercommutative+superalgebras">supercommutative superalgebras</a>.</p> <p>In <em><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+superalgebra">geometry of physics – superalgebra</a></em> we discussed why it is mathematically compelling to pass to <a class="existingWikiWord" href="/nlab/show/supercommutative+superalgebra">supercommutative superalgebra</a> and how this implies a dual concept of <a class="existingWikiWord" href="/nlab/show/superspace">superspace</a> in terms of <a class="existingWikiWord" href="/nlab/show/affine+superschemes">affine superschemes</a>. Here we discuss how to build a fully-fledged theory of <a class="existingWikiWord" href="/nlab/show/geometry">geometry</a> on these affine <a class="existingWikiWord" href="/nlab/show/superspaces">superspaces</a> – <a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a> – in parallel to the discussion of ordinary <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a> in <em><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+smooth+sets">geometry of physics – smooth sets</a></em>.</p> <div style="float:right;margin:0 20px 10px 20px;"> <a href="#ProgressionOfIdempotentEndofunctors"> <img src="http://ncatlab.org/schreiber/files/ProgressionOfModalities.jpg" width="200" /> </a> </div> <p>Apart from the abstract mathematical motivation for supergeometry, it is also an <a class="existingWikiWord" href="/nlab/show/experimental+observation">experimental</a> fact that the <a class="existingWikiWord" href="/nlab/show/observable+universe">observable universe</a> is fundamentally described by <a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a>. Namely the <a class="existingWikiWord" href="/nlab/show/Pauli+exclusion+principle">Pauli exclusion principle</a>, in its refined form of the <a class="existingWikiWord" href="/nlab/show/spin-statistics+theorem">spin-statistics theorem</a>, implies that the <a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a> of a <a class="existingWikiWord" href="/nlab/show/field+theory">field theory</a> (already of a <a class="existingWikiWord" href="/nlab/show/classical+field+theory">classical field theory</a>) with <a class="existingWikiWord" href="/nlab/show/fermion">fermion</a> <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">fields</a> (such as that of <a class="existingWikiWord" href="/nlab/show/electrons">electrons</a> and <a class="existingWikiWord" href="/nlab/show/quarks">quarks</a>) is a <a class="existingWikiWord" href="/nlab/show/superspace">superspace</a> whose even-graded <a class="existingWikiWord" href="/nlab/show/coordinates">coordinates</a> are the configurations of the <a class="existingWikiWord" href="/nlab/show/boson">boson</a> fields, while the odd-graded coordinates correspond to the configurations of the <a class="existingWikiWord" href="/nlab/show/fermion">fermion</a> fields. It is impossible to have an <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a> for fermion fields as a function on a non-super (<a class="existingWikiWord" href="/nlab/show/infinite-dimensional+manifold">infinite-dimensional</a>)-<a class="existingWikiWord" href="/nlab/show/manifold">manifold</a>.</p> <p>This is an old insight: The experimental detection of the special properties of <a class="existingWikiWord" href="/nlab/show/fermions">fermions</a> that show their super-geometric nature goes back all the way to the <a class="existingWikiWord" href="/nlab/show/Stern-Gerlach+experiment">Stern-Gerlach experiment</a> in 1922, which revealed that <a class="existingWikiWord" href="/nlab/show/electrons">electrons</a> are <a class="existingWikiWord" href="/nlab/show/spinors">spinors</a>. The <a class="existingWikiWord" href="/nlab/show/Pauli+exclusion+principle">Pauli exclusion principle</a> (<a href="#Pauli25">Pauli 1925</a>) – <a class="existingWikiWord" href="/nlab/show/deduction">deduced</a> from the nature of energy levels of <a class="existingWikiWord" href="/nlab/show/electrons">electrons</a> in <a class="existingWikiWord" href="/nlab/show/atoms">atoms</a> – says that no two such <a class="existingWikiWord" href="/nlab/show/spinors">spinors</a> may occupy the exact same <a class="existingWikiWord" href="/nlab/show/quantum+state">quantum state</a>. Technically this says that the spinor <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field</a> variable <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ψ</mi></mrow><annotation encoding="application/x-tex">\psi</annotation></semantics></math> has to satisfy</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ψ</mi><mi>ψ</mi><mo>=</mo><mn>0</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \psi \psi = 0 \,. </annotation></semantics></math></div> <p>A little later it was realized that the <a class="existingWikiWord" href="/nlab/show/axioms">axioms</a> of <a class="existingWikiWord" href="/nlab/show/local+field+theory">local field theory</a> <em>imply</em> that all <a class="existingWikiWord" href="/nlab/show/spinor">spinor</a> fields need to be <a class="existingWikiWord" href="/nlab/show/fermions">fermions</a> in that for any two <a class="existingWikiWord" href="/nlab/show/classical+field+theory">classical</a> spinor field variables <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ψ</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\psi_1</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ψ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\psi_2</annotation></semantics></math> the Grassmann sign rule holds (<a href="geometry+of+physics+--+-superalgebra#Grassmann1844">Grassmann 1844</a>):</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ψ</mi> <mn>1</mn></msub><msub><mi>ψ</mi> <mn>2</mn></msub><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mi>ψ</mi> <mn>2</mn></msub><msub><mi>ψ</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex"> \psi_1 \psi_2 = - \psi_2 \psi_1 </annotation></semantics></math></div> <p>(which of course immediately implies the <a class="existingWikiWord" href="/nlab/show/Pauli+exclusion+principle">Pauli exclusion principle</a>). This is the celebrated <em><a class="existingWikiWord" href="/nlab/show/spin-statistics+theorem">spin-statistics theorem</a></em>, whose formulation goes back to <a href="#Fierz39">Fierz in 1939</a> and <a href="#Pauli40">Pauli in 1940</a>. And that sign rule is of course precisely the sign rule in a <a class="existingWikiWord" href="/nlab/show/supercommutative+superalgebra">supercommutative superalgebra</a>, for the fermion field observables <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ψ</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\psi_i</annotation></semantics></math> being odd-graded functions on a supergeometric <a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a>.</p> <p>Notice that this phenomenon is not a negligible subtlety: the <a class="existingWikiWord" href="/nlab/show/Pauli+exclusion+principle">Pauli exclusion principle</a> is what implies <a class="existingWikiWord" href="/nlab/show/stability+of+matter">stability of matter</a> by forcing <a class="existingWikiWord" href="/nlab/show/electrons">electrons</a> in an <a class="existingWikiWord" href="/nlab/show/atom">atom</a> to fill up “orbitals” consecutively as opposed to all falling into the ground state, as a bosonic <a class="existingWikiWord" href="/nlab/show/condensate">condensate</a> would do. There would be no <a class="existingWikiWord" href="/nlab/show/solid+state+physics">solid state physics</a> without supergeometry of <a class="existingWikiWord" href="/nlab/show/phase+spaces">phase spaces</a>, no <a class="existingWikiWord" href="/nlab/show/solidity">solid</a> <a class="existingWikiWord" href="/nlab/show/matter">matter</a>. (In addition, the Pauli <a class="existingWikiWord" href="/nlab/show/degeneracy+pressure">degeneracy pressure</a> controls more exotic phenomena, for instance the stability of neutron stars against their gravitational collapse.)</p> <p>As a slogan:</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,\,\,\,\,\;</annotation></semantics></math> <em><strong>The <a class="existingWikiWord" href="/nlab/show/geometry">geometry</a> of <a class="existingWikiWord" href="/nlab/show/phase+spaces">phase spaces</a> for <a class="existingWikiWord" href="/nlab/show/fermions">fermions</a>,</strong></em></p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,\,\,\,\,\;</annotation></semantics></math> <em><strong>hence for <a class="existingWikiWord" href="/nlab/show/matter">matter</a> fields admitting <a class="existingWikiWord" href="/nlab/show/solid+state+physics">solid states</a>,</strong></em></p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,\,\,\,\,\;</annotation></semantics></math> <em><strong>is <a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a>.</strong></em></p> <p>Nevertheless, few textbooks make the supergeometric aspect in the physics of fermions explicit. Some discuss it only after <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a> (and some will even claim that fermion fields do not exist classically); some only discuss it in the context of <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> <a class="existingWikiWord" href="/nlab/show/supersymmetry">supersymmetry</a>. But notice that <em><a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a></em> is a subject in itself even without presence or mentioning of spacetime <a class="existingWikiWord" href="/nlab/show/supersymmetry">supersymmetry</a> (which is <a class="existingWikiWord" href="/nlab/show/super+Poincar%C3%A9+group">super Poincaré group</a> <a class="existingWikiWord" href="/nlab/show/symmetry">symmetry</a>), just as <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a> is a subject in itself even without the presence or mentioning of <a class="existingWikiWord" href="/nlab/show/Poincar%C3%A9+group">Poincaré group</a> symmetry. Hence this we discuss independently in <em><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+supersymmetry">geometry of physics – supersymmetry</a></em>.</p> <p>Texts which do make the super-geometric nature of <a class="existingWikiWord" href="/nlab/show/fermion+fields">fermion fields</a> explicit include</p> <ul> <li> <p><a href="#DeligneFreed99">Deligne-Freed 99, §3.4</a>, <a href="#Freed01">Freed 01, lecture 4</a>,</p> </li> <li> <p><a href="#GiachettaMangiarottiSardanashvily09">Giachetta-Mangiarotti-Sardanashvily 09, chapter 3</a>,</p> </li> <li> <p><a href="#Sardanashvily12">Sardanashvily 12</a>, <a href="#Sardanashvily16">Sardanashvily 16</a>.</p> </li> </ul> <p>This we discuss elsewhere.</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><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> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="supergeometry">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> </div> </div> <h1 id="supergeometry_2">Supergeometry</h1> <div class='maruku_toc'> <ul> <li><a href='#CoordinareSystemsSuperCartesianSpaces'>Super Cartesian spaces</a></li> <li><a href='#SuperSmoothSets'>Super smooth sets</a></li> <li><a href='#SuperMappingSpaces'>Super mapping spaces</a></li> <li><a href='#Supermanifolds'>Supermanifolds</a></li> <li><a href='#DeRhamComplexOfSuperDifferentialForms'>Super differential forms</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#general_theory'>General theory</a></li> <li><a href='#for_classical_field_theories_with_fermions'>For classical field theories with fermions</a></li> <li><a href='#ReferencesOverSuperpoints'>In the topos over superpoints</a></li> <li><a href='#in_the_topos_over_super_cartesian_spaces'>In the topos over super Cartesian spaces</a></li> </ul> </ul> </div> <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><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>In the section <em><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+superalgebra">geometry of physics – superalgebra</a></em> we had discussed the <a class="existingWikiWord" href="/nlab/show/category">category</a> of <strong><a class="existingWikiWord" href="/nlab/show/affine+super+schemes">affine super schemes</a></strong> (in <a href="geometry+of+physics+--+superalgebra#AffineSuperSchemes">this definition</a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Var</mi><mo stretchy="false">(</mo><msub><mi>sVect</mi> <mi>ℝ</mi></msub><mo stretchy="false">)</mo><mo>≔</mo><msubsup><mi>sCalg</mi> <mi>k</mi> <mi>op</mi></msubsup><mo>=</mo><mi>CMon</mi><mo stretchy="false">(</mo><msub><mi>sVect</mi> <mi>ℝ</mi></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Var(sVect_{\mathbb{R}}) \coloneqq sCalg_k^{op} = CMon(sVect_{\mathbb{R}}) \,. </annotation></semantics></math></div> <p>Here we use these as local model spaces to develop <a class="existingWikiWord" href="/nlab/show/geometry">geometry</a> locally modeled on affine superschemes. Since we are interested in <a class="existingWikiWord" href="/nlab/show/physics">physics</a>, and since <a class="existingWikiWord" href="/nlab/show/spacetimes">spacetimes</a> and <a class="existingWikiWord" href="/nlab/show/phase+spaces">phase spaces</a> in physics are objects in <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a>, not in <a class="existingWikiWord" href="/nlab/show/algebraic+geometry">algebraic geometry</a> (even if sometimes they may come from there), we consider the <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> of <strong><a class="existingWikiWord" href="/nlab/show/super+Cartesian+spaces">super Cartesian spaces</a></strong> (<a href="geometry+of+physics+--+superalgebra#SuperCartesianSpace">this example</a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>SuperCartSp</mi><mo>=</mo><mo stretchy="false">{</mo><msup><mi>ℝ</mi> <mrow><mi>p</mi><mo stretchy="false">|</mo><mi>q</mi></mrow></msup><msub><mo stretchy="false">}</mo> <mrow><mi>p</mi><mo>,</mo><mi>q</mi><mo>∈</mo><mi>ℕ</mi></mrow></msub><mo>↪</mo><mi>Aff</mi><mo stretchy="false">(</mo><msub><mi>sVect</mi> <mi>k</mi></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> SuperCartSp = \{\mathbb{R}^{p\vert q}\}_{p,q \in \mathbb{N}} \hookrightarrow Aff(sVect_k) \,, </annotation></semantics></math></div> <p>whose <a class="existingWikiWord" href="/nlab/show/algebras+of+functions">algebras of functions</a> by definition are <a class="existingWikiWord" href="/nlab/show/tensor+products">tensor products</a> (over the <a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a> <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">\mathbb{R}</annotation></semantics></math>) 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">\mathbb{R}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/associative+algebras">algebras</a> of <a class="existingWikiWord" href="/nlab/show/smooth+functions">smooth functions</a> with <a class="existingWikiWord" href="/nlab/show/Grassmann+algebras">Grassmann algebras</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝒪</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mi>p</mi><mo stretchy="false">|</mo><mi>q</mi></mrow></msup><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>p</mi></msup><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mi>ℝ</mi></msub><msup><mo>∧</mo> <mo>•</mo></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mo>*</mo></msup><msup><mo stretchy="false">)</mo> <mi>q</mi></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathcal{O}(\mathbb{R}^{p\vert q}) \;\coloneqq\; C^\infty(\mathbb{R}^p) \otimes_{\mathbb{R}} \wedge^\bullet (\mathbb{R}^\ast)^q \,. </annotation></semantics></math></div> <p>These serve as our “abstract super-coordinate systems” that define <a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a> in direct analogy to how ordinary <a class="existingWikiWord" href="/nlab/show/Cartesian+spaces">Cartesian spaces</a> serve as the abstract coordinate systems that define <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a> as found at <em><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+coordinate+systems">geometry of physics – coordinate systems</a></em> and_<a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+smooth+sets">geometry of physics – smooth sets</a><em>.</em></p> <p>Then a general superspace is modeled as a <a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a> on the category of super Cartesian spaces, possibly satisfying some suitable properties. see remark <a class="maruku-ref" href="#ASheafAsASpace"></a> below for explanation of this perspective.</p> <p>This means that we follow the perspective of “<strong><a class="existingWikiWord" href="/nlab/show/functorial+geometry">functorial geometry</a></strong>” due to <a href="#Grothendieck65">Grothendieck 65</a>, where a <a class="existingWikiWord" href="/nlab/show/scheme">scheme</a> is regarded as a <a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a> over the category of <a class="existingWikiWord" href="/nlab/show/affine+schemes">affine schemes</a> (its “<a class="existingWikiWord" href="/nlab/show/functor+of+points">functor of points</a>”) satisfying the condition that it is covered by (<a class="existingWikiWord" href="/nlab/show/representable+functor">representables</a> of) affines via <a class="existingWikiWord" href="/nlab/show/formally+%C3%A9tale+morphism">formally étale morphism</a>. We explain all this below.</p> <p>Beware that – despite the urging in <a href="#Grothendieck73">Grothendieck 73</a> that the definition of <a class="existingWikiWord" href="/nlab/show/schemes">schemes</a> as <a class="existingWikiWord" href="/nlab/show/locally+ringed+spaces">locally ringed spaces</a> should be abandoned in favour of the perspective of <a class="existingWikiWord" href="/nlab/show/functorial+geometry">functorial geometry</a> – most textbooks on <a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a> do stick with the point of view of <a class="existingWikiWord" href="/nlab/show/locally+ringed+spaces">locally ringed spaces</a>. (One might argue that in smooth supergeometry this is a particularly heavy violation of <a href="#Grothendieck73">Grothendieck’s urging</a>, since, in contrast to algebraic schemes, every supermanifold is already affine.)</p> <p>An exception is the approach propagated in <a href="#Schwarz84">Schwarz 84</a>, <a href="#Molotkov84">Molotkov 84</a>, <a href="KonechnySchwarz97">Konechny-Schwarz 97</a> of which a clean account is given in <a href="#Sachse08">Sachse 08</a>. These authors consider (pre-)sheaves on the category of <a class="existingWikiWord" href="/nlab/show/superpoints">superpoints</a>. This gives the “super” in “super-geometry” a functorial interpretation, but the (smooth) “geometry” still needs to be added in by hand. Hence this approach satisfies <a href="#Grothendieck73">Grothendieck’s urging</a> half-way.</p> <p>The full application of the perspective of <a class="existingWikiWord" href="/nlab/show/functorial+geometry">functorial geometry</a> to <a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a> is known as <em><a class="existingWikiWord" href="/nlab/show/synthetic+differential+supergeometry">synthetic differential supergeometry</a></em>, where one considers sheaves over the full category of formal <a class="existingWikiWord" href="/nlab/show/super+Cartesian+spaces">super Cartesian spaces</a> (<a href="#Yetter88">Yetter 88, section 3</a>). This is essentially the perspective which we adopt here. We do however not refer to the (super-)<a class="existingWikiWord" href="/nlab/show/Kock-Lawvere+axioms">Kock-Lawvere axioms</a> for <a class="existingWikiWord" href="/nlab/show/synthetic+differential+geometry">synthetic differential geometry</a> but instead use the axiomatics of “<a class="existingWikiWord" href="/nlab/show/differential+cohesion">differential cohesion</a>” (<a href="#Schreiber">Schreiber 13</a>). This we explain <a href="#SuperSmoothSets">below</a>.</p> <h2 id="CoordinareSystemsSuperCartesianSpaces">Super Cartesian spaces</h2> <p>For reference, recall:</p> <div class="num_defn" id="SuperCartesianSpace"> <h6 id="definition">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/Grassmann+algebra">Grassmann algebra</a>)</strong></p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">q\in \mathbb{N}</annotation></semantics></math>, the real <a class="existingWikiWord" href="/nlab/show/Grassmann+algebra">Grassmann algebra</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mn>0</mn><mo stretchy="false">|</mo><mi>q</mi></mrow></msup><mo stretchy="false">)</mo><mo>≔</mo><msup><mo>∧</mo> <mo>•</mo></msup><mo stretchy="false">⟨</mo><msub><mi>θ</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>θ</mi> <mi>q</mi></msub><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex"> C^\infty(\mathbb{R}^{0|q}) \coloneqq \wedge^\bullet \langle \theta_1, \cdots, \theta_q\rangle </annotation></semantics></math></div> <p>is the <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">\mathbb{R}</annotation></semantics></math>-algebra <a class="existingWikiWord" href="/nlab/show/free+functor">freely</a> generated from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi></mrow><annotation encoding="application/x-tex">q</annotation></semantics></math> generators <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>θ</mi> <mi>i</mi></msub><msubsup><mo stretchy="false">}</mo> <mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow> <mi>q</mi></msubsup></mrow><annotation encoding="application/x-tex">\{\theta_i\}_{i = 1}^q</annotation></semantics></math> (now regarded as being in odd degree), subject to the relations</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>θ</mi> <mi>i</mi></msub><msub><mi>θ</mi> <mi>j</mi></msub><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mi>θ</mi> <mi>j</mi></msub><msub><mi>θ</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex"> \theta_i \theta_j = - \theta_j \theta_i </annotation></semantics></math></div> <p>for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>∈</mo><mo stretchy="false">{</mo><mn>1</mn><mo>,</mo><mi>⋯</mi><mo>,</mo><mi>q</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">i,j \in \{1,\cdots, q\}</annotation></semantics></math>. In particular</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>θ</mi> <mi>i</mi></msub><msub><mi>θ</mi> <mi>i</mi></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> \theta_i \theta_i = 0 </annotation></semantics></math></div> <p>for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>∈</mo><mo stretchy="false">{</mo><mn>1</mn><mo>,</mo><mi>⋯</mi><mo>,</mo><mi>q</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">i \in \{1, \cdots, q\}</annotation></semantics></math> (since <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">\mathbb{R}</annotation></semantics></math> has <a class="existingWikiWord" href="/nlab/show/characteristic+zero">characteristic zero</a> and in particular <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>char</mi><mo stretchy="false">(</mo><mi>ℝ</mi><mo stretchy="false">)</mo><mo>≠</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">char(\mathbb{R}) \neq 2</annotation></semantics></math>).</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>,</mo><mi>q</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">p,q \in \mathbb{N}</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/super-Cartesian+space">super-Cartesian space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mi>p</mi><mo stretchy="false">|</mo><mi>q</mi></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{p|q}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/formal+dual">formal dual</a> of the <a class="existingWikiWord" href="/nlab/show/supercommutative+superalgebra">supercommutative superalgebra</a> written <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒪</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mi>p</mi><mo stretchy="false">|</mo><mi>q</mi></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{O}(\mathbb{R}^{p\vert q})</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mi>p</mi><mo stretchy="false">|</mo><mi>q</mi></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^\infty(\mathbb{R}^{p|q})</annotation></semantics></math> whose underlying <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><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}/2\mathbb{Z}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/graded+vector+space">graded vector space</a> is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mi>p</mi><mo stretchy="false">|</mo><mi>q</mi></mrow></msup><mo stretchy="false">)</mo><mo>≔</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>p</mi></msup><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mi>ℝ</mi></msub><msup><mo>∧</mo> <mo>•</mo></msup><mo stretchy="false">⟨</mo><msub><mi>θ</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>θ</mi> <mi>q</mi></msub><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex"> C^\infty(\mathbb{R}^{p|q}) \coloneqq C^\infty(\mathbb{R}^p) \otimes_{\mathbb{R}} \wedge^\bullet\langle\theta_1, \cdots, \theta_q\rangle </annotation></semantics></math></div> <p>with the product given by the relations</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><mrow><mo>(</mo><mi>f</mi><msub><mi>θ</mi> <mrow><msub><mi>i</mi> <mn>1</mn></msub></mrow></msub><mi>⋯</mi><msub><mi>θ</mi> <mrow><msub><mi>i</mi> <mi>k</mi></msub></mrow></msub><mo>)</mo></mrow><mrow><mo>(</mo><mi>g</mi><msub><mi>θ</mi> <mrow><msub><mi>j</mi> <mn>1</mn></msub></mrow></msub><mi>⋯</mi><msub><mi>θ</mi> <mrow><msub><mi>j</mi> <mi>l</mi></msub></mrow></msub><mo>)</mo></mrow></mtd> <mtd><mo>=</mo><mi>f</mi><mo>⋅</mo><mi>g</mi><mspace width="thickmathspace"></mspace><msub><mi>θ</mi> <mrow><msub><mi>i</mi> <mn>1</mn></msub></mrow></msub><mi>⋯</mi><msub><mi>θ</mi> <mrow><msub><mi>i</mi> <mi>k</mi></msub></mrow></msub><msub><mi>θ</mi> <mrow><msub><mi>j</mi> <mn>1</mn></msub></mrow></msub><mi>⋯</mi><msub><mi>θ</mi> <mrow><msub><mi>j</mi> <mi>l</mi></msub></mrow></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><mi>k</mi><mi>l</mi></mrow></msup><mi>g</mi><mo>⋅</mo><mi>f</mi><mspace width="thickmathspace"></mspace><msub><mi>θ</mi> <mrow><msub><mi>j</mi> <mn>1</mn></msub></mrow></msub><mi>⋯</mi><msub><mi>θ</mi> <mrow><msub><mi>j</mi> <mi>l</mi></msub></mrow></msub><msub><mi>θ</mi> <mrow><msub><mi>i</mi> <mn>1</mn></msub></mrow></msub><mi>⋯</mi><msub><mi>θ</mi> <mrow><msub><mi>i</mi> <mi>k</mi></msub></mrow></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \left( f \theta_{i_1}\cdots \theta_{i_k} \right) \left( g \theta_{j_1}\cdots \theta_{j_l} \right) & = f \cdot g \; \theta_{i_1}\cdots \theta_{i_k} \theta_{j_1}\cdots \theta_{j_l} \\ & = (-1)^{k l} g\cdot f \; \theta_{j_1}\cdots \theta_{j_l} \theta_{i_1}\cdots \theta_{i_k} \end{aligned} </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>⋅</mo><mi>g</mi></mrow><annotation encoding="application/x-tex">f \cdot g</annotation></semantics></math> is the ordinary pointwise product of <a class="existingWikiWord" href="/nlab/show/smooth+functions">smooth functions</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mi>p</mi><mo stretchy="false">|</mo><mi>q</mi></mrow></msup><mo>≔</mo><mi>Spec</mi><mo stretchy="false">(</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>p</mi></msup><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mi>ℝ</mi></msub><msup><mo>∧</mo> <mo>•</mo></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>q</mi></msup><msup><mo stretchy="false">)</mo> <mo>*</mo></msup><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbb{R}^{p \vert q} \coloneqq Spec( C^\infty(\mathbb{R}^p) \otimes_{\mathbb{R}} \wedge^\bullet (\mathbb{R}^q)^\ast ) \,. </annotation></semantics></math></div> <p>Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>SuperCartSp</mi><mo>↪</mo><msubsup><mi>sCAlg</mi> <mi>ℝ</mi> <mi>op</mi></msubsup></mrow><annotation encoding="application/x-tex"> SuperCartSp \hookrightarrow sCAlg_{\mathbb{R}}^{op} </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> of the <a class="existingWikiWord" href="/nlab/show/opposite+category">opposite category</a> of <a class="existingWikiWord" href="/nlab/show/commutative+superalgebras">commutative superalgebras</a> on those of this form. We write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mi>p</mi><mo stretchy="false">|</mo><mi>q</mi></mrow></msup><mo>∈</mo><mi>SuperCartSp</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}^{p|q} \in SuperCartSp</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/formal+dual">formal dual</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mi>p</mi><mo stretchy="false">|</mo><mi>q</mi></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^\infty(\mathbb{R}^{p|q})</annotation></semantics></math>.</p> </div> <p>Moreover, we write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>CartSp</mi><mover><mo>↪</mo><mphantom><mi>AAAA</mi></mphantom></mover><mi>SuperCartSp</mi></mrow><annotation encoding="application/x-tex"> CartSp \overset{\phantom{AAAA}}{\hookrightarrow} SuperCartSp </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> on ordinary <a class="existingWikiWord" href="/nlab/show/Cartesian+spaces">Cartesian spaces</a> with <a class="existingWikiWord" href="/nlab/show/smooth+functions">smooth functions</a> between them. These are the “abstract coordinate charts” from the discussion at <em><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+smooth+sets">geometry of physics – smooth sets</a></em>, and so we are evidently entitled to think of the objects in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SuperCartSp</mi></mrow><annotation encoding="application/x-tex">SuperCartSp</annotation></semantics></math> as <strong>abstract super coordinate systems</strong> and to develop a geometry induced from these.</p> <p>Recall the three magic algebraic properties of <a class="existingWikiWord" href="/nlab/show/smooth+functions">smooth functions</a> that make the above algebraic description of <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a> work:</p> <div class="num_prop" id="FirstTwoMagicPropertiesOfAlgebrasOfSmoothFunctions"> <h6 id="proposition">Proposition</h6> <p><strong>(first two magic properties of <a class="existingWikiWord" href="/nlab/show/algebra+of+functions">algebras of</a> <a class="existingWikiWord" href="/nlab/show/smooth+functions">smooth functions</a>)</strong></p> <ol> <li> <p>(<strong><a class="existingWikiWord" href="/nlab/show/embedding+of+smooth+manifolds+into+formal+duals+of+R-algebras">embedding of smooth manifolds into formal duals of R-algebras</a></strong>)</p> <p>The <a class="existingWikiWord" href="/nlab/show/functor">functor</a> that assigns <a class="existingWikiWord" href="/nlab/show/algebra+of+functions">algebras of</a> <a class="existingWikiWord" href="/nlab/show/smooth+functions">smooth functions</a> to <a class="existingWikiWord" href="/nlab/show/smooth+manifolds">smooth manifolds</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>SmthMfd</mi><mover><mo>↪</mo><mphantom><mi>AAAA</mi></mphantom></mover><msubsup><mi>CAlg</mi> <mi>ℝ</mi> <mi>op</mi></msubsup><mover><mo>↪</mo><mphantom><mi>AAAA</mi></mphantom></mover><msubsup><mi>sCAlg</mi> <mi>ℝ</mi> <mi>op</mi></msubsup></mrow><annotation encoding="application/x-tex"> C^\infty(-) \;\colon\; SmthMfd \overset{\phantom{AAAA}}{\hookrightarrow} CAlg_{\mathbb{R}}^{op} \overset{\phantom{AAAA}}{\hookrightarrow} sCAlg_{\mathbb{R}}^{op} </annotation></semantics></math></div> <p>is <a class="existingWikiWord" href="/nlab/show/fully+faithful+functor">fully faithful</a> (<a href="geometry+of+physics+--+categories+and+toposes#FullyFaithfulFunctor">this Def.</a>).</p> </li> <li> <p>(<strong><a class="existingWikiWord" href="/nlab/show/smooth+Serre-Swan+theorem">smooth Serre-Swan theorem</a></strong>)</p> <p>The functor that assigns smooth <a class="existingWikiWord" href="/nlab/show/sections">sections</a> of smooth <a class="existingWikiWord" href="/nlab/show/vector+bundles">vector bundles</a> of <a class="existingWikiWord" href="/nlab/show/finite+number">finite</a> <a class="existingWikiWord" href="/nlab/show/rank">rank</a></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>X</mi></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>SmthVectBund</mi> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow></msub><mo>↪</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mi>Mod</mi></mrow><annotation encoding="application/x-tex"> \Gamma_X(-) \;\colon\; SmthVectBund_{/X} \hookrightarrow C^\infty(X) Mod </annotation></semantics></math></div> <p>is <a class="existingWikiWord" href="/nlab/show/fully+faithful+functor">fully faithful</a> (its <a class="existingWikiWord" href="/nlab/show/essential+image">essential image</a> being the <a class="existingWikiWord" href="/nlab/show/finitely+generated+object">finitely generated</a> <a class="existingWikiWord" href="/nlab/show/projective+modules">projective modules</a> over the <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">\mathbb{R}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/algebra+of+functions">algebra of</a> <a class="existingWikiWord" href="/nlab/show/smooth+functions">smooth functions</a>).</p> <p>(<a href="smooth+Serre-Swan+theorem#Nestruev03">Nestruev 03, theorem 11.32</a>)</p> <p>Moreover, the <a class="existingWikiWord" href="/nlab/show/modules">modules</a> over the <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">\mathbb{R}</annotation></semantics></math>-algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^\infty(X)</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/smooth+functions">smooth functions</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> which arise this way as <a class="existingWikiWord" href="/nlab/show/sections">sections</a> of <a class="existingWikiWord" href="/nlab/show/smooth+vector+bundles">smooth vector bundles</a> over a <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><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> are precisely the <a class="existingWikiWord" href="/nlab/show/finitely+generated+module">finitely generated</a> <a class="existingWikiWord" href="/nlab/show/free+modules">free modules</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^\infty(X)</annotation></semantics></math>.</p> <p>(<a href="smooth+Serre-Swan+theorem#Nestruev03">Nestruev 03, theorem 11.32</a>)</p> </li> </ol> </div> <p>There is a third such magic algebraic property of smooth functions, which plays a role now:</p> <div class="num_defn" id="DerivationsOfSmoothFunctions"> <h6 id="proposition_2">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/derivations+of+smooth+functions+are+vector+fields">derivations of smooth functions are vector fields</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a>. Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>der</mi> <mi>X</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>Γ</mi><mo stretchy="false">(</mo><mi>T</mi><mi>X</mi><mo stretchy="false">)</mo><mo>⟶</mo><mi>Der</mi><mo stretchy="false">(</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> der_X \;\colon\; \Gamma(T X) \longrightarrow Der(C^\infty(X)) </annotation></semantics></math></div> <p>for the function that sends a smooth <a class="existingWikiWord" href="/nlab/show/vector+field">vector field</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi><mo>∈</mo><mi>Γ</mi><mo stretchy="false">(</mo><mi>T</mi><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">v \in \Gamma(T X)</annotation></semantics></math> to the <a class="existingWikiWord" href="/nlab/show/derivation">derivation</a> of the <a class="existingWikiWord" href="/nlab/show/algebra+of+functions">algebra of</a> <a class="existingWikiWord" href="/nlab/show/smooth+functions">smooth functions</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> given by forming <a class="existingWikiWord" href="/nlab/show/derivatives">derivatives</a>: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>der</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>≔</mo><mi>v</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">der_X(v)(f) \coloneqq v(f)</annotation></semantics></math>. This is a <a class="existingWikiWord" href="/nlab/show/derivation">derivation</a> by the <a class="existingWikiWord" href="/nlab/show/chain+rule">chain rule</a>.</p> <p>Then this function is a <a class="existingWikiWord" href="/nlab/show/bijection">bijection</a>, hence every <a class="existingWikiWord" href="/nlab/show/derivation">derivation</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>∈</mo><msub><mi>CAlg</mi> <mi>ℝ</mi></msub></mrow><annotation encoding="application/x-tex">C^\infty(X) \in CAlg_{\mathbb{R}}</annotation></semantics></math> comes from <a class="existingWikiWord" href="/nlab/show/differentiation">differentiation</a> along some smooth vector field, which is uniquely defined thereby.</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>By the existence of <a class="existingWikiWord" href="/nlab/show/partitions+of+unity">partitions of unity</a> we may restrict to the situation where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>=</mo><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">X = \mathbb{R}^n</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a>. By the <a class="existingWikiWord" href="/nlab/show/Hadamard+lemma">Hadamard lemma</a> every <a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>∈</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f \in C^\infty(\mathbb{R}^n)</annotation></semantics></math> may be written as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>+</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mi>i</mi></munder><msub><mi>x</mi> <mi>i</mi></msub><msub><mi>g</mi> <mi>i</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> f(x) = f(0) + \sum_i x_i g_i(x) </annotation></semantics></math></div> <p>for <a class="existingWikiWord" href="/nlab/show/smooth+functions">smooth functions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>g</mi> <mi>i</mi></msub><mo>∈</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{g_i \in C^\infty(X)\}</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mi>i</mi></msub><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mo>∂</mo><mi>f</mi></mrow><mrow><mo>∂</mo><msub><mi>x</mi> <mi>i</mi></msub></mrow></mfrac><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g_i(0) = \frac{\partial f}{\partial x_i}(0)</annotation></semantics></math>. Since any derivation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>δ</mi><mo>:</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\delta : C^\infty(X) \to C^\infty(X)</annotation></semantics></math> by definition satisfies the Leibniz rule, it follows that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>δ</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mi>i</mi></munder><mi>δ</mi><mo stretchy="false">(</mo><msub><mi>x</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mfrac><mrow><mo>∂</mo><mi>f</mi></mrow><mrow><mo>∂</mo><msub><mi>x</mi> <mi>i</mi></msub></mrow></mfrac><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \delta(f)(0) = \sum_i \delta(x_i) \frac{\partial f}{\partial x_i}(0) \,. </annotation></semantics></math></div> <p>Similarly, by translation, at all other points. Therefore <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>δ</mi></mrow><annotation encoding="application/x-tex">\delta</annotation></semantics></math> is already fixed by its action of the coordinate functions <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>i</mi></msub><mo>∈</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{x_i \in C^\infty(X)\}</annotation></semantics></math>. Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mi>δ</mi></msub><mo>∈</mo><mi>T</mi><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">v_\delta \in T \mathbb{R}^n</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/vector+field">vector field</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mi>δ</mi></msub><mo>≔</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mi>i</mi></munder><mi>δ</mi><mo stretchy="false">(</mo><msub><mi>x</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mfrac><mo>∂</mo><mrow><mo>∂</mo><msub><mi>x</mi> <mi>i</mi></msub></mrow></mfrac></mrow><annotation encoding="application/x-tex"> v_\delta \coloneqq \sum_i \delta(x_i) \frac{\partial}{\partial x_i} </annotation></semantics></math></div> <p>then it follows that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>δ</mi></mrow><annotation encoding="application/x-tex">\delta</annotation></semantics></math> is the derivation coming from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mi>δ</mi></msub></mrow><annotation encoding="application/x-tex">v_\delta</annotation></semantics></math> under <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Γ</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><mi>T</mi><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Der</mi><mo stretchy="false">(</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Gamma_X(T X) \to Der(C^\infty(X))</annotation></semantics></math>.</p> </div> <p>Recall further from <em><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+superalgebra">geometry of physics – superalgebra</a></em> that the category of <a class="existingWikiWord" href="/nlab/show/supercommutative+superalgebras">supercommutative superalgebras</a> is related to that of ordinary <a class="existingWikiWord" href="/nlab/show/commutative+algebras">commutative algebras</a> over <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">\mathbb{R}</annotation></semantics></math> by an <a class="existingWikiWord" href="/nlab/show/adjoint+cylinder">adjoint cylinder</a> (<a href="geometry+of+physics+--+superalgebra#InclusionOfCAlgIntosCAlg">this prop.</a>):</p> <div class="num_prop" id="AdjointCylinderOnSuperAffines"> <h6 id="proposition_3">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/adjoint+modality">adjoint modality</a> of <a class="existingWikiWord" href="/nlab/show/fermionic+modality">even fermionic</a> and <a class="existingWikiWord" href="/nlab/show/bosonic+modality">bosonic</a> in <a class="existingWikiWord" href="/nlab/show/superalgebras">superalgebras</a>)</strong></p> <p>The canonical inclusion of <a class="existingWikiWord" href="/nlab/show/commutative+algebras">commutative algebras</a> into <a class="existingWikiWord" href="/nlab/show/supercommutative+superalgebra">supercommutative superalgebras</a> is part of an <a class="existingWikiWord" href="/nlab/show/fully+faithful+functor">fully faithful</a> <a class="existingWikiWord" href="/nlab/show/adjoint+triple">adjoint triple</a> (<a href="geometry+of+physics+--+categories+and+toposes#AdjointModality">this Def.</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><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mo stretchy="false">)</mo> <mi>even</mi></msub><mspace width="thickmathspace"></mspace><mo>⊣</mo><mspace width="thickmathspace"></mspace><msup><mi>ι</mi> <mi>op</mi></msup><mspace width="thickmathspace"></mspace><mo>⊣</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mo stretchy="false">)</mo> <mi>odd</mi></msub><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msub><mi>CAlg</mi> <mi>k</mi></msub><munderover><mover><mo>↪</mo><mrow><mphantom><mi>AAAA</mi></mphantom><msup><mi>ι</mi> <mi>op</mi></msup><mphantom><mi>AAAA</mi></mphantom></mrow></mover><munder><mo>⟵</mo><mrow><mphantom><mi>AA</mi></mphantom><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mo stretchy="false">)</mo> <mi>even</mi></msub><mphantom><mi>AA</mi></mphantom></mrow></munder><mover><mo>⟵</mo><mrow><mphantom><mi>AA</mi></mphantom><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mo stretchy="false">)</mo> <mi>odd</mi></msub><mphantom><mi>AA</mi></mphantom></mrow></mover></munderover><msub><mi>sCAlg</mi> <mi>k</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (-)_{even} \;\dashv\; \iota^{op} \;\dashv\; (-)/(-)_{odd} \;\;\colon\;\; CAlg_k \underoverset {\underset{\phantom{AA}(-)_{even}\phantom{AA}}{\longleftarrow}} {\overset{\phantom{AA}(-)/(-)_{odd}\phantom{AA}}{\longleftarrow}} {\overset{\phantom{AAAA}\iota^{op}\phantom{AAAA}}{\hookrightarrow}} sCAlg_k \,. </annotation></semantics></math></div> <p>(Here and in the following we display pairs of <a class="existingWikiWord" href="/nlab/show/adjoint+functors">adjoint functors</a> (<a href="geometry+of+physics+–+basic+notions+of+category+theory#AdjointFunctorsInTermsOfNaturalBijectionOfHomSets">this Def.</a>) such that the <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> is on top and the right adjoint is on the bottom.)</p> <p>The <a class="existingWikiWord" href="/nlab/show/formal+dual">formal dual</a> (<a href="geometry+of+physics+--+categories+and+toposes#OppositeCategory">this Def.</a>) of this statement is that <a class="existingWikiWord" href="/nlab/show/affine+superschemes">affine superschemes</a> are related to ordinary <a class="existingWikiWord" href="/nlab/show/affine+schemes">affine schemes</a> over <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">\mathbb{R}</annotation></semantics></math> by an <a class="existingWikiWord" href="/nlab/show/adjoint+modality">adjoint modality</a> of this form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>even</mi><mspace width="thickmathspace"></mspace><mo>⊣</mo><mspace width="thickmathspace"></mspace><msub><mi>ι</mi> <mi>sup</mi></msub><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msub><mi>Disc</mi> <mi>sup</mi></msub><mi>Aff</mi><mo stretchy="false">(</mo><msub><mi>Vect</mi> <mi>k</mi></msub><mo stretchy="false">)</mo><munderover><mover><mo>↪</mo><mrow><mphantom><mi>AAA</mi></mphantom><mi>ι</mi><mphantom><mi>AAA</mi></mphantom></mrow></mover><munder><mo>⟵</mo><mrow><mphantom><mi>AA</mi></mphantom><msub><mi>Π</mi> <mi>sup</mi></msub><mphantom><mi>AA</mi></mphantom></mrow></munder><mover><mo>⟵</mo><mrow><mphantom><mi>AA</mi></mphantom><mi>even</mi><mphantom><mi>AA</mi></mphantom></mrow></mover></munderover><mi>Aff</mi><mo stretchy="false">(</mo><msub><mi>sVect</mi> <mi>k</mi></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> even \;\dashv\; \iota_{sup} \;\;\colon\;\; Disc_{sup} Aff(Vect_k) \underoverset {\underset { \phantom{AA} \Pi_{sup} \phantom{AA} } {\longleftarrow} } {\overset { \phantom{AA} even \phantom{AA} } {\longleftarrow} } {\overset{ \phantom{AAA} \iota \phantom{AAA} }{\hookrightarrow}} Aff(sVect_k) \,. </annotation></semantics></math></div></div> <p>Beware that</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mi>sup</mi></msub></mrow><annotation encoding="application/x-tex">\Pi_{sup}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/formal+dual">formal dual</a> of <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><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mo stretchy="false">)</mo> <mi>odd</mi></msub></mrow><annotation encoding="application/x-tex">(-)/(-)_{odd}</annotation></semantics></math>,</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>even</mi></mrow><annotation encoding="application/x-tex">even</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/formal+dual">formal dual</a> of <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><msub><mo stretchy="false">)</mo> <mi>even</mi></msub></mrow><annotation encoding="application/x-tex">(-)_{even}</annotation></semantics></math>.</p> </li> </ol> <p>That they change position in the diagrams is because we always draw <a class="existingWikiWord" href="/nlab/show/left+adjoints">left adjoints</a> on top of <a class="existingWikiWord" href="/nlab/show/right+adjoints">right adjoints</a> and the handedness of <a class="existingWikiWord" href="/nlab/show/adjoints">adjoints</a> changes as we pass to <a class="existingWikiWord" href="/nlab/show/opposite+categories">opposite categories</a>.</p> <div class="num_example" id="EvenPartOfDimTwoSuperpoint"> <h6 id="example">Example</h6> <p>The <a class="existingWikiWord" href="/nlab/show/algebra+of+functions">algebra of functions</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>even</mi><mo stretchy="false">(</mo><mrow><msup><mi>ℝ</mi> <mrow><mn>0</mn><mo stretchy="false">|</mo><mn>2</mn></mrow></msup></mrow><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">even({\mathbb{R}^{0\vert 2}})</annotation></semantics></math> is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mi>𝒪</mi><mrow><mo>(</mo><mi>even</mi><mo stretchy="false">(</mo><mrow><msup><mi>ℝ</mi> <mrow><mn>0</mn><mo stretchy="false">|</mo><mn>2</mn></mrow></msup></mrow><mo stretchy="false">)</mo><mo>)</mo></mrow></mtd> <mtd><mo>=</mo><mi>𝒪</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mn>0</mn><mo stretchy="false">|</mo><mn>2</mn></mrow></msup><msub><mo stretchy="false">)</mo> <mi>even</mi></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo stretchy="false">(</mo><msup><mo>∧</mo> <mo>•</mo></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mn>2</mn></msup><msup><mo stretchy="false">)</mo> <mo>*</mo></msup><msub><mo stretchy="false">)</mo> <mi>even</mi></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><msub><mrow><mo>{</mo><msub><mi>a</mi> <mn>0</mn></msub><mo>+</mo><msub><mi>a</mi> <mn>1</mn></msub><mspace width="thinmathspace"></mspace><msub><mi>θ</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>a</mi> <mn>2</mn></msub><mspace width="thinmathspace"></mspace><msub><mi>θ</mi> <mn>2</mn></msub><mo>+</mo><msub><mi>a</mi> <mn>12</mn></msub><msub><mi>θ</mi> <mn>1</mn></msub><msub><mi>θ</mi> <mn>2</mn></msub><mspace width="thinmathspace"></mspace><mo stretchy="false">|</mo><msub><mi>a</mi> <mo>•</mo></msub><mo>∈</mo><mi>ℝ</mi><mspace width="thinmathspace"></mspace><mo>}</mo></mrow> <mi>even</mi></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mrow><mo>{</mo><msub><mi>a</mi> <mn>0</mn></msub><mo>+</mo><msub><mi>a</mi> <mn>12</mn></msub><msub><mi>θ</mi> <mn>1</mn></msub><msub><mi>θ</mi> <mn>2</mn></msub><mspace width="thinmathspace"></mspace><mo stretchy="false">|</mo><msub><mi>a</mi> <mn>12</mn></msub><mo>∈</mo><mi>ℝ</mi><mspace width="thinmathspace"></mspace><mo>}</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>ℝ</mi><mo stretchy="false">[</mo><mi>ϵ</mi><mo stretchy="false">]</mo><mo stretchy="false">/</mo><mo stretchy="false">(</mo><msup><mi>ϵ</mi> <mn>2</mn></msup><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \mathcal{O}\left( even({\mathbb{R}^{0\vert 2}})\right) & = \mathcal{O}(\mathbb{R}^{0\vert 2})_{even} \\ & = (\wedge^\bullet(\mathbb{R}^2)^\ast)_{even} \\ &= \left\{a_0 + a_1 \, \theta_1 + a_2 \, \theta_2 + a_{12} \theta_1 \theta_2 \,\vert a_\bullet \in \mathbb{R}\,\right\}_{even} \\ &= \left\{a_0 + a_{12} \theta_1 \theta_2 \,\vert a_{12} \in \mathbb{R}\,\right\} \\ &= \mathbb{R}[\epsilon]/(\epsilon^2) \end{aligned} </annotation></semantics></math></div> <p>where in the last line we renamed <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>θ</mi> <mn>1</mn></msub><msub><mi>θ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\theta_1 \theta_2</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">\epsilon</annotation></semantics></math>.</p> <p>This algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi><mo stretchy="false">[</mo><mi>ϵ</mi><mo stretchy="false">]</mo><mo stretchy="false">/</mo><mo stretchy="false">(</mo><msup><mi>ϵ</mi> <mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{R}[\epsilon]/(\epsilon^2)</annotation></semantics></math> is known as the <strong><a class="existingWikiWord" href="/nlab/show/algebra+of+dual+numbers">algebra of dual numbers</a></strong> over <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">\mathbb{R}</annotation></semantics></math>. It is to be thought of as the <a class="existingWikiWord" href="/nlab/show/algebra+of+functions">algebra of functions</a> on a bosonic but <a class="existingWikiWord" href="/nlab/show/infinitesimally+thickened+point">infinitesimally thickened point</a>, an <a class="existingWikiWord" href="/nlab/show/infinitesimal+neighbourhood">infinitesimal neighbourhood</a> of a point inside the <a class="existingWikiWord" href="/nlab/show/real+line">real line</a> which is “so very small” that the canonical <a class="existingWikiWord" href="/nlab/show/coordinate">coordinate</a> function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">\epsilon</annotation></semantics></math> on it takes values “so tiny” that its square, which is bound to be even tinier, is actually indistinguishable from <a class="existingWikiWord" href="/nlab/show/zero">zero</a>.</p> <p>We will write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>𝔻</mi> <mn>1</mn></msup><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mi>Spec</mi><mo stretchy="false">(</mo><mi>ℝ</mi><mo stretchy="false">[</mo><mi>ϵ</mi><mo stretchy="false">]</mo><mo stretchy="false">/</mo><mo stretchy="false">(</mo><msup><mi>ϵ</mi> <mn>2</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>∈</mo><mspace width="thickmathspace"></mspace><mi>Aff</mi><mo stretchy="false">(</mo><msub><mi>sVect</mi> <mi>k</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathbb{D}^1 \;\coloneqq\; Spec(\mathbb{R}[\epsilon]/(\epsilon^2)) \;\in\; Aff(sVect_k) </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/space">space</a> <a class="existingWikiWord" href="/nlab/show/formal+dual">formally dual</a> to this <a class="existingWikiWord" href="/nlab/show/algebra+of+dual+numbers">algebra of dual numbers</a> and think of it as the <em>1-dimensional first order <a class="existingWikiWord" href="/nlab/show/infinitesimal+disk">infinitesimal disk</a></em>.</p> </div> <p>In generalization of this we make the following definitions:</p> <div class="num_defn" id="FormalCartSp"> <h6 id="definition_2">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/formal+Cartesian+spaces">formal Cartesian spaces</a>)</strong></p> <p>Write</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>InfPoint</mi></mtd> <mtd><mover><mo>↪</mo><mphantom><mi>AAA</mi></mphantom></mover></mtd> <mtd><msubsup><mi>CAlg</mi> <mi>ℝ</mi> <mi>op</mi></msubsup></mtd></mtr> <mtr><mtd><msub><mi>𝔻</mi> <mi>V</mi></msub></mtd> <mtd><mo>↦</mo></mtd> <mtd><mi>ℝ</mi><mo>⊕</mo><mi>V</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ InfPoint &\overset{\phantom{AAA}}{\hookrightarrow}& CAlg_{\mathbb{R}}^{op} \\ \mathbb{D}_V &\mapsto& \mathbb{R} \oplus V } </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> (<a href="geometry+of+physics+--+categories+and+toposes#FullSubcategoryOnClassOfObjects">this Example</a>) of the <a class="existingWikiWord" href="/nlab/show/opposite+category">opposite category</a> (<a href="geometry+of+physics+--+categories+and+toposes#OppositeCategory">this Example</a>) of <a class="existingWikiWord" href="/nlab/show/commutative+algebras">commutative algebras</a> over <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">\mathbb{R}</annotation></semantics></math> on <a class="existingWikiWord" href="/nlab/show/formal+duals">formal duals</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔻</mi></mrow><annotation encoding="application/x-tex">\mathbb{D}</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/commutative+algebras">commutative algebras</a> over the <a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝒪</mi><mo stretchy="false">(</mo><msub><mi>𝔻</mi> <mi>V</mi></msub><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><mi>ℝ</mi><mo>⊕</mo><mi>V</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \mathcal{O}(\mathbb{D}_V) \;\coloneqq\; (\mathbb{R}\oplus V) \,, </annotation></semantics></math></div> <p>with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/nilpotent+ideal">nilpotent ideal</a> of <a class="existingWikiWord" href="/nlab/show/finite+number">finite</a>-<a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> over <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">\mathbb{R}</annotation></semantics></math> (the <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a> on the right is that of underlying <a class="existingWikiWord" href="/nlab/show/vector+spaces">vector spaces</a>, not of algebras).</p> <p>We call this the <a class="existingWikiWord" href="/nlab/show/category">category</a> of <em><a class="existingWikiWord" href="/nlab/show/infinitesimally+thickened+points">infinitesimally thickened points</a></em>.</p> <p>Alternative terminology:</p> <ol> <li> <p>In <a class="existingWikiWord" href="/nlab/show/synthetic+differential+geometry">synthetic differential geometry</a> these algebras are called <a class="existingWikiWord" href="/nlab/show/infinitesimally+thickened+point">Weil algebras</a><strong>,</strong></p> </li> <li> <p>in <a class="existingWikiWord" href="/nlab/show/algebraic+geometry">algebraic geometry</a> they are known as <em><a class="existingWikiWord" href="/nlab/show/local+ring">local</a> <a class="existingWikiWord" href="/nlab/show/real+numbers">real</a> <a class="existingWikiWord" href="/nlab/show/Artin+algebras">Artin algebras</a></em>.</p> </li> </ol> <p>Write moreover</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>FormalCartSp</mi></mtd> <mtd><mover><mo>↪</mo><mphantom><mi>AAAA</mi></mphantom></mover></mtd> <mtd><msubsup><mi>CAlg</mi> <mi>ℝ</mi> <mi>op</mi></msubsup></mtd></mtr> <mtr><mtd><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>×</mo><msub><mi>𝔻</mi> <mi>V</mi></msub></mtd> <mtd><mo>↦</mo></mtd> <mtd><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mi>ℝ</mi></msub><mo stretchy="false">(</mo><mi>ℝ</mi><mo>⊕</mo><mi>V</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ FormalCartSp &\overset{\phantom{AAAA}}{\hookrightarrow}& CAlg_{\mathbb{R}}^{op} \\ \mathbb{R}^n \times \mathbb{D}_V &\mapsto& C^\infty(\mathbb{R}^n) \otimes_{\mathbb{R}} (\mathbb{R} \oplus V) } </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> on <a class="existingWikiWord" href="/nlab/show/formal+duals">formal duals</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><mo>×</mo><mi>𝔻</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}^n \times \mathbb{D}</annotation></semantics></math> of those algebras which are <a class="existingWikiWord" href="/nlab/show/tensor+products">tensor products</a> of commutative <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">\mathbb{R}</annotation></semantics></math>-algebras of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝒪</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>×</mo><mi>𝔻</mi><mo stretchy="false">)</mo><mo>≔</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mi>ℝ</mi></msub><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>𝔻</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathcal{O}(\mathbb{R}^n \times \mathbb{D}) \coloneqq C^\infty(\mathbb{R}^n) \otimes_{\mathbb{R}} C^\infty(\mathbb{D}) </annotation></semantics></math></div> <p>of algebras <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>p</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^\infty(\mathbb{R}^p)</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/smooth+functions">smooth functions</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> with algebras corresponding to infinitesimally thickened points <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔻</mi></mrow><annotation encoding="application/x-tex">\mathbb{D}</annotation></semantics></math> as above.</p> </div> <div class="num_remark" id="FormalSchemesAndSDG"> <h6 id="remark">Remark</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/formal+schemes">formal schemes</a> and <a class="existingWikiWord" href="/nlab/show/synthetic+differential+geometry">synthetic differential geometry</a>)</strong></p> <p>This kind of construction in Def. <a class="maruku-ref" href="#FormalCartSp"></a> is traditionally more familiar from the theory of <a class="existingWikiWord" href="/nlab/show/formal+schemes">formal schemes</a>, but the same kind of general abstract theory goes through in the context of <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a>, a point of view known as <em><a class="existingWikiWord" href="/nlab/show/synthetic+differential+geometry">synthetic differential geometry</a></em> (<a class="existingWikiWord" href="/nlab/show/Toposes+of+laws+of+motion">Lawvere 97</a>), preconfigured already in the prespective of <em><a class="existingWikiWord" href="/nlab/show/functorial+geometry">functorial geometry</a></em> of <a href="functorial+geometry#Grothendieck65">Grothendieck 65</a>.</p> </div> <p>The crucial property of <a class="existingWikiWord" href="/nlab/show/infinitesimally+thickened+points">infinitesimally thickened points</a> (def. <a class="maruku-ref" href="#FormalCartSp"></a>) is that they co-represent <a class="existingWikiWord" href="/nlab/show/tangent+vectors">tangent vectors</a> and <a class="existingWikiWord" href="/nlab/show/jets">jets</a>:</p> <div class="num_example" id="HomsOutOfFirstOrderInfinitesimalLine"> <h6 id="example_2">Example</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> out of <a class="existingWikiWord" href="/nlab/show/infinitesimally+thickened+point">infinitesimally thickened point</a> are <a class="existingWikiWord" href="/nlab/show/tangent+vectors">tangent vectors</a>)</strong></p> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝔻</mi> <mn>1</mn></msup><mo>=</mo><mi>Spec</mi><mo stretchy="false">(</mo><mi>ℝ</mi><mo stretchy="false">[</mo><mi>ϵ</mi><mo stretchy="false">]</mo><mo stretchy="false">/</mo><mo stretchy="false">(</mo><msup><mi>ϵ</mi> <mn>2</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{D}^1 = Spec(\mathbb{R}[\epsilon]/(\epsilon^2))</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/formal+dual">formal dual</a> of the <a class="existingWikiWord" href="/nlab/show/algebra+of+dual+numbers">algebra of dual numbers</a> (example <a class="maruku-ref" href="#EvenPartOfDimTwoSuperpoint"></a>). Then morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>FormalCartSp</mi></mrow><annotation encoding="application/x-tex">FormalCartSp</annotation></semantics></math> (def. <a class="maruku-ref" href="#FormalCartSp"></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><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>×</mo><msup><mi>𝔻</mi> <mn>1</mn></msup><mo>⟶</mo><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex"> \mathbb{R}^n \times \mathbb{D}^1 \longrightarrow \mathbb{R}^n </annotation></semantics></math></div> <p>which are the identity after restriction along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>→</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>×</mo><msup><mi>𝔻</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n \to \mathbb{R}^n \times \mathbb{D}^1</annotation></semantics></math>, are in <a class="existingWikiWord" href="/nlab/show/natural+bijection">natural bijection</a> with smooth <a class="existingWikiWord" href="/nlab/show/vector+fields">vector fields</a> 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>.</p> <p>Moreover, morphisms of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>𝔻</mi> <mn>1</mn></msup><mo>⟶</mo><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex"> \mathbb{D}^1 \longrightarrow \mathbb{R}^n </annotation></semantics></math></div> <p>are equivalently single <a class="existingWikiWord" href="/nlab/show/tangent+vectors">tangent vectors</a>, hence for every <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> there is a <a class="existingWikiWord" href="/nlab/show/natural+bijection">natural bijection</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mi>FormalCartSp</mi></msub><mo stretchy="false">(</mo><msup><mi>𝔻</mi> <mn>1</mn></msup><mo>,</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo>≃</mo><munder><munder><mrow><mi>T</mi><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><mo>⏟</mo></munder><mfrac linethickness="0"><mrow><mtext>underlying</mtext></mrow><mrow><mtext>set</mtext></mrow></mfrac></munder></mrow><annotation encoding="application/x-tex"> Hom_{FormalCartSp}(\mathbb{D}^1, \mathbb{R}^n) \simeq \underset{\text{underlying} \atop \text{set}}{\underbrace{T \mathbb{R}^n}} </annotation></semantics></math></div> <p>between the <a class="existingWikiWord" href="/nlab/show/hom-set">hom-set</a> from the formal dual of the <a class="existingWikiWord" href="/nlab/show/ring+of+dual+numbers">ring of dual numbers</a> and the set of <a class="existingWikiWord" href="/nlab/show/tangent+vectors">tangent vectors</a>.</p> </div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>By definition, morphisms</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>×</mo><mi>𝔻</mi><mo>⟶</mo><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex"> \mathbb{R}^n \times \mathbb{D}\longrightarrow \mathbb{R}^n </annotation></semantics></math></div> <p>which are the identity after restriction along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>→</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>×</mo><mi>𝔻</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}^n \to \mathbb{R}^n \times \mathbb{D}</annotation></semantics></math>, are equivalently algebra homomorphisms of the form</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><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo>⊕</mo><mi>ϵ</mi><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⟵</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (C^\infty(\mathbb{R}^n) \oplus \epsilon C^\infty(\mathbb{R}^n)) \longleftarrow C^\infty(\mathbb{R}^n) </annotation></semantics></math></div> <p>which are the identity modulo <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">\epsilon</annotation></semantics></math>. Such a morphism has to take any function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>∈</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f \in C^\infty(\mathbb{R}^n)</annotation></semantics></math> to</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>+</mo><mo stretchy="false">(</mo><mo>∂</mo><mi>f</mi><mo stretchy="false">)</mo><mi>ϵ</mi></mrow><annotation encoding="application/x-tex"> f + (\partial f) \epsilon </annotation></semantics></math></div> <p>for some smooth function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo>∂</mo><mi>f</mi><mo stretchy="false">)</mo><mo>∈</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\partial f) \in C^\infty(\mathbb{R}^n)</annotation></semantics></math>. The condition that this assignment makes an algebra homomorphism is equivalent to the statement that for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>f</mi> <mn>2</mn></msub><mo>∈</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f_1,f_2 \in C^\infty(\mathbb{R}^n)</annotation></semantics></math></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>f</mi> <mn>1</mn></msub><msub><mi>f</mi> <mn>2</mn></msub><mo>+</mo><mo stretchy="false">(</mo><mo>∂</mo><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>1</mn></msub><msub><mi>f</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi>ϵ</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>1</mn></msub><mo>+</mo><mo stretchy="false">(</mo><mo>∂</mo><msub><mi>f</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mi>ϵ</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>2</mn></msub><mo>+</mo><mo stretchy="false">(</mo><mo>∂</mo><msub><mi>f</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mi>ϵ</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (f_1 f_2 + (\partial (f_1 f_2))\epsilon ) = (f_1 + (\partial f_1) \epsilon) (f_2 + (\partial f_2) \epsilon) \,. </annotation></semantics></math></div> <p>Multiplying this out and using that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ϵ</mi> <mn>2</mn></msup><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\epsilon^2 = 0</annotation></semantics></math> this in turn is equivalent to</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>∂</mo><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>1</mn></msub><msub><mi>f</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mo>∂</mo><msub><mi>f</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><msub><mi>f</mi> <mn>2</mn></msub><mo>+</mo><msub><mi>f</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mo>∂</mo><msub><mi>f</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \partial(f_1 f_2) = (\partial f_1) f_2 + f_1 (\partial f_2) \,. </annotation></semantics></math></div> <p>This means equivalently that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∂</mo><mo lspace="verythinmathspace">:</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo>→</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\partial\colon C^\infty(\mathbb{R}^n)\to C^\infty(\mathbb{R}^n)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/derivation">derivation</a>. But <a class="existingWikiWord" href="/nlab/show/derivations+of+smooth+functions+are+vector+fields">derivations of smooth functions are vector fields</a> (prop. <a class="maruku-ref" href="#DerivationsOfSmoothFunctions"></a>).</p> </div> <p>We now explore further the relations between <a class="existingWikiWord" href="/nlab/show/Cartesian+spaces">Cartesian spaces</a>, <a class="existingWikiWord" href="/nlab/show/formal+Cartesian+spaces">formal Cartesian spaces</a>, and <a class="existingWikiWord" href="/nlab/show/super+formal+Cartesian+spaces">super formal Cartesian spaces</a>.</p> <div class="num_prop" id="CartSpCoreflectiveInclusion"> <h6 id="proposition_4">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/coreflective+subcategory">co-reflection</a> of <a class="existingWikiWord" href="/nlab/show/Cartesian+spaces">Cartesian spaces</a> inside <a class="existingWikiWord" href="/nlab/show/formal+Cartesian+spaces">formal Cartesian spaces</a>)</strong></p> <p>The canonical inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ι</mi></mrow><annotation encoding="application/x-tex">\iota</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/category">category</a> of ordinary <a class="existingWikiWord" href="/nlab/show/Cartesian+spaces">Cartesian spaces</a> into that of <a class="existingWikiWord" href="/nlab/show/formal+manifold">formal</a> <a class="existingWikiWord" href="/nlab/show/Cartesian+spaces">Cartesian spaces</a> has a <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> (<a href="geometry+of+physics+–+basic+notions+of+category+theory#AdjointFunctorsInTermsOfNaturalBijectionOfHomSets">this Def.</a>) <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">\Re</annotation></semantics></math> (“<a class="existingWikiWord" href="/nlab/show/reduction+modality">reduction</a>”)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>CartSp</mi><munderover><mo>⊥</mo><munder><mo>⟵</mo><mrow><mphantom><mi>AA</mi></mphantom><msub><mi>Π</mi> <mi>inf</mi></msub><mphantom><mi>AA</mi></mphantom></mrow></munder><mover><mo>↪</mo><mrow><mphantom><mi>AA</mi></mphantom><msub><mi>ι</mi> <mi>inf</mi></msub><mphantom><mi>AA</mi></mphantom></mrow></mover></munderover><mi>FormalCartSp</mi></mrow><annotation encoding="application/x-tex"> CartSp \underoverset {\underset{\phantom{AA}\Pi_{inf} \phantom{AA}}{\longleftarrow}} {\overset{\phantom{AA}\iota_{inf} \phantom{AA}}{\hookrightarrow}} {\bot} FormalCartSp </annotation></semantics></math></div> <p>given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mi>inf</mi></msub><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>×</mo><mi>𝔻</mi><mo stretchy="false">)</mo><mo>≔</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Pi_{inf}(\mathbb{R}^n \times \mathbb{D}) \coloneqq \mathbb{R}^n \,. </annotation></semantics></math></div> <p>Hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ι</mi></mrow><annotation encoding="application/x-tex">\iota</annotation></semantics></math> exhibits <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CartSp</mi></mrow><annotation encoding="application/x-tex">CartSp</annotation></semantics></math> as a <a class="existingWikiWord" href="/nlab/show/coreflective+subcategory">coreflective subcategory</a> (<a href="geometry+of+physics+--+categories+and+toposes#ReflectiveSubcategory">this Def.</a>) of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>FormalCartSp</mi></mrow><annotation encoding="application/x-tex">FormalCartSp</annotation></semantics></math></p> <p>We say that <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> is the <strong><a class="existingWikiWord" href="/nlab/show/reduced+scheme">reduced scheme</a></strong> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>×</mo><mi>𝔻</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}^n \times \mathbb{D}</annotation></semantics></math>.</p> </div> <div class="proof"> <h6 id="proof_3">Proof</h6> <p>We check the <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural isomorphism</a> on <a class="existingWikiWord" href="/nlab/show/hom-sets">hom-sets</a> that characterizes a pair of <a class="existingWikiWord" href="/nlab/show/adjoint+functors">adjoint functors</a> (<a href="geometry+of+physics+--+categories+and+toposes#eq:HomIsomorphismForAdjointFunctors">here</a>):</p> <p>By definition, a morphism of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>i</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msup><mo stretchy="false">)</mo><mo>⟶</mo><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup><mo>×</mo><mi>𝔻</mi></mrow><annotation encoding="application/x-tex"> f \;\colon\; i(\mathbb{R}^{n_1}) \longrightarrow \mathbb{R}^{n_2} \times \mathbb{D} </annotation></semantics></math></div> <p>is equivalently a <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> of <a class="existingWikiWord" href="/nlab/show/commutative+algebras">commutative algebras</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><msup><mi>f</mi> <mo>*</mo></msup><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msup><mo stretchy="false">)</mo><mo>⟵</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mi>ℝ</mi></msub><mo stretchy="false">(</mo><mi>ℝ</mi><mo>⊕</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> f^\ast \;\colon\; C^\infty(\mathbb{R}^{n_1}) \longleftarrow C^\infty(\mathbb{R}^{n_2}) \otimes_{\mathbb{R}} (\mathbb{R} \oplus V) </annotation></semantics></math></div> <p>where all elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi><mo>∈</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">v \in V</annotation></semantics></math> are nilpotent, in that there exists <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>n</mi> <mi>v</mi></msub><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n_v \in \mathbb{N}</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>v</mi><msup><mo stretchy="false">)</mo> <mrow><msub><mi>n</mi> <mi>v</mi></msub></mrow></msup><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">(v)^{n_v} = 0</annotation></semantics></math>. Every algebra homomorphism needs to preserve this equation, and hence needs to send nilpotent elements to nilpotent elements. But the only nilpotent element in the ordinary function algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^\infty(\mathbb{R}^n)</annotation></semantics></math> is the zero-function, and so it follows that the above homomorphism has to vanish on all of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>, hence has to factor (necessarily uniquely) through a homomorphism of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mover><mi>f</mi><mo stretchy="false">˜</mo></mover> <mo>*</mo></msup><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msup><mo stretchy="false">)</mo><mo>⟵</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \tilde f^\ast \;\colon\; C^\infty(\mathbb{R}^{n_1}) \longleftarrow C^\infty(\mathbb{R}^{n_2}) \,. </annotation></semantics></math></div> <p>This is dually a morphism of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mi>f</mi><mo stretchy="false">˜</mo></mover><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msup><mo>⟶</mo><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup></mrow><annotation encoding="application/x-tex"> \tilde f \;\colon\; \mathbb{R}^{n_1} \longrightarrow \mathbb{R}^{n_2} </annotation></semantics></math></div> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CartSp</mi></mrow><annotation encoding="application/x-tex">CartSp</annotation></semantics></math>. This establishes a natural bijection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>↔</mo><mover><mi>f</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">f \leftrightarrow \tilde f</annotation></semantics></math>.</p> </div> <p>The above discussion following prop. <a class="maruku-ref" href="#AdjointCylinderOnSuperAffines"></a> means that in passing to commutative superalgebras there are <em>two</em> stages of generalizations of plain differential geometry involved:</p> <ol> <li> <p><a class="existingWikiWord" href="/nlab/show/Cartesian+spaces">Cartesian spaces</a> are generalized to <a class="existingWikiWord" href="/nlab/show/formal+manifold">formal</a> <a class="existingWikiWord" href="/nlab/show/Cartesian+spaces">Cartesian spaces</a>;</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/formal+manifold">formal</a><a class="existingWikiWord" href="/nlab/show/Cartesian+spaces">Cartesian spaces</a> are further generalized to <a class="existingWikiWord" href="/nlab/show/super+formal+Cartesian+spaces">super formal Cartesian spaces</a>.</p> </li> </ol> <p>In order to make this explicit, it is convenient to introduce the following slight generalization of <a class="existingWikiWord" href="/nlab/show/super+Cartesian+spaces">super Cartesian spaces</a> (def. <a class="maruku-ref" href="#SuperCartesianSpace"></a>), which are simply <a class="existingWikiWord" href="/nlab/show/Cartesian+products">Cartesian products</a> of ordinary <a class="existingWikiWord" href="/nlab/show/Cartesian+spaces">Cartesian spaces</a> with an <a class="existingWikiWord" href="/nlab/show/infinitesimally+thickened+point">infinitesimally thickened point</a> that may have both even and odd graded elements in its <a class="existingWikiWord" href="/nlab/show/algebra+of+functions">algebra of functions</a>.</p> <div class="num_defn" id="SuperFormalCartSp"> <h6 id="definition_3">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/super+formal+Cartesian+spaces">super formal Cartesian spaces</a>)</strong></p> <p>Write</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>SuperFormalCartSp</mi></mtd> <mtd><mover><mo>↪</mo><mrow><mphantom><mi>AA</mi></mphantom><mi>ι</mi><mphantom><mi>AA</mi></mphantom></mrow></mover></mtd> <mtd><msubsup><mi>sCAlg</mi> <mi>ℝ</mi> <mi>op</mi></msubsup></mtd></mtr> <mtr><mtd><msup><mi>ℝ</mi> <mrow><mi>n</mi><mo stretchy="false">|</mo><mi>q</mi></mrow></msup><mo>×</mo><msub><mi>𝔻</mi> <mi>V</mi></msub><mo>=</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>×</mo><msup><mi>ℝ</mi> <mrow><mn>0</mn><mo stretchy="false">|</mo><mi>q</mi></mrow></msup><mo>×</mo><msub><mi>𝔻</mi> <mi>V</mi></msub></mtd> <mtd><mo>↦</mo></mtd> <mtd><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mi>ℝ</mi></msub><msup><mo>∧</mo> <mo>•</mo></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>q</mi></msup><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mi>ℝ</mi></msub><mo stretchy="false">(</mo><mi>ℝ</mi><mo>⊕</mo><mi>V</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ SuperFormalCartSp &\overset{ \phantom{AA} \iota \phantom{AA} }{\hookrightarrow}& sCAlg_{\mathbb{R}}^{op} \\ \mathbb{R}^{n\vert q} \times \mathbb{D}_V = \mathbb{R}^n \times \mathbb{R}^{0 \vert q} \times \mathbb{D}_V & \mapsto & C^\infty( \mathbb{R}^{n} ) \otimes_{\mathbb{R}} \wedge^\bullet( \mathbb{R}^q ) \otimes_{\mathbb{R}} (\mathbb{R} \oplus V) } </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> (<a href="geometry+of+physics+--+categories+and+toposes#FullSubcategoryOnClassOfObjects">this Example</a>) of that of the <a class="existingWikiWord" href="/nlab/show/opposite+category">opposite category</a> of <a class="existingWikiWord" href="/nlab/show/supercommutative+superalgebras">supercommutative superalgebras</a> on those which are <a class="existingWikiWord" href="/nlab/show/tensor+product+of+algebras">tensor products of commutative algebras</a> of</p> <ol> <li> <p>the <a class="existingWikiWord" href="/nlab/show/algebra+of+functions">algebra of</a> <a class="existingWikiWord" href="/nlab/show/smooth+functions">smooth functions</a> on a <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a>,</p> </li> <li> <p>a <a class="existingWikiWord" href="/nlab/show/Grassmann+algebra">Grassmann algebra</a> (Def. <a class="maruku-ref" href="#SuperCartesianSpace"></a>),</p> </li> <li> <p>a <a class="existingWikiWord" href="/nlab/show/Weil+algebra">Weil algebra</a> (Def. <a class="maruku-ref" href="#FormalCartSp"></a>)</p> </li> </ol> </div> <p>One place in the literature where such <a class="existingWikiWord" href="/nlab/show/super+formal+Cartesian+spaces">super formal Cartesian spaces</a> are made explicit is in <a href="#KonechnySchwarz97">Konechny-Schwarz 97</a>.</p> <p>Just as formal Cartesian spaces may be thought of as local model spaces for <a class="existingWikiWord" href="/nlab/show/synthetic+differential+geometry">synthetic differential geometry</a>, <a class="existingWikiWord" href="/nlab/show/super+formal+Cartesian+spaces">super formal Cartesian spaces</a> may be thought of as a model for <a class="existingWikiWord" href="/nlab/show/synthetic+differential+supergeometry">synthetic differential supergeometry</a>. This we come to <a href="#SuperSmoothSets">below</a>.</p> <p>For completeness it is useful to compare the <a class="existingWikiWord" href="/nlab/show/coreflective+subcategory">coreflection</a> of prop. <a class="maruku-ref" href="#CartSpCoreflectiveInclusion"></a> to the following simple <a class="existingWikiWord" href="/nlab/show/reflective+subcategory">reflection</a>.</p> <p>Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>*</mo><mo>∈</mo><mi>Cat</mi></mrow><annotation encoding="application/x-tex"> \ast \in Cat </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/category">category</a> with a single object and a single morphism (the <a class="existingWikiWord" href="/nlab/show/identity">identity</a>) on that object. We will think of this as the category containing just the point space, which we want to think of as the local model space for <em><a class="existingWikiWord" href="/nlab/show/discrete+topological+space">discrete geometry</a></em>.</p> <div class="num_prop" id="ReflectionOfPointInCartSp"> <h6 id="proposition_5">Proposition</h6> <p>There is a pair of <a class="existingWikiWord" href="/nlab/show/adjoint+functors">adjoint functors</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>*</mo><mspace width="thickmathspace"></mspace><munderover><mrow></mrow><munder><mo>↪</mo><mrow></mrow></munder><mover><mo>⟵</mo><mi>Π</mi></mover></munderover><mspace width="thickmathspace"></mspace><mi>CartSp</mi></mrow><annotation encoding="application/x-tex"> \ast \; \underoverset {\underset{}{\hookrightarrow}} {\overset{\Pi}{\longleftarrow}} {} \;CartSp </annotation></semantics></math></div> <p>into the category of <a class="existingWikiWord" href="/nlab/show/Cartesian+spaces">Cartesian spaces</a>, where the bottom functor sends the unique object to the point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>0</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^0</annotation></semantics></math>, and where its <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> on top is, necessarily, the unique functor constant on the unique object on the left.</p> </div> <div class="proof"> <h6 id="proof_4">Proof</h6> <p>The <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural isomorphism</a> between <a class="existingWikiWord" href="/nlab/show/hom+sets">hom sets</a> characterizing this pair of <a class="existingWikiWord" href="/nlab/show/adjoint+functors">adjoint functors</a> simply expresses the fact that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>0</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^0</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CartSp</mi></mrow><annotation encoding="application/x-tex">CartSp</annotation></semantics></math>, hence that for <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> any <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a>, then there is a unique <a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</a> of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>→</mo><msup><mi>ℝ</mi> <mn>0</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n \to \mathbb{R}^0</annotation></semantics></math>.</p> </div> <p>In conclusion, the various <a class="existingWikiWord" href="/nlab/show/extra+structures">extra structures</a> on local model spaces (abstract coordinate systems) which we considered are organized in the following <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a>:</p> <div class="num_prop" id="SystemOfSites"> <h6 id="proposition_6">Proposition</h6> <p><strong>(progression of (<a class="existingWikiWord" href="/nlab/show/coreflective+subcategory">co-</a>)<a class="existingWikiWord" href="/nlab/show/reflective+subcategory">reflective</a> <a class="existingWikiWord" href="/nlab/show/site">site</a>-inclusions)</strong></p> <p>The <a class="existingWikiWord" href="/nlab/show/coreflective+subcategory">coreflective subcategory</a> inclusion of <a class="existingWikiWord" href="/nlab/show/Cartesian+spaces">Cartesian spaces</a> into <a class="existingWikiWord" href="/nlab/show/formal+manifold">formal</a> <a class="existingWikiWord" href="/nlab/show/Cartesian+spaces">Cartesian spaces</a> from prop. <a class="maruku-ref" href="#CartSpCoreflectiveInclusion"></a> and the coreflective as well as <a class="existingWikiWord" href="/nlab/show/reflective+subcategory">reflective subcategory</a> inclusion of affine schemes into <a class="existingWikiWord" href="/nlab/show/affine+superschemes">affine superschemes</a> from prop. <a class="maruku-ref" href="#AdjointCylinderOnSuperAffines"></a> and the terminal inclusion of prop. <a class="maruku-ref" href="#ReflectionOfPointInCartSp"></a> combine to give the following system of <a class="existingWikiWord" href="/nlab/show/adjoint+functors">adjoint functors</a> on our local model spaces</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><mfrac linethickness="0"><mrow><mtext></mtext></mrow><mrow><mfrac linethickness="0"><mrow><mtext>discrete</mtext></mrow><mrow><mtext>geometry</mtext></mrow></mfrac></mrow></mfrac></mtd> <mtd></mtd> <mtd><mfrac linethickness="0"><mrow><mtext></mtext></mrow><mrow><mfrac linethickness="0"><mrow><mtext>differential</mtext></mrow><mrow><mtext>geometry</mtext></mrow></mfrac></mrow></mfrac></mtd> <mtd></mtd> <mtd><mfrac linethickness="0"><mrow><mtext>formal</mtext></mrow><mrow><mfrac linethickness="0"><mrow><mtext>differential</mtext></mrow><mrow><mi>geometry</mi></mrow></mfrac></mrow></mfrac></mtd> <mtd></mtd> <mtd><mfrac linethickness="0"><mrow><mtext>super</mtext></mrow><mrow><mfrac linethickness="0"><mrow><mtext>differential</mtext></mrow><mrow><mtext>geometry</mtext></mrow></mfrac></mrow></mfrac></mtd></mtr> <mtr><mtd></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd><mrow><mtable><mtr><mtd><mover><mo>⟵</mo><mrow><mphantom><mi>AA</mi></mphantom><mi>Π</mi><mphantom><mi>AA</mi></mphantom></mrow></mover></mtd></mtr> <mtr><mtd><mover><mo>↪</mo><mrow><mphantom><mi>AA</mi></mphantom><mi>Disc</mi><mphantom><mi>AA</mi></mphantom></mrow></mover></mtd></mtr></mtable></mrow></mtd> <mtd><mi>CartSp</mi></mtd> <mtd><mrow><mtable><mtr><mtd><mover><mo>↪</mo><mrow><mphantom><mi>AA</mi></mphantom><msub><mi>ι</mi> <mi>inf</mi></msub><mphantom><mi>AA</mi></mphantom></mrow></mover></mtd></mtr> <mtr><mtd><mover><mo>⟵</mo><mrow><mphantom><mi>AA</mi></mphantom><msub><mi>Π</mi> <mi>inf</mi></msub><mphantom><mi>AA</mi></mphantom></mrow></mover></mtd></mtr> <mtr><mtd><mphantom><mi>A</mi></mphantom></mtd></mtr> <mtr><mtd><mphantom><mfrac linethickness="0"><mrow><mi>A</mi></mrow><mrow><mi>A</mi></mrow></mfrac></mphantom></mtd></mtr></mtable></mrow></mtd> <mtd><mi>FormalCartSp</mi></mtd> <mtd><mrow><mtable><mtr><mtd><mover><mo>⟵</mo><mrow><mphantom><mi>AA</mi></mphantom><mi>even</mi><mphantom><mi>AA</mi></mphantom></mrow></mover></mtd></mtr> <mtr><mtd><mover><mo>↪</mo><mrow><mphantom><mi>AA</mi></mphantom><msub><mi>ι</mi> <mi>sup</mi></msub><mphantom><mi>AA</mi></mphantom></mrow></mover></mtd></mtr> <mtr><mtd><mover><mo>⟵</mo><mrow><mphantom><mi>AA</mi></mphantom><msub><mi>Π</mi> <mi>sup</mi></msub><mphantom><mi>AA</mi></mphantom></mrow></mover></mtd></mtr> <mtr><mtd><mphantom><mi>A</mi></mphantom></mtd></mtr> <mtr><mtd><mphantom><mfrac linethickness="0"><mrow><mi>A</mi></mrow><mrow><mi>A</mi></mrow></mfrac></mphantom></mtd></mtr> <mtr><mtd><mphantom><mfrac linethickness="0"><mrow><mi>A</mi></mrow><mrow><mi>A</mi></mrow></mfrac></mphantom></mtd></mtr></mtable></mrow></mtd> <mtd><mi>SuperFormalCartSp</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ { \text{} \atop { \text{discrete} \atop {\text{geometry}} } } && {\text{} \atop {\text{differential} \atop \text{geometry}}} && {\text{formal} \atop {\text{differential} \atop {geometry}}} && {\text{super} \atop {\text{differential} \atop \text{geometry}}} \\ \\ \ast & \array{ \overset{\phantom{AA}\Pi\phantom{AA}}{\longleftarrow} \\ \overset{\phantom{AA}Disc\phantom{AA}}{\hookrightarrow} } & CartSp & \array{ \overset{\phantom{AA}\iota_{inf}\phantom{AA}}{\hookrightarrow} \\ \overset{\phantom{AA}\Pi_{inf}\phantom{AA}}{\longleftarrow} \\ \phantom{A} \\ \phantom{A \atop A} } & FormalCartSp & \array{ \overset{\phantom{AA}even\phantom{AA}}{\longleftarrow} \\ \overset{\phantom{AA}\iota_{sup}\phantom{AA}}{\hookrightarrow} \\ \overset{\phantom{AA}\Pi_{sup}\phantom{AA}}{\longleftarrow} \\ \phantom{A} \\ \phantom{A \atop A} \\ \phantom{A \atop A} } & SuperFormalCartSp } \,. </annotation></semantics></math></div></div> <h2 id="SuperSmoothSets">Super smooth sets</h2> <p><a href="#CoordinareSystemsSuperCartesianSpaces">Above</a> we discussed (<a class="existingWikiWord" href="/nlab/show/formal+manifold">formal</a>) <a class="existingWikiWord" href="/nlab/show/super+Cartesian+spaces">super Cartesian spaces</a>. What we are after is geometry of <a class="existingWikiWord" href="/nlab/show/generalized+space">generalized space</a> which is “locally modeled” on these, in the sense explained in the chapter <em><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+categories+and+toposes">on categories and toposes</a></em> . In order to do so we</p> <ol> <li> <p>consider <a class="existingWikiWord" href="/nlab/show/generalized+space">generalized</a> <a class="existingWikiWord" href="/nlab/show/superspaces">superspaces</a> whose collection forms a “good category” (the <a class="existingWikiWord" href="/nlab/show/sheaf+topos">sheaf topos</a> over super Cartesian space) in that lots of <a class="existingWikiWord" href="/nlab/show/universal+constructions">universal constructions</a> on general superspaces (<a class="existingWikiWord" href="/nlab/show/limits">limits</a>, <a class="existingWikiWord" href="/nlab/show/colimits">colimits</a>) still yield general superspaces;</p> </li> <li> <p>use special properties of this nice category (<a class="existingWikiWord" href="/nlab/show/differential+cohesion">differential cohesion</a>) in order to characterize and construct the good, tame superspaces, such as actual <a class="existingWikiWord" href="/nlab/show/supermanifolds">supermanifolds</a>.</p> </li> </ol> <p>The construction proceeds in direct anology to the non-super version discussed in the chapters <em><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+smooth+sets">on smooth sets</a></em> and <em><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+manifolds+and+orbifolds">on manifolds and orbifolds</a></em>. For reference we first briefly recall this <a class="existingWikiWord" href="/nlab/show/bosonic+modality">bosonic</a> situation.</p> <p>The following simple definitions <a class="maruku-ref" href="#SmoothSet"></a> and <a class="maruku-ref" href="#FormalSmoothSets"></a> are key to the whole theory. They embody the perspective of <em><a class="existingWikiWord" href="/nlab/show/functorial+geometry">functorial geometry</a></em> (<a href="#Grothendieck65">Grothendieck 65</a>). See remark <a class="maruku-ref" href="#ASheafAsASpace"></a> below for exegesis and illustration.</p> <div class="num_defn" id="SiteCartSp"> <h6 id="definition_4">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/CartSp">CartSp</a>)</strong></p> <p>Write <a class="existingWikiWord" href="/nlab/show/CartSp">CartSp</a> for the <a class="existingWikiWord" href="/nlab/show/category">category</a> of <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> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in\mathbb{N}</annotation></semantics></math> with <a class="existingWikiWord" href="/nlab/show/smooth+functions">smooth functions</a> between them. Say that a collection of morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><mi>X</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{U_i \to X\}</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CartSp</mi></mrow><annotation encoding="application/x-tex">CartSp</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/covering">covering</a> if this is a <a class="existingWikiWord" href="/nlab/show/good+open+cover">good open cover</a> in that every finite non-empty intersection of the charts is <a class="existingWikiWord" href="/nlab/show/diffeomorphism">diffeomorphic</a> to a Cartesian space. This defines a <a class="existingWikiWord" href="/nlab/show/coverage">coverage</a> on <a class="existingWikiWord" href="/nlab/show/CartSp">CartSp</a> and hence makes it a <a class="existingWikiWord" href="/nlab/show/site">site</a>.</p> </div> <div class="num_defn" id="SmoothSet"> <h6 id="definition_5">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/smooth+set">smooth set</a>)</strong></p> <p>We say a <em><a class="existingWikiWord" href="/nlab/show/smooth+set">smooth set</a></em> (this Def.ets#CategoryOfSmoothSets)) is, equivalently, a <a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a> on <a class="existingWikiWord" href="/nlab/show/CartSp">CartSp</a> (Def. <a class="maruku-ref" href="#SiteCartSp"></a>), according to this Prop.ets#SmoothSetsAreSheavesOnCartSp). We write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>SmoothSet</mi><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mi>Sh</mi><mo stretchy="false">(</mo><mi>CartSp</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> SmoothSet \;\coloneqq\; Sh(CartSp) </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/sheaf+topos">sheaf topos</a> (<a href="geometry+of+physics+--+categories+and+toposes#Sheaf">this Def.</a>) of smooth set.</p> </div> <div class="num_remark" id="ASheafAsASpace"> <h6 id="remark_2">Remark</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/functorial+geometry">functorial geometry</a>)</strong></p> <p>The useful way to think of def. <a class="maruku-ref" href="#SmoothSet"></a> in the present context is as defining a kind of <a class="existingWikiWord" href="/nlab/show/generalized+smooth+space">generalized smooth space</a> which is <em>defined</em> by which smooth functions from Cartesian spaces it receives (see also at <em><a class="existingWikiWord" href="/nlab/show/motivation+for+sheaves%2C+cohomology+and+higher+stacks">motivation for sheaves, cohomology and higher stacks</a></em> for more exposition of this point).</p> <p>Namely a smooth set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> in the sense of def. <a class="maruku-ref" href="#SmoothSet"></a> is first of all a rule</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>p</mi></msup><mo>↦</mo><mi>X</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>p</mi></msup><mo stretchy="false">)</mo><mo>∈</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex"> \mathbb{R}^p \mapsto X(\mathbb{R}^p) \in Set </annotation></semantics></math></div> <p>which assigns a set to each Cartesian space. We are to think of this set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X(\mathbb{R}^n)</annotation></semantics></math> as the set of smooth functions {“<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup><mover><mo>→</mo><mrow></mrow></mover><mi>X</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}^n \stackrel{}{\to} X</annotation></semantics></math>”}, only that there is no pre-defined concept of smoothness of functions into <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, instead it is defined by that very rule. Moreover, for every smooth function <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> between Cartesian spaces, there is to be a corresponding function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">f^\ast</annotation></semantics></math> between these sets, going in the opposite direction</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>ℝ</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msup></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>f</mi></mpadded></msup></mtd></mtr> <mtr><mtd><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup></mtd></mtr></mtable></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>↦</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mtable><mtr><mtd><mi>X</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msup><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↑</mo> <mpadded width="0"><mrow><msup><mi>f</mi> <mo>*</mo></msup></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>X</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathbb{R}^{n_1} \\ \downarrow^{\mathrlap{f}} \\ \mathbb{R}^{n_2} } \;\;\; \mapsto \;\;\; \array{ X(\mathbb{R}^{n_1}) \\ \uparrow^{\mathrlap{f^\ast}} \\ X(\mathbb{R}^{n_2}) } </annotation></semantics></math></div> <p>which we are to think of as being the precomposition operation</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><mtext>"</mtext><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup><mover><mo>→</mo><mi>ϕ</mi></mover><mi>X</mi><mtext>"</mtext><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>↦</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><mtext>"</mtext><msup><mi>f</mi> <mo>*</mo></msup><mi>ϕ</mi><mo lspace="verythinmathspace">:</mo><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msup><mover><mo>→</mo><mi>f</mi></mover><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup><mover><mo>→</mo><mi>ϕ</mi></mover><mi>X</mi><mtext>"</mtext><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (\text{"}\mathbb{R}^{n_2} \stackrel{\phi}{\to} X\text{"}) \;\mapsto\; (\text{"}f^\ast \phi \colon \mathbb{R}^{n_1} \overset{f}{\to} \mathbb{R}^{n_2} \overset{\phi}{\to} X\text{"}) \,. </annotation></semantics></math></div> <p>This is required to satisfy the evident conditions that <a class="existingWikiWord" href="/nlab/show/composition">composition</a> and <a class="existingWikiWord" href="/nlab/show/identity">identity</a> is respected, in that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>g</mi><mo>∘</mo><mi>f</mi><msup><mo stretchy="false">)</mo> <mo>*</mo></msup><mo>=</mo><msup><mi>g</mi> <mo>*</mo></msup><mo>∘</mo><msup><mi>f</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">(g \circ f)^\ast = g^\ast \circ f^\ast </annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>id</mi> <mo>*</mo></msup><mo>=</mo><mi>id</mi></mrow><annotation encoding="application/x-tex">id^\ast = id</annotation></semantics></math>. Together these conditions say that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/presheaf">presheaf</a> on the category of test spaces – the “<a class="existingWikiWord" href="/nlab/show/functor+of+points">functor of points</a>” (<a href="#Grothendieck65">Grothendieck 65</a>).</p> <p>In addition, the requirement that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a> on the <a class="existingWikiWord" href="/nlab/show/site">site</a> of Cartesian spaces from def. <a class="maruku-ref" href="#SiteCartSp"></a> means that the assignment <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>↦</mo><mi>X</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{R}^n \mapsto X(\mathbb{R}^n)</annotation></semantics></math> knows that one coordinate system <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> may be covered by other coordinate systems. Namely let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mover><mo>→</mo><mrow><msub><mi>ϕ</mi> <mi>i</mi></msub></mrow></mover><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{U_i \overset{\phi_i}{\to}\}</annotation></semantics></math> be an <a class="existingWikiWord" href="/nlab/show/open+cover">open cover</a> by <a class="existingWikiWord" href="/nlab/show/open+balls">open balls</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">U_i</annotation></semantics></math> (each of which may be identified with <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> itself, by a <a class="existingWikiWord" href="/nlab/show/diffeomorphism">diffeomorphism</a>) then the condition is that the function</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><mtext>"</mtext><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>→</mo><mi>X</mi><mtext>"</mtext><mo stretchy="false">)</mo><mo>↦</mo><mrow><mo>{</mo><mtext>"</mtext><msub><mi>U</mi> <mi>i</mi></msub><mover><mo>→</mo><mrow><msub><mi>ϕ</mi> <mi>i</mi></msub></mrow></mover><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>→</mo><mi>X</mi><mtext>"</mtext><mo>}</mo></mrow></mrow><annotation encoding="application/x-tex"> (\text{"}\mathbb{R}^n \to X\text{"}) \mapsto \left\{ \text{"}U_i \overset{\phi_i}{\to} \mathbb{R}^n \to X\text{"} \right\} </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/bijection">bijection</a> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X(\mathbb{R}^n)</annotation></semantics></math> to the subset of the <a class="existingWikiWord" href="/nlab/show/Cartesian+product">Cartesian product</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo><mi>i</mi></munder><mi>X</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\underset{i}{\prod} X(U_i)</annotation></semantics></math> of those <a class="existingWikiWord" href="/nlab/show/tuples">tuples</a> of functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub><mover><mo>→</mo><mrow><msub><mi>f</mi> <mi>i</mi></msub></mrow></mover><mi>X</mi></mrow><annotation encoding="application/x-tex">U_i \stackrel{f_i}{\to} X</annotation></semantics></math> which coincide on all <a class="existingWikiWord" href="/nlab/show/intersections">intersections</a>: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mi>i</mi></msub><msub><mo stretchy="false">|</mo> <mrow><msub><mi>U</mi> <mi>i</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>j</mi></msub></mrow></msub><mo>=</mo><msub><mi>f</mi> <mi>j</mi></msub><msub><mo stretchy="false">|</mo> <mrow><msub><mi>U</mi> <mi>i</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>j</mi></msub></mrow></msub></mrow><annotation encoding="application/x-tex">f_i |_{U_i \cap U_j} = f_j |_{U_i \cap U_j}</annotation></semantics></math> (“<a class="existingWikiWord" href="/nlab/show/matching+families">matching families</a>”).</p> <p>For example every <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><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> defines a smooth set by the rule</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>↦</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathbb{R}^n \mapsto C^\infty(\mathbb{R}^n ,X) </annotation></semantics></math></div> <p>which says that the set of would-be smooth functions into <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is the actual set of smooth functions into <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>. One also says that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/representable+functor">represents</a> a sheaf on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CartSp</mi></mrow><annotation encoding="application/x-tex">CartSp</annotation></semantics></math>,</p> <p>This defines a <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> inclusion of Cartesian spaces into smooth sets</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>y</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>CartSp</mi><mo>↪</mo><mi>SmoothSet</mi></mrow><annotation encoding="application/x-tex"> y \;\colon\; CartSp \hookrightarrow SmoothSet </annotation></semantics></math></div> <p>called the <em><a class="existingWikiWord" href="/nlab/show/Yoneda+embedding">Yoneda embedding</a></em>.</p> <p>The same construction works for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> any <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a>: it is regarded as a smooth set defined by the rule</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>↦</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathbb{R}^n \mapsto C^\infty(\mathbb{R}^n ,X) </annotation></semantics></math></div> <p>which assigns actual sets of smooth functions.</p> <p>Notice that since Cartesian spaces (and smooth manifolds) themselves are understood as special cases of smooth sets, there now appears an actual concept of smooth functions of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}^n \to X</annotation></semantics></math>, for every smooth set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, without quotation marks: namely this is a morphism in the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SmoothSet</mi></mrow><annotation encoding="application/x-tex">SmoothSet</annotation></semantics></math> of smooth sets.</p> <p>Now there might be a worry: given any smooth set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> and any Cartesian space <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>, we seem to have <em>two different</em> concepts of what the set of smooth functions from <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> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is: the set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X(\mathbb{R}^n)</annotation></semantics></math> of “smooth functions by fiat” (maps with quotation marks) and the actual <a class="existingWikiWord" href="/nlab/show/hom-set">hom-set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mi>SmoothSet</mi></msub><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Hom_{SmoothSet}(\mathbb{R}^n, X)</annotation></semantics></math> (maps without quotation mark).</p> <p>That these two sets are in fact in <a class="existingWikiWord" href="/nlab/show/natural+bijection">natural bijection</a>, hence that the interpretation of a sheaf <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> as a generalized smooth space is consistent, is the statement of the <em><a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a></em>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mi>SmoothSet</mi></msub><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Hom_{SmoothSet}(\mathbb{R}^n, X) \; \simeq \; X(\mathbb{R}^n) \,. </annotation></semantics></math></div> <p>Hence the <a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a> says that we may remove the quotation marks:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo>≃</mo><mrow><mo>{</mo><mi>morphisms</mi><mspace width="thickmathspace"></mspace><mi>of</mi><mspace width="thickmathspace"></mspace><mi>smooth</mi><mspace width="thickmathspace"></mspace><mi>spaces</mi><mspace width="thickmathspace"></mspace><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>→</mo><mi>X</mi><mo>}</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> X(\mathbb{R}^n) \simeq \left\{ morphisms\;of\;smooth\;spaces\; \mathbb{R}^n \to X \right\} \,. </annotation></semantics></math></div></div> <div class="num_remark" id="GeneralizedSpaces"> <h6 id="remark_3">Remark</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/generalized+spaces">generalized spaces</a>)</strong></p> <p>The strategy is now to work in the nice category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SmoothSets</mi></mrow><annotation encoding="application/x-tex">SmoothSets</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/generalized+smooth+spaces">generalized smooth spaces</a> (a <a class="existingWikiWord" href="/nlab/show/topos">topos</a>), and find in there <a class="existingWikiWord" href="/nlab/show/full+subcategories">full subcategories</a> of more specific types of smooth spaces having extra properties which one may need in given applications. There is a long list of such subcategories of relevance, some of these we briefly discuss now:</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo></mrow><annotation encoding="application/x-tex">\{</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Cartesian+spaces">Cartesian spaces</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\}</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↪</mo></mrow><annotation encoding="application/x-tex">\hookrightarrow</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo></mrow><annotation encoding="application/x-tex">\{</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/smooth+manifolds">smooth manifolds</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\}</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↪</mo></mrow><annotation encoding="application/x-tex">\hookrightarrow</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo></mrow><annotation encoding="application/x-tex">\{</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Hilbert+manifolds">Hilbert manifolds</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\}</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↪</mo></mrow><annotation encoding="application/x-tex">\hookrightarrow</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo></mrow><annotation encoding="application/x-tex">\{</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Banach+manifolds">Banach manifolds</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\}</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↪</mo></mrow><annotation encoding="application/x-tex">\hookrightarrow</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo></mrow><annotation encoding="application/x-tex">\{</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Fr%C3%A9chet+manifolds">Fréchet manifolds</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\}</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↪</mo></mrow><annotation encoding="application/x-tex">\hookrightarrow</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo></mrow><annotation encoding="application/x-tex">\{</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/diffeological+spaces">diffeological spaces</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\}</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↪</mo></mrow><annotation encoding="application/x-tex">\hookrightarrow</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>⋯</mi></mrow><annotation encoding="application/x-tex">\cdots</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↪</mo></mrow><annotation encoding="application/x-tex">\hookrightarrow</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo></mrow><annotation encoding="application/x-tex">\{</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/smooth+sets">smooth sets</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\}</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↪</mo></mrow><annotation encoding="application/x-tex">\hookrightarrow</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo></mrow><annotation encoding="application/x-tex">\{</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/formal+smooth+sets">formal smooth sets</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\}</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↪</mo></mrow><annotation encoding="application/x-tex">\hookrightarrow</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo></mrow><annotation encoding="application/x-tex">\{</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/super+formal+smooth+sets">super formal smooth sets</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\}</annotation></semantics></math></p> <p>and similarly for their supergeometric version (which we turn to below, def. <a class="maruku-ref" href="#FormalSmoothSets"></a>)</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo></mrow><annotation encoding="application/x-tex">\{</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Cartesian+spaces">Cartesian spaces</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\}</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↪</mo></mrow><annotation encoding="application/x-tex">\hookrightarrow</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo></mrow><annotation encoding="application/x-tex">\{</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/super+Cartesian+spaces">super Cartesian spaces</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\}</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↪</mo></mrow><annotation encoding="application/x-tex">\hookrightarrow</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo></mrow><annotation encoding="application/x-tex">\{</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/supermanifolds">supermanifolds</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\}</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↪</mo></mrow><annotation encoding="application/x-tex">\hookrightarrow</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>⋯</mi></mrow><annotation encoding="application/x-tex">\cdots</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↪</mo></mrow><annotation encoding="application/x-tex">\hookrightarrow</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo></mrow><annotation encoding="application/x-tex">\{</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/super+formal+smooth+sets">super smooth sets</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\}</annotation></semantics></math> .</p> </div> <p>By the strategy of remark <a class="maruku-ref" href="#GeneralizedSpaces"></a> we now pass to generalized spaces which are locally modeled not just on plain Cartesian spaces, but also on <a class="existingWikiWord" href="/nlab/show/formal+Cartesian+spaces">formal Cartesian spaces</a> and <a class="existingWikiWord" href="/nlab/show/super+formal+Cartesian+spaces">super formal Cartesian spaces</a></p> <div class="num_defn" id="FormalSmoothSets"> <h6 id="definition_6">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/super+formal+smooth+sets">super formal smooth sets</a>)</strong></p> <p>Define a <a class="existingWikiWord" href="/nlab/show/coverage">coverage</a> on the categories <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>FormalCartSp</mi></mrow><annotation encoding="application/x-tex">FormalCartSp</annotation></semantics></math> (def. <a class="maruku-ref" href="#FormalCartSp"></a>) and on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SuperFormalCartSp</mi></mrow><annotation encoding="application/x-tex">SuperFormalCartSp</annotation></semantics></math> (def. <a class="maruku-ref" href="#SuperFormalCartSp"></a>) by declaring the <a class="existingWikiWord" href="/nlab/show/covering">covering</a> families of any object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>×</mo><mi>𝔻</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}^n \times \mathbb{D}</annotation></semantics></math> (hence for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔻</mi></mrow><annotation encoding="application/x-tex">\mathbb{D}</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒪</mi><mo stretchy="false">(</mo><mi>𝔻</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>ℝ</mi><mo>⊕</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{O}(\mathbb{D}) = (\mathbb{R} \oplus V)</annotation></semantics></math> any infinitesimally thickened superpoint) to be those of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>×</mo><mi>𝔻</mi><mover><mo>⟶</mo><mrow><msub><mi>ϕ</mi> <mi>i</mi></msub><mo>×</mo><mi>id</mi></mrow></mover><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>×</mo><mi>𝔻</mi><mo>}</mo></mrow></mrow><annotation encoding="application/x-tex"> \left\{ U_i \times \mathbb{D} \overset{\phi_i \times id}{\longrightarrow} \mathbb{R}^n \times \mathbb{D} \right\} </annotation></semantics></math></div> <p>for</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>{</mo><msub><mi>U</mi> <mi>i</mi></msub><mover><mo>⟶</mo><mrow><msub><mi>ϕ</mi> <mi>i</mi></msub></mrow></mover><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>}</mo></mrow></mrow><annotation encoding="application/x-tex"> \left\{ U_i \overset{\phi_i }{\longrightarrow} \mathbb{R}^n \right\} </annotation></semantics></math></div> <p>a <a class="existingWikiWord" href="/nlab/show/good+open+cover">good open cover</a> as in def. <a class="maruku-ref" href="#SiteCartSp"></a>.</p> <p>In analogy with def. <a class="maruku-ref" href="#SmoothSet"></a> we say that</p> <ol> <li>a <a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>FormalCartSp</mi></mrow><annotation encoding="application/x-tex">FormalCartSp</annotation></semantics></math> is a <strong><a class="existingWikiWord" href="/nlab/show/formal+smooth+set">formal smooth set</a></strong> or <strong>formal smooth <a class="existingWikiWord" href="/nlab/show/0-type">0-type</a></strong> and we write</li> </ol> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>FormalSmoothSet</mi><mo>≔</mo><mi>FormalSmoothSet</mi><mo>≔</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>FormalCartSp</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> FormalSmoothSet \coloneqq FormalSmoothSet \coloneqq Sh(FormalCartSp) </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/sheaf+topos">sheaf topos</a> of all of these;</p> <ol> <li>a <a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>FormalCartSp</mi></mrow><annotation encoding="application/x-tex">FormalCartSp</annotation></semantics></math> is a <strong><a class="existingWikiWord" href="/nlab/show/super+formal+smooth+set">super formal smooth set</a></strong> or <strong>super formal smooth <a class="existingWikiWord" href="/nlab/show/0-type">0-type</a></strong> and we write</li> </ol> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>SuperFormalSmoothSet</mi><mo>≔</mo><mi>SuperFormalSmoothSet</mi><mo>≔</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>FormalCartSp</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> SuperFormalSmoothSet \coloneqq SuperFormalSmoothSet \coloneqq Sh(FormalCartSp) </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/sheaf+topos">sheaf topos</a> of all of these;</p> </div> <p>The category of <a class="existingWikiWord" href="/nlab/show/formal+smooth+sets">formal smooth sets</a> from def. <a class="maruku-ref" href="#FormalSmoothSets"></a> is often known as the <em><a class="existingWikiWord" href="/nlab/show/Cahiers+topos">Cahiers topos</a></em>. It was introduced in (<a href="Cahiers+topos#Dubuc79">Dubuc 79</a>) as a well-adapted model for the <a class="existingWikiWord" href="/nlab/show/Kock-Lawvere+axioms">Kock-Lawvere axioms</a> for <a class="existingWikiWord" href="/nlab/show/synthetic+differential+geometry">synthetic differential geometry</a>. The category of <a class="existingWikiWord" href="/nlab/show/super+formal+smooth+sets">super formal smooth sets</a> from def. <a class="maruku-ref" href="#FormalSmoothSets"></a> was considered in <a href="#Yetter88">Yetter 88</a>, called the <em>super Dubuc topos</em> there.</p> <p>We have now defined four sites and considered the corresponding <a class="existingWikiWord" href="/nlab/show/categories+of+sheaves">categories of sheaves</a> (noticing that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sh</mi><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo><mo>=</mo></mrow><annotation encoding="application/x-tex">Sh(\ast) = </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Set">Set</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>𝒞</mi><mo>=</mo></mtd> <mtd><mo>*</mo></mtd> <mtd><mi>CartSp</mi></mtd> <mtd><mi>FormalCartSp</mi></mtd> <mtd><mi>SuperFormalCartSp</mi></mtd></mtr> <mtr><mtd></mtd></mtr> <mtr><mtd><mi>Sh</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo><mo>=</mo></mtd> <mtd><mi>Set</mi></mtd> <mtd><mi>SmoothSet</mi></mtd> <mtd><mi>FormalSmoothSet</mi></mtd> <mtd><mi>SuperFormalSmoothSet</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \mathcal{C} = & \ast & CartSp & FormalCartSp & SuperFormalCartSp \\ \\ Sh(\mathcal{C}) = & Set & SmoothSet & FormalSmoothSet & SuperFormalSmoothSet } \,. </annotation></semantics></math></div> <p>Moreover, by prop. <a class="maruku-ref" href="#SystemOfSites"></a> these four sites form a <a class="existingWikiWord" href="/nlab/show/filtration">filtration</a> of the site <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SuperFormalCartSp</mi></mrow><annotation encoding="application/x-tex">SuperFormalCartSp</annotation></semantics></math> by consecutively smaller subsites, where each inclusion is either <a class="existingWikiWord" href="/nlab/show/reflective+subcategory">reflective</a> or <a class="existingWikiWord" href="/nlab/show/coreflective+subcategory">coreflective</a> or both. The following states that this filtration of sites extends to their categories of sheaves.</p> <div class="num_prop" id="SuperSmoothSetsSystemOfAdjunctions"> <h6 id="proposition_7">Proposition</h6> <p><strong>(progression of (<a class="existingWikiWord" href="/nlab/show/coreflective+subcategory">co-</a>)<a class="existingWikiWord" href="/nlab/show/reflective+subcategories">reflective subcategories</a> of <a class="existingWikiWord" href="/nlab/show/SuperFormalSmoothSet">SuperFormalSmoothSet</a>)</strong></p> <p>The <a class="existingWikiWord" href="/nlab/show/sheaf+topos">sheaf topos</a> <a class="existingWikiWord" href="/nlab/show/SuperFormalSmoothSet">SuperFormalSmoothSet</a> (Def. <a class="maruku-ref" href="#FormalSmoothSets"></a>) is a <em><a class="existingWikiWord" href="/nlab/show/solid+topos">solid topos</a></em> over <a class="existingWikiWord" href="/nlab/show/FormalSmoothSet">FormalSmoothSet</a>, in that:</p> <p>There exists an essentially unique system of <a class="existingWikiWord" href="/nlab/show/functors">functors</a> between the categories of <a class="existingWikiWord" href="/nlab/show/sets">sets</a>, <a class="existingWikiWord" href="/nlab/show/smooth+sets">smooth sets</a> (def. <a class="maruku-ref" href="#SmoothSet"></a>), <a class="existingWikiWord" href="/nlab/show/formal+smooth+sets">formal smooth sets</a> and <a class="existingWikiWord" href="/nlab/show/super+formal+smooth+sets">super formal smooth sets</a> (def. <a class="maruku-ref" href="#FormalSmoothSets"></a>) as shown in the second and third row of the following diagram, such that</p> <ol> <li> <p>every <a class="existingWikiWord" href="/nlab/show/pair">pair</a> of consecutive <a class="existingWikiWord" href="/nlab/show/functors">functors</a> is an <a class="existingWikiWord" href="/nlab/show/adjoint+pair">adjoint pair</a> (<a href="geometry+of+physics+--+categories+and+toposes#AdjointFunctorsInTermsOfNaturalBijectionOfHomSets">this Def.</a>), with the functor above being <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> and the functor below being <a class="existingWikiWord" href="/nlab/show/right+adjoint">right adjoint</a></p> </li> <li> <p>on <a class="existingWikiWord" href="/nlab/show/representables">representables</a> the top two functors on the left, the top two functors in the middle, and the top three functors on the right coincide with the corresponding functors between <a class="existingWikiWord" href="/nlab/show/sites">sites</a> shown in the first row (from prop. <a class="maruku-ref" href="#SystemOfSites"></a>):</p> </li> </ol> <center> <img src="https://ncatlab.org/nlab/files/AjunctionsForSuperCohesion.png" width="600" /> </center> <p>In such a situation we also say that in the third row</p> <ol> <li> <p>the bottom four functors exhibit <a class="existingWikiWord" href="/nlab/show/SuperFormalSmoothSet">SuperFormalSmoothSet</a> as a <a class="existingWikiWord" href="/nlab/show/cohesive+topos">cohesive topos</a> (<a href="geometry+of+physics+--+categories+and+toposes#CohesiveTopos">this Def.</a>)</p> </li> <li> <p>the middle four functors exhibit <a class="existingWikiWord" href="/nlab/show/SuperFormalSmoothSet">SuperFormalSmoothSet</a> as <a class="existingWikiWord" href="/nlab/show/differentially+cohesive+topos">differentially cohesive topos</a> (elastic topos) relative to <a class="existingWikiWord" href="/nlab/show/SmoothSet">SmoothSet</a> (<a href="https://ncatlab.org/nlab/show/geometry+of+physics+--+categories+and+toposes#DifferentialCohesion">this Def.</a>);</p> </li> <li> <p>the top four functors exhibit <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SuperFormalSmoothSet</mi></mrow><annotation encoding="application/x-tex">SuperFormalSmoothSet</annotation></semantics></math> as <em>solid topos</em> relative to <a class="existingWikiWord" href="/nlab/show/FormalSmoothSet">FormalSmoothSet</a> (<a href="geometry+of+physics+--+categories+and+toposes#SuperDifferentialCohesion">this Def.</a>)</p> </li> </ol> </div> <div class="proof"> <h6 id="proof_5">Proof</h6> <p>The system of functors between sites in Prop. <a class="maruku-ref" href="#SystemOfSites"></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><mfrac linethickness="0"><mrow><mtext></mtext></mrow><mrow><mfrac linethickness="0"><mrow><mtext>discrete</mtext></mrow><mrow><mtext>geometry</mtext></mrow></mfrac></mrow></mfrac></mtd> <mtd></mtd> <mtd><mfrac linethickness="0"><mrow><mtext></mtext></mrow><mrow><mfrac linethickness="0"><mrow><mtext>differential</mtext></mrow><mrow><mtext>geometry</mtext></mrow></mfrac></mrow></mfrac></mtd> <mtd></mtd> <mtd><mfrac linethickness="0"><mrow><mtext>formal</mtext></mrow><mrow><mfrac linethickness="0"><mrow><mtext>differential</mtext></mrow><mrow><mi>geometry</mi></mrow></mfrac></mrow></mfrac></mtd> <mtd></mtd> <mtd><mfrac linethickness="0"><mrow><mtext>super</mtext></mrow><mrow><mfrac linethickness="0"><mrow><mtext>differential</mtext></mrow><mrow><mtext>geometry</mtext></mrow></mfrac></mrow></mfrac></mtd></mtr> <mtr><mtd></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd><mrow><mtable><mtr><mtd><mover><mo>⟵</mo><mrow><mphantom><mi>AA</mi></mphantom><mi>Π</mi><mphantom><mi>AA</mi></mphantom></mrow></mover></mtd></mtr> <mtr><mtd><mover><mo>↪</mo><mrow><mphantom><mi>AA</mi></mphantom><mi>Disc</mi><mphantom><mi>AA</mi></mphantom></mrow></mover></mtd></mtr></mtable></mrow></mtd> <mtd><mi>CartSp</mi></mtd> <mtd><mrow><mtable><mtr><mtd><mover><mo>↪</mo><mrow><mphantom><mi>AA</mi></mphantom><msub><mi>ι</mi> <mi>inf</mi></msub><mphantom><mi>AA</mi></mphantom></mrow></mover></mtd></mtr> <mtr><mtd><mover><mo>⟵</mo><mrow><mphantom><mi>AA</mi></mphantom><msub><mi>Π</mi> <mi>inf</mi></msub><mphantom><mi>AA</mi></mphantom></mrow></mover></mtd></mtr> <mtr><mtd><mphantom><mi>A</mi></mphantom></mtd></mtr> <mtr><mtd><mphantom><mfrac linethickness="0"><mrow><mi>A</mi></mrow><mrow><mi>A</mi></mrow></mfrac></mphantom></mtd></mtr></mtable></mrow></mtd> <mtd><mi>FormalCartSp</mi></mtd> <mtd><mrow><mtable><mtr><mtd><mover><mo>⟵</mo><mrow><mphantom><mi>AA</mi></mphantom><mi>even</mi><mphantom><mi>AA</mi></mphantom></mrow></mover></mtd></mtr> <mtr><mtd><mover><mo>↪</mo><mrow><mphantom><mi>AA</mi></mphantom><msub><mi>ι</mi> <mi>sup</mi></msub><mphantom><mi>AA</mi></mphantom></mrow></mover></mtd></mtr> <mtr><mtd><mover><mo>⟵</mo><mrow><mphantom><mi>AA</mi></mphantom><msub><mi>Π</mi> <mi>sup</mi></msub><mphantom><mi>AA</mi></mphantom></mrow></mover></mtd></mtr> <mtr><mtd><mphantom><mi>A</mi></mphantom></mtd></mtr> <mtr><mtd><mphantom><mfrac linethickness="0"><mrow><mi>A</mi></mrow><mrow><mi>A</mi></mrow></mfrac></mphantom></mtd></mtr> <mtr><mtd><mphantom><mfrac linethickness="0"><mrow><mi>A</mi></mrow><mrow><mi>A</mi></mrow></mfrac></mphantom></mtd></mtr></mtable></mrow></mtd> <mtd><mi>SuperFormalCartSp</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ { \text{} \atop { \text{discrete} \atop {\text{geometry}} } } && {\text{} \atop {\text{differential} \atop \text{geometry}}} && {\text{formal} \atop {\text{differential} \atop {geometry}}} && {\text{super} \atop {\text{differential} \atop \text{geometry}}} \\ \\ \ast & \array{ \overset{\phantom{AA}\Pi\phantom{AA}}{\longleftarrow} \\ \overset{\phantom{AA}Disc\phantom{AA}}{\hookrightarrow} } & CartSp & \array{ \overset{\phantom{AA}\iota_{inf}\phantom{AA}}{\hookrightarrow} \\ \overset{\phantom{AA}\Pi_{inf}\phantom{AA}}{\longleftarrow} \\ \phantom{A} \\ \phantom{A \atop A} } & FormalCartSp & \array{ \overset{\phantom{AA}even\phantom{AA}}{\longleftarrow} \\ \overset{\phantom{AA}\iota_{sup}\phantom{AA}}{\hookrightarrow} \\ \overset{\phantom{AA}\Pi_{sup}\phantom{AA}}{\longleftarrow} \\ \phantom{A} \\ \phantom{A \atop A} \\ \phantom{A \atop A} } & SuperFormalCartSp } \,. </annotation></semantics></math></div> <p>induces the claimed <a class="existingWikiWord" href="/nlab/show/adjoint+quadruples">adjoint quadruples</a> between <a class="existingWikiWord" href="/nlab/show/presheaf+toposes">presheaf toposes</a> in the second row, by <a class="existingWikiWord" href="/nlab/show/Kan+extension">Kan extension</a> (<a href="geometry+of+physics+--+categories+and+toposes#KanExtensionOfAdjointPairIsAdjointQuadruple">this Example</a> in the chapter <em><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+categories+and+toposes">on categories and toposes</a></em>).</p> <p>That the <a class="existingWikiWord" href="/nlab/show/adjoint+quadruple">adjoint quadruple</a> on the left (<a class="existingWikiWord" href="/nlab/show/corestriction">co-</a>)<a class="existingWikiWord" href="/nlab/show/restriction">restricts</a> to <a class="existingWikiWord" href="/nlab/show/sheaves">sheaves</a>, exhibiting <a class="existingWikiWord" href="/nlab/show/SmoothSet">SmoothSet</a> as a <a class="existingWikiWord" href="/nlab/show/cohesive+topos">cohesive topos</a>, is <a href="geometry+of+physics+--+smooth+sets#SmoothSetsFormACohesiveTopos">this Prop.</a> in the chapter <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+smooth+sets">on smooth sets</a>.</p> <p>For the other two <a class="existingWikiWord" href="/nlab/show/adjoint+quadruples">adjoint quadruples</a> the (<a class="existingWikiWord" href="/nlab/show/corestriction">co-</a>)<a class="existingWikiWord" href="/nlab/show/restriction">restricts</a> to <a class="existingWikiWord" href="/nlab/show/sheaves">sheaves</a> is trivial, since this concerns only the <a class="existingWikiWord" href="/nlab/show/coverages">coverages</a> along the infinitsimal directions in <a class="existingWikiWord" href="/nlab/show/FormalCartSp">FormalCartSp</a> and <a class="existingWikiWord" href="/nlab/show/SuperFormalCartSp">SuperFormalCartSp</a>, which are <a class="existingWikiWord" href="/nlab/show/trivial+coverage">trivial</a> (<a href="geometry+of+physics+--+categories+and+toposes#TrivialCoverage">this Example</a>), by definition.</p> <p>This establishes the system of <a class="existingWikiWord" href="/nlab/show/adjoint+quadruples">adjoint quadruples</a> between <a class="existingWikiWord" href="/nlab/show/sheaf+toposes">sheaf toposes</a> in the second row.</p> <p>The diagram in the third row states that two extra adjoints to <em>composites</em> of the functors appear. Here</p> <ol> <li> <p>the <a class="existingWikiWord" href="/nlab/show/right+adjoint">right adjoint</a> functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>SmoothSet</mi><mo>⟵</mo><mi>SuperFormalSmoothSet</mi></mrow><annotation encoding="application/x-tex"> SmoothSet \longleftarrow SuperFormalSmoothSet </annotation></semantics></math></div> <p>exists as part of the <a class="existingWikiWord" href="/nlab/show/adjoint+quadruple">adjoint quadruple</a> which is induced (<a href="geometry+of+physics+--+categories+and+toposes#KanExtensionOfAdjointPairIsAdjointQuadruple">this Example</a>) from the <em><a class="existingWikiWord" href="/nlab/show/composition">composite</a></em> <a class="existingWikiWord" href="/nlab/show/coreflective+subcategory">coreflective inclusion</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>CartSp</mi><mrow><mtable><mtr><mtd><mover><mo>↪</mo><mrow><mphantom><mi>A</mi></mphantom><msub><mi>ι</mi> <mi>inf</mi></msub><mphantom><mi>A</mi></mphantom></mrow></mover></mtd></mtr> <mtr><mtd><mover><mo>⟵</mo><mrow><mphantom><mi>AA</mi></mphantom><msub><mi>Π</mi> <mi>inf</mi></msub><mphantom><mi>AA</mi></mphantom></mrow></mover></mtd></mtr> <mtr><mtd><mphantom><mover><mo>⟵</mo><mrow><mphantom><mi>AA</mi></mphantom><msub><mi>Π</mi> <mi>sup</mi></msub><mphantom><mi>AA</mi></mphantom></mrow></mover></mphantom></mtd></mtr></mtable></mrow><mi>FormalCartSp</mi><mrow><mtable><mtr><mtd><mover><mo>↪</mo><mrow><mphantom><mi>A</mi></mphantom><msub><mi>ι</mi> <mi>sup</mi></msub><mphantom><mi>A</mi></mphantom></mrow></mover></mtd></mtr> <mtr><mtd><mover><mo>⟵</mo><mrow><mphantom><mi>AA</mi></mphantom><msub><mi>Π</mi> <mi>sup</mi></msub><mphantom><mi>AA</mi></mphantom></mrow></mover></mtd></mtr> <mtr><mtd><mphantom><mover><mo>⟵</mo><mrow><mphantom><mi>AA</mi></mphantom><mover><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><mo>⇉</mo></mover><mphantom><mi>AA</mi></mphantom></mrow></mover></mphantom></mtd></mtr></mtable></mrow><mi>SuperFormalCartSp</mi></mrow><annotation encoding="application/x-tex"> CartSp \array{ \overset{\phantom{A}\iota_{inf}\phantom{A}}{\hookrightarrow} \\ \overset{\phantom{AA} \Pi_{inf} \phantom{AA}}{\longleftarrow} \\ \phantom{\overset{\phantom{AA} \Pi_{sup} \phantom{AA}}{\longleftarrow}} } FormalCartSp \array{ \overset{\phantom{A}\iota_{sup}\phantom{A}}{\hookrightarrow} \\ \overset{\phantom{AA} \Pi_{sup} \phantom{AA}}{\longleftarrow} \\ \phantom{\overset{\phantom{AA} \overset{\rightrightarrows}{(-)} \phantom{AA}}{\longleftarrow}} } SuperFormalCartSp </annotation></semantics></math></div> <p>and using that composites of adjoints are adjoint, and that adjoints are unique, when they exist (<a href="geometry+of+physics+--+categories+and+toposes#UniquenessOfAdjoints">this prop.</a>)</p> </li> <li> <p>Notice that also <a class="existingWikiWord" href="/nlab/show/SuperFormalCartSp">SuperFormalCartSp</a> is a <a class="existingWikiWord" href="/nlab/show/cohesive+site">cohesive site</a>: since the <a class="existingWikiWord" href="/nlab/show/coverage">coverage</a> along the infinitesimal directions is <a class="existingWikiWord" href="/nlab/show/trivial+coverage">trivial</a>, while that along the finite directions is the same as that on <a class="existingWikiWord" href="/nlab/show/CartSp">CartSp</a>, this follows with the same <a class="existingWikiWord" href="/nlab/show/proof">proof</a> as for <a class="existingWikiWord" href="/nlab/show/CartSp">CartSp</a> (<a href="geometry+of+physics+--+smooth+sets#SmoothSetsFormACohesiveTopos">this Prop.</a>).</p> <p>Moreover, the <a class="existingWikiWord" href="/nlab/show/composition">composite</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Disc</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>Set</mi><mo>↪</mo><mi>SmoothSet</mi><mo>↪</mo><mi>FormalSmoothSet</mi><mo>↪</mo><mi>SuperFormalSmoothSet</mi></mrow><annotation encoding="application/x-tex"> Disc \;\colon\; Set \hookrightarrow SmoothSet \hookrightarrow FormalSmoothSet \hookrightarrow SuperFormalSmoothSet </annotation></semantics></math></div> <p>from the second row above equals the functor</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>Set</mi></mtd> <mtd><mover><mo>⟶</mo><mi>Disc</mi></mover></mtd> <mtd><mi>SuperFormalSmoothSet</mi></mtd></mtr> <mtr><mtd><mi>S</mi></mtd> <mtd><mo>↦</mo></mtd> <mtd><msub><mi>const</mi> <mi>S</mi></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ Set &\overset{Disc}{\longrightarrow}& SuperFormalSmoothSet \\ S &\mapsto& const_S } </annotation></semantics></math></div> <p>which is part of the <a class="existingWikiWord" href="/nlab/show/adjoint+quadruple">adjoint quadruple</a> of the <a class="existingWikiWord" href="/nlab/show/cohesion">cohesion</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SuperFormalSmoothSet</mi></mrow><annotation encoding="application/x-tex">SuperFormalSmoothSet</annotation></semantics></math> (via <a href="geometry+of+physics+--+smooth+sets#SmoothSetsFormACohesiveTopos">that Prop.</a>)): This is because all these inclusion functors are <a class="existingWikiWord" href="/nlab/show/right+adjoints">right adjoints</a>, and hence <a class="existingWikiWord" href="/nlab/show/preserved+limit">preserve</a> the <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</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">\ast</annotation></semantics></math> (<a href="geometry+of+physics+--+categories+and+toposes#LeftAdjointFunctorPreservesEpi">this Prop.</a>), but also <a class="existingWikiWord" href="/nlab/show/left+adjoints">left adjoints</a>, and hence <a class="existingWikiWord" href="/nlab/show/preserved+limit">preserved limit</a> <a class="existingWikiWord" href="/nlab/show/coproducts">coproducts</a> (<a href="geometry+of+physics+--+categories+and+toposes#AdjointsPreserveCoLimits">this Prop.</a>). This implies the claim because every <a class="existingWikiWord" href="/nlab/show/set">set</a> is a <a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a> of <a class="existingWikiWord" href="/nlab/show/generalized+the">the</a> <a class="existingWikiWord" href="/nlab/show/singleton">singleton</a>, which is the <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a> in <a class="existingWikiWord" href="/nlab/show/Set">Set</a>.</p> <p>This implies, again by uniqueness of adjoints (<a href="geometry+of+physics+--+categories+and+toposes#UniquenessOfAdjoints">this Prop.</a>) that the composite functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Set</mi><mo>⟵</mo><mi>SuperFormalSmoothSet</mi></mrow><annotation encoding="application/x-tex">Set\longleftarrow SuperFormalSmoothSet</annotation></semantics></math> from the previous item is in fact the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Γ</mi> <mi>SuperFormalSmoothSet</mi></msub></mrow><annotation encoding="application/x-tex">\Gamma_{SuperFormalSmoothSet}</annotation></semantics></math> from the <a class="existingWikiWord" href="/nlab/show/cohesive+topos">cohesive topos</a>-structure on <a class="existingWikiWord" href="/nlab/show/SuperFormalSmoothSet">SuperFormalSmoothSet</a> and hence, finally, that there is the bottom right adjoint</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>coDisc</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>Set</mi><mover><mo>↪</mo><mphantom><mi>AA</mi></mphantom></mover><mi>SuperFormalSmoothSet</mi></mrow><annotation encoding="application/x-tex"> coDisc \;\colon\; Set \overset{\phantom{AA}}{\hookrightarrow} SuperFormalSmoothSet </annotation></semantics></math></div> <p>as claimed.</p> </li> </ol> <p>Finally, that every second functor is a <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> inclusion (a <a class="existingWikiWord" href="/nlab/show/fully+faithful+functor">fully faithful functor</a>) as shown follows because</p> <ol> <li> <p>left Kan extension along <a class="existingWikiWord" href="/nlab/show/fully+faithful+functors">fully faithful functors</a> is itself fully faithful (<a href="Kan+extension#LeftKanExtensionBasicProp">this prop.</a>);</p> </li> <li> <p>in an <a class="existingWikiWord" href="/nlab/show/adjoint+triple">adjoint triple</a> the leftmost adjoint is <a class="existingWikiWord" href="/nlab/show/fully+faithful+functor">fully faithful</a> precisely if the rightmost adjoint is (<a href="adjoint+triple#FullyFaithful">this prop.</a>).</p> </li> </ol> </div> <p>Proposition <a class="maruku-ref" href="#SuperSmoothSetsSystemOfAdjunctions"></a> says that the basic operations on local model spaces, such as</p> <ol> <li> <p>forming the underlying bosonic space,</p> </li> <li> <p>forming the underlying formal bosonic space of bi-fermions,</p> </li> <li> <p>forming the underlying reduced space</p> </li> </ol> <p>extend from local model spaces to the generalized spaces modeled on them, while retaining their relation to each other and to the respective inclusions, and such that yet further operations accompanying them are induced.</p> <p>To record what all these operations are, it is useful to compose the functors above in pairs, with one functor projecting down to the left, and the next one including back with either a left or right adjoint of the projection.</p> <p>For example given a <a class="existingWikiWord" href="/nlab/show/generalized+space">generalized</a> <a class="existingWikiWord" href="/nlab/show/superspace">superspace</a>, then applying the top-most functor in prop. <a class="maruku-ref" href="#SuperSmoothSetsSystemOfAdjunctions"></a> to it yields its underlying bi-fermionic formal smooth space, regarded as an object in <a class="existingWikiWord" href="/nlab/show/FormalSmoothSet">FormalSmoothSet</a>. But if we work in <a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a>, then we want this result to be understood as a superspace again, one that just so happens to have no odd directions, so we re-include it by the inclusion right adjoint to the top projection. This composite operation of projection and re-embedding defines a <em><a class="existingWikiWord" href="/nlab/show/modal+operator">modal operator</a></em> (<a href="geometry+of+physics+--+categories+and+toposes#ModalOperator">this Def.</a>), on <a class="existingWikiWord" href="/nlab/show/SuperFormalSmoothSet">SuperFormalSmoothSet</a>, which we denote by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><mo>⇉</mo></mover><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><msub><mi>ι</mi> <mi>sup</mi></msub><mo>∘</mo><mi>even</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>SuperFormalSmoothSet</mi><mover><mo>⟶</mo><mphantom><mi>AAAA</mi></mphantom></mover><mi>SuperFormalSmoothSet</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \overset{\rightrightarrows}{(-)} \;\coloneqq\; \iota_{sup} \circ even \;\colon\; SuperFormalSmoothSet \overset{\phantom{AAAA}}{\longrightarrow} SuperFormalSmoothSet \,. </annotation></semantics></math></div> <p>Similarly, composing the projection which is the left Kan extension of the underlying bosonic space operation with the canonical re-embedding yields an endofunctor that has the interpretation of sending any generalized superspace to its underlying generalized bosonic space</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><mo>⇝</mo></mover><mo>≔</mo><msub><mi>ι</mi> <mi>sup</mi></msub><mo>∘</mo><msub><mi>Π</mi> <mi>sup</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>SuperFormalSmoothSet</mi><mo>⟶</mo><mi>SuperFormalSmoothSet</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \overset{\rightsquigarrow}{(-)} \coloneqq \iota_{sup} \circ \Pi_{sup} \;\colon\; SuperFormalSmoothSet \longrightarrow SuperFormalSmoothSet \,. </annotation></semantics></math></div> <p>In fact, since these two <a class="existingWikiWord" href="/nlab/show/endofunctors">endofunctors</a> are obtained from the two possible composites in an <a class="existingWikiWord" href="/nlab/show/adjoint+triple">adjoint triple</a> whose middle functor is <a class="existingWikiWord" href="/nlab/show/fully+faithful+functor">fully faithful</a> – corresponding to an <a class="existingWikiWord" href="/nlab/show/adjoint+modality">adjoint modality</a> (<a href="geometry+of+physics+--+categories+and+toposes#AdjointModality">this Def.</a>) – it is immediate to see that:</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><mo>⇝</mo></mover></mrow><annotation encoding="application/x-tex">\overset{\rightsquigarrow}{(-)}</annotation></semantics></math> has the structure of a <a class="existingWikiWord" href="/nlab/show/comodal+operator">comodal operator</a></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><mo>⇉</mo></mover></mrow><annotation encoding="application/x-tex">\overset{\rightrightarrows}{(-)}</annotation></semantics></math> has the structure of an <a class="existingWikiWord" href="/nlab/show/modal+operator">modal operator</a></p> </li> <li> <p>together they form an <a class="existingWikiWord" href="/nlab/show/adjoint+modality">adjoint modality</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><mo>⇉</mo></mover><mo>⊣</mo><mover><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><mo>⇝</mo></mover></mrow><annotation encoding="application/x-tex">\overset{\rightrightarrows}{(-)} \dashv \overset{\rightsquigarrow}{(-)}</annotation></semantics></math>.</p> </li> </ol> <p>Moreover, by prop. <a class="maruku-ref" href="#SuperSmoothSetsSystemOfAdjunctions"></a>, this adjoint triple between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SuperFormalSmoothSet</mi></mrow><annotation encoding="application/x-tex">SuperFormalSmoothSet</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>FormalSmoothSet</mi></mrow><annotation encoding="application/x-tex">FormalSmoothSet</annotation></semantics></math> extends to an <a class="existingWikiWord" href="/nlab/show/adjoint+quadruple">adjoint quadruple</a>, there is yet one more <a class="existingWikiWord" href="/nlab/show/endofunctor">endofunctor</a> which is a further <a class="existingWikiWord" href="/nlab/show/right+adjoint">right adjoint</a> <a class="existingWikiWord" href="/nlab/show/idempotent+comonad">idempotent comonad</a>. Later we will see that this further opration is related to the concept of “<a class="existingWikiWord" href="/nlab/show/rheonomy">rheonomy</a>” in supergravity, and therefore we denote it by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Rh</mi><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><msub><mi>Disc</mi> <mi>sup</mi></msub><mo>∘</mo><msub><mi>Π</mi> <mi>sup</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>SuperFormalSmoothSet</mi><mo>⟶</mo><mi>SuperFormalSmoothSet</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Rh \;\coloneqq\; Disc_{sup} \circ \Pi_{sup} \;\colon\; SuperFormalSmoothSet \longrightarrow SuperFormalSmoothSet \,. </annotation></semantics></math></div> <p>But prop. <a class="maruku-ref" href="#SuperSmoothSetsSystemOfAdjunctions"></a> says that there are yet further adjoints, however these no longer go between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SuperFormalSmoothSet</mi></mrow><annotation encoding="application/x-tex">SuperFormalSmoothSet</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>FormalSmoothSet</mi></mrow><annotation encoding="application/x-tex">FormalSmoothSet</annotation></semantics></math>, but between the former and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SmoothSet</mi></mrow><annotation encoding="application/x-tex">SmoothSet</annotation></semantics></math> or even just between the former and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Set</mi></mrow><annotation encoding="application/x-tex">Set</annotation></semantics></math>. For instance there is the composite projection</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>SmoothSet</mi><mo>⟵</mo><mi>FormalSmoothSet</mi><mo>⟵</mo><mi>SuperFormalSmoothSet</mi></mrow><annotation encoding="application/x-tex"> SmoothSet \longleftarrow FormalSmoothSet \longleftarrow SuperFormalSmoothSet </annotation></semantics></math></div> <p>which first forms the bosonic underlying space, and then forms the reduced underlying space of what remains. The total result of this operation is just plain reduction, removing all infinitesimal directions, whether odd graded or even graded. Therefore, the result of composing this operation with its right adjoint canonical re-embedding yields an endofunctor which we should call</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ℜ</mi><mo>≔</mo><msub><mi>ι</mi> <mi>inf</mi></msub><mo>∘</mo><msub><mi>Π</mi> <mi>inf</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>SuperFormalSmoothSet</mi><mo>⟶</mo><mi>SuperFormalSmoothSet</mi></mrow><annotation encoding="application/x-tex"> \Re \coloneqq \iota_{inf} \circ \Pi_{inf} \;\colon\; SuperFormalSmoothSet \longrightarrow SuperFormalSmoothSet </annotation></semantics></math></div> <p>and think of as the operation of reduction on generalized <a class="existingWikiWord" href="/nlab/show/super+formal+smooth+sets">super formal smooth sets</a>.</p> <p>Notice that <a class="existingWikiWord" href="/nlab/show/reflective+subcategory">reflective subcategory</a> embeddings</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munderover><mrow></mrow><munder><mo>↪</mo><mi>i</mi></munder><mover><mo>⟵</mo><mi>L</mi></mover></munderover></mrow><annotation encoding="application/x-tex"> \underoverset {\underset{i}{\hookrightarrow}} {\overset{L}{\longleftarrow}} {} </annotation></semantics></math></div> <p>are <a class="existingWikiWord" href="/nlab/show/localizations">localizations</a>, in that first including an object and then projecting it back to the subcategory is the identity operation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo>∘</mo><mi>i</mi><mo>≃</mo><mi>id</mi></mrow><annotation encoding="application/x-tex">L \circ i \simeq id</annotation></semantics></math>; and analogously for a <a class="existingWikiWord" href="/nlab/show/coreflective+subcategory">coreflection</a> (<a href="adjoint+functor#FullyFaithfulAndInvertibleAdjoints">this prop.</a>).</p> <p>In our situation this first of all means that all of the <a class="existingWikiWord" href="/nlab/show/endofunctors">endofunctors</a> above are idempotent, as already mentioned. But next it implies that first applying a “deep” projection in the diagram in prop. <a class="maruku-ref" href="#SuperSmoothSetsSystemOfAdjunctions"></a>, and then applying a “more shallow” projection with functors at the same vertical stage in the diagram does not change the result further. For example there is a <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural isomorphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mrow><mi>ℜ</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><mo>⇝</mo></mover><mo>≃</mo><mi>ℜ</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \overset{\rightsquigarrow}{\Re(X)} \simeq \Re(X) </annotation></semantics></math></div> <p>saying that if a space is reduced, then it has no infinitesimal directions whatsoever, in particular no odd-graded ones, hence it is already bosonic.</p> <p>We will denote this situation by an inclusion sign</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ℜ</mi><mspace width="thickmathspace"></mspace><mo><</mo><mspace width="thickmathspace"></mspace><mover><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><mo>⇝</mo></mover></mrow><annotation encoding="application/x-tex"> \Re \;\lt\; \overset{\rightsquigarrow}{(-)} </annotation></semantics></math></div> <p>to be read “reduced implies bosonic”. This is an example of the <em><a class="existingWikiWord" href="/nlab/show/preorder">preorder</a> on <a class="existingWikiWord" href="/nlab/show/modalities">modalities</a></em> (<a href="geometry+of+physics+--+categories+and+toposes#PreorderOnModalities">this Def.</a>)</p> <p>We may proceed this way with all the remaining functors in prop. <a class="maruku-ref" href="#SuperSmoothSetsSystemOfAdjunctions"></a>, consecutively turning them into <a class="existingWikiWord" href="/nlab/show/endofunctors">endofunctors</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SuperFormalCartSp</mi></mrow><annotation encoding="application/x-tex">SuperFormalCartSp</annotation></semantics></math> which are all related to each other by <a class="existingWikiWord" href="/nlab/show/adjunctions">adjunctions</a> or by this inclusion relation of projection operators.</p> <p>The result is a system of 9 endofunctors, or 12 if we inclue the <a class="existingWikiWord" href="/nlab/show/bottom">bottom</a> <a class="existingWikiWord" href="/nlab/show/adjoint+modality">adjoint modality</a> and the <a class="existingWikiWord" href="/nlab/show/top">top</a> <a class="existingWikiWord" href="/nlab/show/adjoint+modality">adjoint modality</a> (<a href="geometry+of+physics+--+categories+and+toposes#InitialAndFinalAdjointModality">this Example</a>):</p> <table style="margin:auto"><thead><tr><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/shape+modality">shape modality</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/flat+modality">flat modality</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/sharp+modality">sharp modality</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ʃ</mi><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mi>Disc</mi><mo>∘</mo><msub><mi>Π</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">&#643; \;\coloneqq\; Disc \circ \Pi_0</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>♭</mo><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mi>Disc</mi><mo>∘</mo><mi>Γ</mi></mrow><annotation encoding="application/x-tex">\flat \;\coloneqq\; Disc \circ \Gamma</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>♯</mo><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mi>coDisc</mi><mo>∘</mo><mi>Γ</mi></mrow><annotation encoding="application/x-tex">\sharp \;\coloneqq\; coDisc \circ \Gamma </annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/reduction+modality">reduction modality</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/infinitesimal+shape+modality">infinitesimal shape modality</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/infinitesimal+flat+modality">infinitesimal flat modality</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℜ</mi><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><msub><mi>ι</mi> <mi>inf</mi></msub><mo>∘</mo><msub><mi>Π</mi> <mi>inf</mi></msub></mrow><annotation encoding="application/x-tex">\Re \;\coloneqq\; \iota_{inf} \circ \Pi_{inf}</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℑ</mi><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><msub><mi>Disc</mi> <mi>inf</mi></msub><mo>∘</mo><msub><mi>Π</mi> <mi>inf</mi></msub></mrow><annotation encoding="application/x-tex">\Im \;\coloneqq\; Disc_{inf} \circ \Pi_{inf}</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>&</mi><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><msub><mi>Disc</mi> <mi>inf</mi></msub><mo>∘</mo><msub><mi>Γ</mi> <mi>inf</mi></msub></mrow><annotation encoding="application/x-tex"> \& \;\coloneqq\; Disc_{inf} \circ \Gamma_{inf} </annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> </tbody></table> <div class="num_prop" id="ProgressionOfIdempotentEndofunctors"> <h6 id="proposition_8">Proposition</h6> <p><strong>(progression of <a class="existingWikiWord" href="/nlab/show/adjoint+modalities">adjoint modalities</a> on <a class="existingWikiWord" href="/nlab/show/SuperFormalSmoothSet">SuperFormalSmoothSet</a>)</strong></p> <p>We have a system of <a class="existingWikiWord" href="/nlab/show/adjoint+modalities">adjoint modalities</a> (<a href="geometry+of+physics+--+categories+and+toposes#AdjointModality">this Def.</a>) and their <a class="existingWikiWord" href="/nlab/show/preordering">preordering</a> (<a href="geometry+of+physics+--+categories+and+toposes#PreorderOnModalities">this Def.</a>) on <a class="existingWikiWord" href="/nlab/show/SuperFormalSmoothSet">SuperFormalSmoothSet</a> (Def. <a class="maruku-ref" href="#FormalSmoothSets"></a>) that is induced (via <a href="geometry+of+physics+--+categories+and+toposes#ModalOperatorsEquivalentToReflectiveSubcategories">this Prop.</a>) by the system of <a class="existingWikiWord" href="/nlab/show/adjoint+functors">adjoint functors</a> in Prop. <a class="maruku-ref" href="#SuperSmoothSetsSystemOfAdjunctions"></a>, as follows:</p> <div class="maruku-equation" id="eq:ProgressionOfModalitiesOnSuperFormalSmoothSet"><span class="maruku-eq-number">(1)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>id</mi></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mi>id</mi></mtd></mtr> <mtr><mtd><mo>∨</mo></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mo>∨</mo></mtd></mtr> <mtr><mtd><mo>⇉</mo></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mo>⇝</mo></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mi>Rh</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd> <mtd><mo>\</mo></mtd> <mtd><mo>∨</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>ℜ</mi></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mi>ℑ</mi></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mi>&</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mo>∨</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>ʃ</mi></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mo>♭</mo></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mo>♯</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd> <mtd><mo stretchy="false">/</mo></mtd> <mtd><mo>∨</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>∅</mi></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mo>*</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ id &\dashv& id \\ \vee &\vert& \vee \\ \rightrightarrows &\dashv& \rightsquigarrow &\dashv& Rh \\ && \vee &\backslash& \vee \\ && \Re &\dashv& \Im &\dashv& \& \\ && && \vee &\vert& \vee \\ && && &#643; &\dashv& \flat &\dashv& \sharp \\ && && && \vee &/& \vee \\ && && && \emptyset &\dashv& \ast } </annotation></semantics></math></div> <p>Moreover, the progression exhibits <a class="existingWikiWord" href="/nlab/show/Aufhebung">Aufhebung</a> (<a href="geometry+of+physics+--+categories+and+toposes#Aufhebung">this Def.</a>, <a href="geometry+of+physics+--+categories+and+toposes#TrivialAufhebung">this Remark</a>) at each stage, as indicated.</p> <p>Finally, the <a class="existingWikiWord" href="/nlab/show/rheonomy+modality">rheonomy modality</a> is <a class="existingWikiWord" href="/nlab/show/homotopy+localization">homotopy localization</a> at <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">\mathbb{R}^{0\vert 1}</annotation></semantics></math> (the <a class="existingWikiWord" href="/nlab/show/superpoint">superpoint</a>) in the sense of <a href="geometry+of+physics+--+categories+and+toposes#HomotopyLocalizationOn1Categories">this Def.</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Rh</mi><mo>≃</mo><mo>◯</mo><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><msup><mi>ℝ</mi> <mrow><mn>0</mn><mo stretchy="false">|</mo><mn>1</mn></mrow></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Rh \simeq \bigcirc\!\!\!\!\!\!\!\!\mathbb{R}^{0\vert 1} \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_6">Proof</h6> <p>The progression of modalities follows with Prop. <a class="maruku-ref" href="#SuperSmoothSetsSystemOfAdjunctions"></a> by <a href="geometry+of+physics+--+categories+and+toposes#ProgressionOfModalitiesOnSolidTopos">this Prop.</a>.</p> <p>The right <a class="existingWikiWord" href="/nlab/show/Aufhebung">Aufhebung</a> at the first stage says that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>♯</mo><mi>∅</mi><mo>≃</mo><mi>∅</mi></mrow><annotation encoding="application/x-tex">\sharp \emptyset \simeq \emptyset</annotation></semantics></math>, which means equivalently that for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mi>n</mi><mo stretchy="false">|</mo><mi>q</mi></mrow></msup><mo>×</mo><msub><mi>𝔻</mi> <mi>V</mi></msub><mo>∈</mo></mrow><annotation encoding="application/x-tex">\mathbb{R}^{n\vert q} \times \mathbb{D}_V \in </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/SuperFormalCartSp">SuperFormalCartSp</a> we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Hom</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mi>n</mi><mo stretchy="false">|</mo><mi>q</mi></mrow></msup><mo>×</mo><msub><mi>𝔻</mi> <mi>V</mi></msub><mo>,</mo><mo>♯</mo><mi>∅</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>∅</mi></mrow><annotation encoding="application/x-tex"> Hom(\mathbb{R}^{n\vert q} \times \mathbb{D}_V, \sharp \emptyset) \simeq \emptyset </annotation></semantics></math></div> <p>But by the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>♭</mo><mo>⊣</mo><mo>♯</mo></mrow><annotation encoding="application/x-tex">\flat \dashv \sharp</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a> hom-isomorphism (<a href="geometry+of+physics+--+categories+and+toposes#eq:HomIsomorphismForAdjointFunctors">here</a>) and the fact that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>♭</mo><mi>X</mi><mo>=</mo><mi>Disc</mi><mi>X</mi><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\flat X = Disc X(\ast)</annotation></semantics></math> (by the proof of <a href="geometry+of+physics+--+categories+and+toposes#CategoriesOfSheavesOnCohesiveSiteIsCohesive">this Prop.</a>) and that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mi>n</mi><mo stretchy="false">|</mo><mi>q</mi></mrow></msup><mo>×</mo><msub><mi>𝔻</mi> <mi>V</mi></msub><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo><mo>≠</mo><mi>∅</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}^{n\vert q} \times \mathbb{D}_V(\ast) \neq \emptyset</annotation></semantics></math> we have indeed</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Hom</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mi>n</mi><mo stretchy="false">|</mo><mi>q</mi></mrow></msup><mo>×</mo><msub><mi>𝔻</mi> <mi>V</mi></msub><mo>,</mo><mo>♯</mo><mi>∅</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>Hom</mi><mo stretchy="false">(</mo><mo>♭</mo><msup><mi>ℝ</mi> <mrow><mi>n</mi><mo stretchy="false">|</mo><mi>q</mi></mrow></msup><mo>×</mo><msub><mi>𝔻</mi> <mi>V</mi></msub><mo>,</mo><mi>∅</mi><mo stretchy="false">)</mo><mo>=</mo><mi>∅</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Hom(\mathbb{R}^{n\vert q} \times \mathbb{D}_V, \sharp \emptyset) \simeq Hom( \flat \mathbb{R}^{n\vert q} \times \mathbb{D}_V , \emptyset) = \emptyset \,. </annotation></semantics></math></div> <p>The left <a class="existingWikiWord" href="/nlab/show/Aufhebung">Aufhebung</a> at the third stage says that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>⇝</mo><mi>ℑ</mi><mo>≃</mo><mi>ℑ</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \rightsquigarrow \Im \simeq \Im \,. </annotation></semantics></math></div> <p>This means equivalenty that for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>SuperFormalSmoothSet</mi></mrow><annotation encoding="application/x-tex">X \in SuperFormalSmoothSet</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mi>n</mi><mo stretchy="false">|</mo><mi>q</mi></mrow></msup><mo>×</mo><msub><mi>𝔻</mi> <mi>V</mi></msub><mo>∈</mo><mi>SuperFormalCartSp</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}^{n\vert q} \times \mathbb{D}_V \in SuperFormalCartSp</annotation></semantics></math> we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Hom</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mi>n</mi><mo stretchy="false">|</mo><mi>q</mi></mrow></msup><mo>×</mo><msub><mi>𝔻</mi> <mi>V</mi></msub><mo>,</mo><mover><mrow><mi>ℑ</mi><mi>X</mi></mrow><mo>⇝</mo></mover><mo stretchy="false">)</mo><mo>≃</mo><mi>Hom</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mi>n</mi><mo stretchy="false">|</mo><mi>q</mi></mrow></msup><mo>×</mo><msub><mi>𝔻</mi> <mi>V</mi></msub><mo>,</mo><mover><mrow><mi>ℑ</mi><mi>X</mi></mrow><mrow></mrow></mover><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Hom(\mathbb{R}^{n\vert q} \times \mathbb{D}_V, \overset{\rightsquigarrow}{\Im X}) \simeq Hom(\mathbb{R}^{n\vert q} \times \mathbb{D}_V, \overset{}{\Im X}) </annotation></semantics></math></div> <p>Again by the adjunction isomorphisms we verify:</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>Hom</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mi>n</mi><mo stretchy="false">|</mo><mi>q</mi></mrow></msup><mo>×</mo><msub><mi>𝔻</mi> <mi>V</mi></msub><mo>,</mo><mover><mrow><mi>ℑ</mi><mi>X</mi></mrow><mo>⇝</mo></mover><mo stretchy="false">)</mo></mtd> <mtd><mo>≃</mo><mi>Hom</mi><mrow><mo>(</mo><mover><mrow><msup><mi>ℝ</mi> <mrow><mi>n</mi><mo stretchy="false">|</mo><mi>q</mi></mrow></msup><mo>×</mo><msub><mi>𝔻</mi> <mi>V</mi></msub></mrow><mo>⇉</mo></mover><mo>,</mo><mover><mrow><mi>ℑ</mi><mi>X</mi></mrow><mrow></mrow></mover><mo>)</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><mi>Hom</mi><mrow><mo>(</mo><mover><mrow><msup><mi>ℝ</mi> <mrow><mi>n</mi><mo stretchy="false">|</mo><mi>q</mi></mrow></msup><mo>×</mo><msub><mi>𝔻</mi> <mi>V</mi></msub></mrow><mo>⇉</mo></mover><mo>,</mo><mover><mrow><mi>ℑ</mi><mi>X</mi></mrow><mrow></mrow></mover><mo>)</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mi>Hom</mi><mrow><mo>(</mo><mi>ℜ</mi><mrow><mo>(</mo><mover><mrow><msup><mi>ℝ</mi> <mrow><mi>n</mi><mo stretchy="false">|</mo><mi>q</mi></mrow></msup><mo>×</mo><msub><mi>𝔻</mi> <mi>V</mi></msub></mrow><mo>⇉</mo></mover><mo>)</mo></mrow><mo>,</mo><mover><mi>X</mi><mrow></mrow></mover><mo>)</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><mi>Hom</mi><mrow><mo>(</mo><mi>ℜ</mi><mrow><mo>(</mo><mover><mrow><msup><mi>ℝ</mi> <mrow><mi>n</mi><mo stretchy="false">|</mo><mi>q</mi></mrow></msup><mo>×</mo><msub><mi>𝔻</mi> <mi>V</mi></msub></mrow><mrow></mrow></mover><mo>)</mo></mrow><mo>,</mo><mover><mi>X</mi><mrow></mrow></mover><mo>)</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><mi>Hom</mi><mrow><mo>(</mo><mover><mrow><msup><mi>ℝ</mi> <mrow><mi>n</mi><mo stretchy="false">|</mo><mi>q</mi></mrow></msup><mo>×</mo><msub><mi>𝔻</mi> <mi>V</mi></msub></mrow><mrow></mrow></mover><mo>,</mo><mover><mrow><mi>ℑ</mi><mi>X</mi></mrow><mrow></mrow></mover><mo>)</mo></mrow></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ Hom(\mathbb{R}^{n\vert q} \times \mathbb{D}_V, \overset{\rightsquigarrow}{\Im X}) & \simeq Hom\left( \overset{\rightrightarrows}{\mathbb{R}^{n\vert q} \times \mathbb{D}_V}, \overset{}{\Im X}\right) \\ & \simeq Hom\left( \overset{\rightrightarrows}{\mathbb{R}^{n\vert q} \times \mathbb{D}_V}, \overset{}{\Im X} \right) \\ & Hom\left( \Re\left(\overset{\rightrightarrows}{\mathbb{R}^{n\vert q} \times \mathbb{D}_V}\right), \overset{}{ X} \right) \\ & \simeq Hom\left( \Re\left(\overset{}{\mathbb{R}^{n\vert q} \times \mathbb{D}_V}\right), \overset{}{ X} \right) \\ & \simeq Hom\left( \overset{}{\mathbb{R}^{n\vert q} \times \mathbb{D}_V}, \overset{}{ \Im X} \right) } </annotation></semantics></math></div> <p>Here we used that on representables</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ℜ</mi><mover><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><mo>⇉</mo></mover><mo>≃</mo><mi>ℜ</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \Re \overset{\rightrightarrows}{(-)} \simeq \Re(-) \,, </annotation></semantics></math></div> <p>which holds by direct inspection: it says that the odd-graded elements in a <a class="existingWikiWord" href="/nlab/show/supercommutative+superalgebra">supercommutative superalgebra</a> are all nilpotent.</p> <p>Finally for the equivalence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Rh</mi><mo>≃</mo><mo>◯</mo><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><msup><mi>ℝ</mi> <mrow><mn>0</mn><mo stretchy="false">|</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex"> Rh \simeq \bigcirc\!\!\!\!\!\!\!\!\mathbb{R}^{0\vert 1} </annotation></semantics></math></div> <p>the proof is directly analogous to that of the analogous statement in the chapter <em><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+smooth+sets">on smooth sets</a></em>, <a href="geometry+of+physics+--+smooth+sets#ShapeModalityOnSmoothSetsIsR1Localization">this Prop</a>:</p> <p>As in that proof, the <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">\mathbb{R}^{0\vert 1}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/local+objects">local objects</a> among all <a class="existingWikiWord" href="/nlab/show/super+formal+smooth+sets">super formal smooth sets</a> are equivalently those which are <a class="existingWikiWord" href="/nlab/show/local+objects">local objects</a> with respect to the following <a class="existingWikiWord" href="/nlab/show/small+set">small set</a> of morphisms:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>{</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>×</mo><msub><mi>𝔻</mi> <mi>V</mi></msub><mo>×</mo><msup><mi>ℝ</mi> <mrow><mn>0</mn><mo stretchy="false">|</mo><mi>q</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>⟶</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>×</mo><msub><mi>𝔻</mi> <mi>V</mi></msub><mo>×</mo><msup><mi>ℝ</mi> <mrow><mn>0</mn><mo stretchy="false">|</mo><mi>q</mi></mrow></msup><mo>}</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \left\{ \mathbb{R}^{n} \times \mathbb{D}_V \times \mathbb{R}^{0\vert q + 1} \longrightarrow \mathbb{R}^{n} \times \mathbb{D}_V \times \mathbb{R}^{0\vert q} \right\} \,. </annotation></semantics></math></div> <p>By <a class="existingWikiWord" href="/nlab/show/induction">induction</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">q \in \mathbb{N}</annotation></semantics></math>, these are equivalently the objects which are local with respect to the following small set:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>{</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>×</mo><msub><mi>𝔻</mi> <mi>V</mi></msub><mo>×</mo><msup><mi>ℝ</mi> <mrow><mn>0</mn><mo stretchy="false">|</mo><mi>q</mi></mrow></msup><mo>⟶</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>×</mo><msub><mi>𝔻</mi> <mi>V</mi></msub><mo>}</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \left\{ \mathbb{R}^{n} \times \mathbb{D}_V \times \mathbb{R}^{0\vert q} \longrightarrow \mathbb{R}^{n} \times \mathbb{D}_V \right\} \,. </annotation></semantics></math></div> <p>But these are manifestly the objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><mi>SuperFormalSmoothSet</mi></mrow><annotation encoding="application/x-tex">A \in SuperFormalSmoothSet</annotation></semantics></math> which are in the image of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Disc</mi> <mi>sup</mi></msub></mrow><annotation encoding="application/x-tex">Disc_{sup}</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><mi>Hom</mi><mrow><mo>(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>×</mo><msub><mi>𝔻</mi> <mi>V</mi></msub><mo>×</mo><msup><mi>ℝ</mi> <mrow><mn>0</mn><mo stretchy="false">|</mo><mi>q</mi></mrow></msup><mo>,</mo><mi>Disc</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo><mo>)</mo></mrow></mtd> <mtd><mo>≃</mo><mi>Hom</mi><mrow><mo>(</mo><msub><mi>Π</mi> <mi>sup</mi></msub><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>×</mo><msub><mi>𝔻</mi> <mi>V</mi></msub><mo>×</mo><msup><mi>ℝ</mi> <mrow><mn>0</mn><mo stretchy="false">|</mo><mi>q</mi></mrow></msup><mo stretchy="false">)</mo><mo>,</mo><mi>B</mi><mo>)</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><mi>Hom</mi><mrow><mo>(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>×</mo><msub><mi>𝔻</mi> <mi>V</mi></msub><mo stretchy="false">)</mo><mo>,</mo><mi>B</mi><mo>)</mo></mrow></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} Hom\left( \mathbb{R}^{n} \times \mathbb{D}_V \times \mathbb{R}^{0\vert q}, Disc(B) \right) & \simeq Hom\left( \Pi_{sup}(\mathbb{R}^{n} \times \mathbb{D}_V \times \mathbb{R}^{0\vert q}), B \right) \\ & \simeq Hom\left(\mathbb{R}^{n} \times \mathbb{D}_V), B\right) \end{aligned} </annotation></semantics></math></div></div> <p>Sometimes it is illuminating to re-arrange the diagram in Prop. <a class="maruku-ref" href="#ProgressionOfIdempotentEndofunctors"></a> equivalently as follows. Here we label each projection operator by the property of superspaces that it “projects out”.</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></mtd> <mtd></mtd> <mtd><mi>id</mi></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mi>id</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>fermionic</mi></mover></mtd> <mtd><mo>⇉</mo></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mo>⇝</mo></mtd> <mtd><mover><mrow></mrow><mi>bosonic</mi></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd> <mtd><mover><mrow></mrow><mi>solidity</mi></mover></mtd> <mtd><mo>⊥</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>bosonic</mi></mover></mtd> <mtd><mo>⇝</mo></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mi>Rh</mi></mtd> <mtd><mover><mrow></mrow><mi>rheonomic</mi></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>reduced</mi></mover></mtd> <mtd><mi>ℜ</mi></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mi>ℑ</mi></mtd> <mtd><mover><mrow></mrow><mi>infinitesimal</mi></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd> <mtd><mover><mrow></mrow><mi>elasticity</mi></mover></mtd> <mtd><mo>⊥</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>infinitesimal</mi></mover></mtd> <mtd><mi>ℑ</mi></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mi>&</mi></mtd> <mtd><mover><mrow></mrow><mtext>étale</mtext></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>connected</mi></mover></mtd> <mtd><mi>ʃ</mi></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mo>♭</mo></mtd> <mtd><mover><mrow></mrow><mi>disconnected</mi></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd> <mtd><mover><mrow></mrow><mi>cohesion</mi></mover></mtd> <mtd><mo>⊥</mo></mtd> <mtd></mtd> <mtd></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>discrete</mi></mover></mtd> <mtd><mo>♭</mo></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mo>♯</mo></mtd> <mtd><mover><mrow></mrow><mi>continuous</mi></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>∅</mi></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mo>*</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot &\stackrel{solidity}{}& \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot &\stackrel{elasticity}{}& \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{&#233;tale}}{} \\ && \vee && \vee \\ &\stackrel{connected}{}& &#643; &\dashv& \flat & \stackrel{disconnected}{} \\ && \bot &\stackrel{cohesion}{}& \bot && \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast } </annotation></semantics></math></div> <p><a href="#Supermanifolds">Below</a> we use these operations to identify within all generalized superspaces those that are <a class="existingWikiWord" href="/nlab/show/supermanifolds">supermanifolds</a>. But first we consider now some general important constructions of <a class="existingWikiWord" href="/nlab/show/super+formal+smooth+sets">super formal smooth sets</a>, such as <a class="existingWikiWord" href="/nlab/show/mapping+spaces">mapping spaces</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> <h2 id="SuperMappingSpaces">Super mapping spaces</h2> <p>We now discuss <a class="existingWikiWord" href="/nlab/show/mapping+spaces">mapping spaces</a> in <a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a>. These are interesting in two ways:</p> <ol> <li> <p>for the theory – mapping spaces nicely exhibit the usage and the power of the <a class="existingWikiWord" href="/nlab/show/functor+of+points">functor of points</a> perspective (remark <a class="maruku-ref" href="#ASheafAsASpace"></a>);</p> </li> <li> <p>for applications – in <a class="existingWikiWord" href="/nlab/show/physics">physics</a> a <a class="existingWikiWord" href="/nlab/show/superfield">superfield</a> is really a <a class="existingWikiWord" href="/nlab/show/generalized+element">generalized element</a> of a mapping space, and hence the <a class="existingWikiWord" href="/nlab/show/phase+spaces">phase spaces</a> of interest in <a class="existingWikiWord" href="/nlab/show/physics">physics</a> are <a class="existingWikiWord" href="/nlab/show/mapping+spaces">mapping spaces</a> (in the generality of spaces of <a class="existingWikiWord" href="/nlab/show/spaces+of+sections">spaces of sections</a>, namely of a <a class="existingWikiWord" href="/nlab/show/field+bundle">field bundle</a>).</p> </li> </ol> <p>The key idea is that <a class="existingWikiWord" href="/nlab/show/sets">sets</a> of functions between sets have the following <a class="existingWikiWord" href="/nlab/show/universal+property">universal property</a>:</p> <div class="num_example" id="FunctionSet"> <h6 id="example_3">Example</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>,</mo><mi>Y</mi><mo>∈</mo></mrow><annotation encoding="application/x-tex">X,Y \in </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Set">Set</a> be two <a class="existingWikiWord" href="/nlab/show/sets">sets</a>, then the set</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Y</mi> <mi>X</mi></msup><mo>≔</mo><mo stretchy="false">{</mo><mi>X</mi><mo>→</mo><mi>Y</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex"> Y^X \coloneqq \{X \to Y\} </annotation></semantics></math></div> <p>of <a class="existingWikiWord" href="/nlab/show/functions">functions</a> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> is characterized by the fact that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Z</mi><mo>∈</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">Z \in Set</annotation></semantics></math> any further set, there is a <a class="existingWikiWord" href="/nlab/show/natural+bijection">natural bijection</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Hom</mi><mo stretchy="false">(</mo><mi>Z</mi><mo>×</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mo>≃</mo></mover><mi>Hom</mi><mo stretchy="false">(</mo><mi>Z</mi><mo>,</mo><msup><mi>Y</mi> <mi>X</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Hom( Z \times X, Y ) \stackrel{\simeq}{\longrightarrow} Hom(Z, Y^X) </annotation></semantics></math></div> <p>between functions of two <a class="existingWikiWord" href="/nlab/show/variables">variables</a> into <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> and functions of one variable into <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Y</mi> <mi>X</mi></msup></mrow><annotation encoding="application/x-tex">Y^X</annotation></semantics></math>. This is given by sending any function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(-,-)</annotation></semantics></math> of two variables to the function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>f</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde f</annotation></semantics></math> which sends any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>z</mi></mrow><annotation encoding="application/x-tex">z</annotation></semantics></math> to the function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>↦</mo><mi>f</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x \mapsto f(z,x)</annotation></semantics></math>. Hence this simply reinterprets “taking two arguments at once” by “taking two arguments consecutively”.</p> </div> <p>It is immediate how to generalize example <a class="maruku-ref" href="#FunctionSet"></a>:</p> <div class="num_defn" id="InternalHom"> <h6 id="definition_7">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/mapping+space">mapping space</a>/<a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom</a>)</strong></p> <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 a <a class="existingWikiWord" href="/nlab/show/category">category</a> with <a class="existingWikiWord" href="/nlab/show/Cartesian+products">Cartesian products</a> of its <a class="existingWikiWord" href="/nlab/show/objects">objects</a>, hence a <a class="existingWikiWord" href="/nlab/show/cartesian+monoidal+category">cartesian monoidal category</a>. Then an <a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom</a>-functor 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> is, if it exists, a functor of the form</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><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>𝒞</mi> <mi>op</mi></msup><mo>×</mo><mi>𝒞</mi><mo>⟶</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex"> [-,-] \;\colon\; \mathcal{C}^{op} \times \mathcal{C} \longrightarrow \mathcal{C} </annotation></semantics></math></div> <p>such that there is a <a class="existingWikiWord" href="/nlab/show/natural+bijection">natural bijection</a> between <a class="existingWikiWord" href="/nlab/show/hom+sets">hom sets</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><msub><mi>Hom</mi> <mi>𝒞</mi></msub><mo stretchy="false">(</mo><mi>Z</mi><mo>×</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>Hom</mi> <mi>𝒞</mi></msub><mo stretchy="false">(</mo><mi>Z</mi><mo>,</mo><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Hom_{\mathcal{C}}(Z \times X, Y) \simeq Hom_{\mathcal{C}}(Z, [X,Y]) </annotation></semantics></math></div> <p>for all <a class="existingWikiWord" href="/nlab/show/objects">objects</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>,</mo><mi>Y</mi><mo>,</mo><mi>Z</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">X,Y,Z \in \mathcal{C}</annotation></semantics></math>.</p> </div> <p>The class of examples that we are interested in is the following:</p> <div class="num_prop" id="SheavesHomInternal"> <h6 id="proposition_9">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mo>=</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>𝒮</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{C} = Sh(\mathcal{S})</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/category+of+sheaves">category of sheaves</a> over some <a class="existingWikiWord" href="/nlab/show/site">site</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">\mathcal{S}</annotation></semantics></math>.</p> <p>Then the <a class="existingWikiWord" href="/nlab/show/Cartesian+product">Cartesian product</a> between any two sheaves <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>,</mo><mi>Y</mi><mo>∈</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X,Y \in Sh(\mathcal{C})</annotation></semantics></math> exists and is given objectwise by the <a class="existingWikiWord" href="/nlab/show/Cartesian+product">Cartesian product</a> of sets:</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><mi>X</mi><mo>×</mo><mi>Y</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>U</mi><mo>↦</mo><mi>X</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo>×</mo><mi>Y</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (X \times Y) \;\colon\; U \mapsto X(U) \times Y(U) </annotation></semantics></math></div> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>∈</mo><mi>𝒮</mi></mrow><annotation encoding="application/x-tex">U \in \mathcal{S}</annotation></semantics></math>.</p> <p>Moreover, an <a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom</a>-functor according to def. <a class="maruku-ref" href="#InternalHom"></a> exists (“generalized <a class="existingWikiWord" href="/nlab/show/mapping+space">mapping space</a>”). It sends any two <a class="existingWikiWord" href="/nlab/show/sheaves">sheaves</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>,</mo><mi>Y</mi><mo>∈</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>𝒮</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X,Y \in Sh(\mathcal{S})</annotation></semantics></math> to the sheaf</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><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">]</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>𝒮</mi> <mi>op</mi></msup><mo>⟶</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex"> [X,Y] \;\colon\; \mathcal{S}^{op} \longrightarrow Set </annotation></semantics></math></div> <p>given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>↦</mo><msub><mi>Hom</mi> <mrow><mi>Sh</mi><mo stretchy="false">(</mo><mi>𝒮</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><mi>y</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> U \mapsto Hom_{Sh(\mathcal{S})}( X \times y(U), Y ) \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo lspace="verythinmathspace">:</mo><mi>𝒮</mi><mo>↪</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>𝒮</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">y \colon \mathcal{S} \hookrightarrow Sh(\mathcal{S})</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/Yoneda+embedding">Yoneda embedding</a>.</p> </div> <p>For the <strong>proof</strong> see at <em><a class="existingWikiWord" href="/nlab/show/closed+monoidal+structure+on+presheaves">closed monoidal structure on presheaves</a></em>.</p> <p>Notice how prop. <a class="maruku-ref" href="#SheavesHomInternal"></a> expresses an intuitively most obvious statement: Applied to geometric sheaf toposes such as <a class="existingWikiWord" href="/nlab/show/smooth+set">SmoothSet</a> or <a class="existingWikiWord" href="/nlab/show/super+formal+smooth+set">SuperFormalSmoothSet</a> (def. <a class="maruku-ref" href="#FormalSmoothSets"></a>) it says that a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math>-parameterized smooth family of points in a <a class="existingWikiWord" href="/nlab/show/mapping+space">mapping space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[X,Y]</annotation></semantics></math> is a smooth map of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><mi>U</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X \times U \to Y</annotation></semantics></math>, hence a family of smooth functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X \to Y</annotation></semantics></math> which is smoothly parameterized by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math>.</p> <p>The following shows formally that the concept of <a class="existingWikiWord" href="/nlab/show/internal+homs">internal homs</a> in the <a class="existingWikiWord" href="/nlab/show/topos">topos</a> of <a class="existingWikiWord" href="/nlab/show/generalized+smooth+spaces">generalized smooth spaces</a> does generalize the traditional concept of smooth <a class="existingWikiWord" href="/nlab/show/mapping+spaces">mapping spaces</a>:</p> <div class="num_example" id="SmoothManifoldsMappingSpace"> <h6 id="example_4">Example</h6> <p>Let <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> be a <a class="existingWikiWord" href="/nlab/show/compact+topological+space">compact</a> <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> be any <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a>. Then the set of <a class="existingWikiWord" href="/nlab/show/smooth+functions">smooth functions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>Σ</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^\infty(\Sigma,X)</annotation></semantics></math> carries the structure of an <a class="existingWikiWord" href="/nlab/show/infinite-dimensional+manifold">infinite dimensional</a> (in general) <a class="existingWikiWord" href="/nlab/show/Fr%C3%A9chet+manifold">Fréchet manifold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Maps</mi><mo stretchy="false">(</mo><mi>Σ</mi><mo>,</mo><mi>X</mi><msub><mo stretchy="false">)</mo> <mi>Frechet</mi></msub></mrow><annotation encoding="application/x-tex">Maps(\Sigma,X)_{Frechet}</annotation></semantics></math>. Under the embedding <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo lspace="verythinmathspace">:</mo><mi>FrechetMfd</mi><mo>↪</mo><mi>SmoothSet</mi></mrow><annotation encoding="application/x-tex">i \colon FrechetMfd \hookrightarrow SmoothSet</annotation></semantics></math> of prop. <a class="maruku-ref" href="#SmoothSetsContainSmoothManifolds"></a> this coincides with <a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>Σ</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\Sigma,X]</annotation></semantics></math> formed in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SmoothSet</mi></mrow><annotation encoding="application/x-tex">SmoothSet</annotation></semantics></math> according to prop. <a class="maruku-ref" href="#SheavesHomInternal"></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><mi>Σ</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo><mo>≃</mo><mi>i</mi><mo stretchy="false">(</mo><mi>Maps</mi><mo stretchy="false">(</mo><mi>Σ</mi><mo>,</mo><mi>X</mi><msub><mo stretchy="false">)</mo> <mi>Frechet</mi></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [\Sigma, X] \simeq i(Maps(\Sigma, X)_{Frechet}) \,. </annotation></semantics></math></div> <p>In particular for instance the smooth <a class="existingWikiWord" href="/nlab/show/loop+space">loop space</a> of a smooth manifold <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is simply the internal hom <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>S</mi> <mn>1</mn></msup><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[S^1,X]</annotation></semantics></math>.</p> </div> <p>A <strong>proof</strong> is given in <a href="diffeological+space#Waldorf09">Waldorf 09, lemma A.1.7</a>.</p> <p>But in the topos <a class="existingWikiWord" href="/nlab/show/super+formal+smooth+set">SuperFormalSmoothSet</a> (def. <a class="maruku-ref" href="#FormalSmoothSets"></a>) we have also mapping spaces much more general than the traditional ones of example <a class="maruku-ref" href="#SmoothManifoldsMappingSpace"></a>. We now look at some examples of these.</p> <div class="num_example" id="CorepresentingTangentSpace"> <h6 id="example_5">Example</h6> <p>Let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>𝔻</mi> <mn>1</mn></msup><mo>=</mo><mi>Spec</mi><mo stretchy="false">(</mo><mi>ℝ</mi><mo stretchy="false">[</mo><mi>ϵ</mi><mo stretchy="false">]</mo><mo stretchy="false">/</mo><mo stretchy="false">(</mo><msup><mi>ϵ</mi> <mn>2</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>∈</mo><mi>InfPoint</mi><mo>↪</mo><mi>SuperFomalSmoothSet</mi></mrow><annotation encoding="application/x-tex"> \mathbb{D}^1 = Spec(\mathbb{R}[\epsilon]/(\epsilon^2)) \in InfPoint \hookrightarrow SuperFomalSmoothSet </annotation></semantics></math></div> <p>be the <a class="existingWikiWord" href="/nlab/show/formal+dual">formal dual</a> of the <a class="existingWikiWord" href="/nlab/show/ring+of+dual+numbers">ring of dual numbers</a> (example <a class="maruku-ref" href="#EvenPartOfDimTwoSuperpoint"></a>). Observe that there is a unique morphism of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo><mo>→</mo><msup><mi>𝔻</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">\ast \to \mathbb{D}^1</annotation></semantics></math>, picking the base point.</p> <p>Then for</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>SmoothMfd</mi><mo>↪</mo><mi>SuperFormalSmoothSet</mi></mrow><annotation encoding="application/x-tex"> X \in SmoothMfd \hookrightarrow SuperFormalSmoothSet </annotation></semantics></math></div> <p>a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a>, regarded as a <a class="existingWikiWord" href="/nlab/show/super+formal+smooth+set">super formal smooth set</a> (this Prop.ets#InclusionOfSmoothManifoldsIntoSmoothSets)), the <a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom</a> out of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝔻</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{D}^1</annotation></semantics></math> with this basepoint is the smooth <a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> (again under the embedding of this Prop.ets#InclusionOfSmoothManifoldsIntoSmoothSets)):</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>𝔻</mi> <mn>1</mn></msup><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mtd> <mtd><mo>≃</mo></mtd> <mtd><mi>T</mi><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo stretchy="false">[</mo><mo>*</mo><mo>→</mo><msup><mi>𝔻</mi> <mn>1</mn></msup><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">[</mo><mo>*</mo><mo>,</mo><mo lspace="0em" rspace="thinmathspace">X</mo><mo stretchy="false">]</mo></mtd> <mtd><mo>≃</mo></mtd> <mtd><mi>X</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ [\mathbb{D}^1, X] &\simeq& T X \\ {}^{\mathllap{[\ast \to \mathbb{D}^1,X]}}\downarrow && \downarrow \\ [\ast,\X] &\simeq& X } \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_7">Proof</h6> <p>By prop. <a class="maruku-ref" href="#SheavesHomInternal"></a> the rule for the <a class="existingWikiWord" href="/nlab/show/smooth+set">smooth set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>𝔻</mi> <mn>1</mn></msup><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\mathbb{D}^1,X]</annotation></semantics></math> is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>↦</mo><msub><mi>Hom</mi> <mi>SmoothSet</mi></msub><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>×</mo><msup><mi>𝔻</mi> <mn>1</mn></msup><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbb{R}^n \mapsto Hom_{SmoothSet}( \mathbb{R}^n \times \mathbb{D}^1, X ) \,. </annotation></semantics></math></div> <p>By prop. <a class="maruku-ref" href="#HomsOutOfFirstOrderInfinitesimalLine"></a>, the set on the right is naturally identified with the set of of smoothly <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>-parameterized families of <a class="existingWikiWord" href="/nlab/show/tangent+vectors">tangent vectors</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>. But this is the set that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">T X</annotation></semantics></math>, regarded as a smooth set, assigns to <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>.</p> <p>Moreover, looking at the proof of prop. <a class="maruku-ref" href="#HomsOutOfFirstOrderInfinitesimalLine"></a> it is immediate that composing a morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>𝔻</mi> <mn>1</mn></msup><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex"> \mathbb{D}^1 \to X </annotation></semantics></math></div> <p>representing some tangent vector in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> with the global point inclusion of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝔻</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{D}^1</annotation></semantics></math> yields the point in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>*</mo><mo>→</mo><msup><mi>𝔻</mi> <mn>1</mn></msup><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex"> \ast \to \mathbb{D}^1 \to X </annotation></semantics></math></div> <p>at which this tangent vector is based. This shows that the vertical map in the above claim is indeed the projection from the <a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a> to the base manifold.</p> </div> <p>Example <a class="maruku-ref" href="#CorepresentingTangentSpace"></a> is a key observation that motivated the development of <a class="existingWikiWord" href="/nlab/show/synthetic+differential+geometry">synthetic differential geometry</a> (<a class="existingWikiWord" href="/nlab/show/Toposes+of+laws+of+motion">Lawvere 97</a>).</p> <p>The following is the supergeometric analog of this situation:</p> <div class="num_example" id="MappingSpaceOddTangentBundle"> <h6 id="example_6">Example</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/odd+tangent+bundle">odd tangent bundle</a> as <a class="existingWikiWord" href="/nlab/show/mapping+space">mapping space</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> be any <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a>. Then the <a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SuperFormalSmoothSet</mi></mrow><annotation encoding="application/x-tex">SuperFormalSmoothSet</annotation></semantics></math> out of the <a class="existingWikiWord" href="/nlab/show/superpoint">superpoint</a> <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">\mathbb{R}^{0\vert 1}</annotation></semantics></math> into <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, according to prop. <a class="maruku-ref" href="#SheavesHomInternal"></a>, is the <a class="existingWikiWord" href="/nlab/show/odd+tangent+bundle">odd tangent bundle</a> from def. <a class="maruku-ref" href="#OddTangentBundle"></a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi><mi>T</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">\Pi T X</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo stretchy="false">[</mo><msup><mi>ℝ</mi> <mrow><mn>0</mn><mo stretchy="false">|</mo><mn>1</mn></mrow></msup><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mtd> <mtd><mo>≃</mo></mtd> <mtd><mi>Π</mi><mi>T</mi><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo stretchy="false">[</mo><mo>*</mo><mo>→</mo><msup><mi>ℝ</mi> <mrow><mn>0</mn><mo stretchy="false">|</mo><mn>1</mn></mrow></msup><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">[</mo><mo>*</mo><mo>,</mo><mo lspace="0em" rspace="thinmathspace">X</mo><mo stretchy="false">]</mo></mtd> <mtd><mo>≃</mo></mtd> <mtd><mi>X</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ [\mathbb{R}^{0\vert 1}, X] &\simeq& \Pi T X \\ {}^{\mathllap{[\ast \to \mathbb{R}^{0\vert 1},X]}}\downarrow && \downarrow \\ [\ast,\X] &\simeq& X } \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_8">Proof</h6> <p>Let <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> be a <a class="existingWikiWord" href="/nlab/show/bosonic+object">bosonic</a> <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a>. By prop. <a class="maruku-ref" href="#SheavesHomInternal"></a> the set of plots of the smooth set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>ℝ</mi> <mrow><mn>0</mn><mo stretchy="false">|</mo><mn>1</mn></mrow></msup><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\mathbb{R}^{0\vert 1}, X]</annotation></semantics></math> on this test space is</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><msup><mi>ℝ</mi> <mrow><mn>0</mn><mo stretchy="false">|</mo><mn>1</mn></mrow></msup><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo>=</mo><msub><mi>Hom</mi> <mi>SuperFormalSmoothSet</mi></msub><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mi>n</mi><mo stretchy="false">|</mo><mn>1</mn></mrow></msup><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [\mathbb{R}^{0\vert 1},X](\mathbb{R}^n) = Hom_{SuperFormalSmoothSet}( \mathbb{R}^{n\vert 1}, X ) \,. </annotation></semantics></math></div> <p>But since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is bosonic (an ordinary smooth manifold), this is equivalently just</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><msup><mi>ℝ</mi> <mrow><mn>0</mn><mo stretchy="false">|</mo><mn>1</mn></mrow></msup><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo>=</mo><msub><mi>Hom</mi> <mi>SuperFormalSmoothSet</mi></msub><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>X</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> [\mathbb{R}^{0\vert 1},X](\mathbb{R}^n) = Hom_{SuperFormalSmoothSet}( \mathbb{R}^{n}, X ) \simeq X(\mathbb{R}^n) \,, </annotation></semantics></math></div> <p>which shows that the bosonic super smooth set underlying <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>ℝ</mi> <mrow><mn>0</mn><mo stretchy="false">|</mo><mn>1</mn></mrow></msup><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\mathbb{R}^{0\vert 1},X]</annotation></semantics></math> is just <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> itself.</p> <p>But then consider probes parameterized by the <a class="existingWikiWord" href="/nlab/show/superpoint">superpoint</a> <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">\mathbb{R}^{0\vert 1}</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><mo stretchy="false">[</mo><msup><mi>ℝ</mi> <mrow><mn>0</mn><mo stretchy="false">|</mo><mn>1</mn></mrow></msup><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mi>n</mi><mo stretchy="false">|</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></mtd> <mtd><mo>≃</mo><mi>Hom</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mi>n</mi><mo stretchy="false">|</mo><mn>2</mn></mrow></msup><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><mi>Hom</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mi>n</mi><mo stretchy="false">|</mo><mn>2</mn></mrow></msup><mo>,</mo><mover><mi>X</mi><mo>⇝</mo></mover><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><mi>Hom</mi><mo stretchy="false">(</mo><mover><mrow><msup><mi>ℝ</mi> <mrow><mi>n</mi><mo stretchy="false">|</mo><mn>2</mn></mrow></msup></mrow><mo>⇉</mo></mover><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><mi>Hom</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>×</mo><msup><mi>𝔻</mi> <mn>1</mn></msup><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><mi>Hom</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>,</mo><mi>T</mi><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} [\mathbb{R}^{0\vert 1},X](\mathbb{R}^{n\vert 1}) & \simeq Hom( \mathbb{R}^{n\vert 2}, X ) \\ & \simeq Hom( \mathbb{R}^{n\vert 2}, \overset{\rightsquigarrow}{X} ) \\ & \simeq Hom( \overset{\rightrightarrows}{\mathbb{R}^{n \vert 2}}, X ) \\ & \simeq Hom( \mathbb{R}^n \times \mathbb{D}^1 , X ) \\ & \simeq Hom(\mathbb{R}^n, T X) \end{aligned} \,, </annotation></semantics></math></div> <p>where we used the <a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⇉</mo><mo>⊣</mo><mo>⇝</mo></mrow><annotation encoding="application/x-tex">\rightrightarrows \dashv \rightsquigarrow</annotation></semantics></math> from prop. <a class="maruku-ref" href="#SuperSmoothSetsSystemOfAdjunctions"></a>, Prop. <a class="maruku-ref" href="#ProgressionOfIdempotentEndofunctors"></a>, then example <a class="maruku-ref" href="#EvenPartOfDimTwoSuperpoint"></a> and finally example <a class="maruku-ref" href="#CorepresentingTangentSpace"></a>.</p> </div> <p>Notice the curious difference between the bosonic and the odd-graded version of the <a class="existingWikiWord" href="/nlab/show/synthetic+tangent+bundle">synthetic tangent bundle</a> as seen by its <a class="existingWikiWord" href="/nlab/show/algebra+of+functions">algebra of functions</a></p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo stretchy="false">[</mo><msup><mi>𝔻</mi> <mn>1</mn></msup><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo><mo>,</mo><mi>ℝ</mi><mo stretchy="false">]</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>T</mi><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">[[\mathbb{D}^1, X], \mathbb{R}] \;\simeq\; C^\infty(T X)</annotation></semantics></math></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo stretchy="false">[</mo><msup><mi>ℝ</mi> <mrow><mn>0</mn><mo stretchy="false">|</mo><mn>1</mn></mrow></msup><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo><mo>,</mo><mi>ℝ</mi><mo stretchy="false">]</mo><mo>≃</mo><msup><mi>Ω</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">[[\mathbb{R}^{0\vert 1}, X], \mathbb{R}] \simeq \Omega^\bullet(X)</annotation></semantics></math></p> </li> </ul> <p>In the first case the, smooth functions on the <a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">T X</annotation></semantics></math> do not know about the linear structure on <a class="existingWikiWord" href="/nlab/show/fibers">fibers</a>. But in the second case, they do: the <a class="existingWikiWord" href="/nlab/show/differential+forms">differential forms</a> in the second case appear as the sub-space of that of all smooth functions on those which are graded <a class="existingWikiWord" href="/nlab/show/polynomials">polynomials</a> (over the algebra of smooth functions on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>) in fiber-wise <em><a class="existingWikiWord" href="/nlab/show/linear+functions">linear functions</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>More generally, this is the concept if <em><a class="existingWikiWord" href="/nlab/show/superfields">superfields</a></em> as used in the <a class="existingWikiWord" href="/nlab/show/physics">physics</a> literature:</p> <div class="num_example" id="Superfields"> <h6 id="example_7">Example</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/superfields">superfields</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/supermanifold">supermanifold</a> (for instance a <a class="existingWikiWord" href="/nlab/show/super+spacetime">super spacetime</a>) and consider the super-<a class="existingWikiWord" href="/nlab/show/mapping+space">mapping space</a> (def. <a class="maruku-ref" href="#InternalHom"></a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mi>ℝ</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[X,\mathbb{R}]</annotation></semantics></math> (or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mi>ℂ</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[X,\mathbb{C}]</annotation></semantics></math>) of real (or complex) valued functions on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> (“<a class="existingWikiWord" href="/nlab/show/scalar+fields">scalar fields</a>”). We may understand the <a class="existingWikiWord" href="/nlab/show/generalized+elements">generalized elements</a> of this superspace via its <a class="existingWikiWord" href="/nlab/show/functor+of+points">functor of points</a> which, via prop. <a class="maruku-ref" href="#SheavesHomInternal"></a>, is given by the assignment</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><mi>q</mi></mrow></msup></mtd> <mtd><mo>↦</mo><mi>Hom</mi><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><msup><mi>ℝ</mi> <mrow><mn>0</mn><mo stretchy="false">|</mo><mi>q</mi></mrow></msup><mo>,</mo><mi>ℝ</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><msup><mi>ℝ</mi> <mrow><mn>0</mn><mo stretchy="false">|</mo><mi>q</mi></mrow></msup><msub><mo stretchy="false">)</mo> <mi>even</mi></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><msub><mrow><mo>(</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mi>ℝ</mi></msub><msup><mo>∧</mo> <mo>•</mo></msup><mo stretchy="false">⟨</mo><msub><mi>θ</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>θ</mi> <mi>q</mi></msub><mo stretchy="false">⟩</mo><mo>)</mo></mrow> <mi>even</mi></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} \mathbb{R}^{0\vert q} & \mapsto Hom( X \times \mathbb{R}^{0\vert q}, \mathbb{R} ) \\ & \simeq C^\infty(X \times \mathbb{R}^{0\vert q})_{even} \\ & \simeq \left( C^\infty(X) \otimes_{\mathbb{R}} \wedge^\bullet \langle \theta_1, \cdots, \theta_q\rangle\right)_{even} \end{aligned} \,. </annotation></semantics></math></div> <p>Here in the first line on the right we have the set of maps of supermanifolds of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><msup><mi>ℝ</mi> <mrow><mn>0</mn><mo stretchy="false">|</mo><mi>q</mi></mrow></msup><mo>→</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">X \times \mathbb{R}^{0\vert q} \to \mathbb{R}</annotation></semantics></math>, which, by the <a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a>, is equivalently just the even subalgebra of the super-algebra of functions on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><msup><mi>ℝ</mi> <mrow><mn>0</mn><mo stretchy="false">|</mo><mi>q</mi></mrow></msup></mrow><annotation encoding="application/x-tex">X \times \mathbb{R}^{0\vert q}</annotation></semantics></math>, which finally is equivalently the even elements in the <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a> of the super-algebra of functions on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> with the <a class="existingWikiWord" href="/nlab/show/Grassmann+algebra">Grassmann algebra</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi></mrow><annotation encoding="application/x-tex">q</annotation></semantics></math> odd generators <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>θ</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\theta_i</annotation></semantics></math>.</p> <p>Now an element in this tensor product is of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>+</mo><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>q</mi></munderover><msub><mi>g</mi> <mi>i</mi></msub><msub><mi>θ</mi> <mi>i</mi></msub><mo>+</mo><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>q</mi></munderover><msub><mi>f</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><msub><mi>θ</mi> <mi>i</mi></msub><msub><mi>θ</mi> <mi>j</mi></msub><mo>+</mo><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mi>q</mi></munderover><msub><mi>g</mi> <mrow><mi>i</mi><mi>j</mi><mi>k</mi></mrow></msub><msub><mi>θ</mi> <mi>i</mi></msub><msub><mi>θ</mi> <mi>j</mi></msub><msub><mi>θ</mi> <mi>j</mi></msub><mo>+</mo><mi>⋯</mi></mrow><annotation encoding="application/x-tex"> f + \underoverset{i = 1}{q}{\sum} g_i \theta_i + \underoverset{i,j = 1}{q}{\sum} f_{i j} \theta_i \theta_j + \underoverset{i,j,k = 1}{q}{\sum} g_{i j k} \theta_i \theta_j \theta_j + \cdots </annotation></semantics></math></div> <p>for any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mi>⋯</mi></msub><mo>∈</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><msub><mo stretchy="false">)</mo> <mi>even</mi></msub></mrow><annotation encoding="application/x-tex">f_{\cdots} \in C^\infty(X)_{even}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mi>⋯</mi></msub><mo>∈</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><msub><mo stretchy="false">)</mo> <mi>odd</mi></msub></mrow><annotation encoding="application/x-tex">g_{\cdots} \in C^\infty(X)_{odd}</annotation></semantics></math>.</p> <p>(Notice that here if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is of superdimension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>d</mi> <mi>X</mi></msub><mo>,</mo><msub><mi>q</mi> <mi>X</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(d_X,q_X)</annotation></semantics></math>, then this expansion becomes redundant for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi><mo>></mo><msub><mi>q</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">q \gt q_X</annotation></semantics></math>: The <a class="existingWikiWord" href="/nlab/show/functor+of+points">functor of points</a> provides an arbitrary supply of auxiliary Grassmann coordinates <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math>, only some of which will typically be of non-redundant use for any given superspace.)</p> <p>In <a class="existingWikiWord" href="/nlab/show/physics">physics</a>, such a <a class="existingWikiWord" href="/nlab/show/linear+combination">linear combination</a> of even and odd component functions multipled with Grassmann algebra elements to yield a homogeneously graded sum is called a <em><a class="existingWikiWord" href="/nlab/show/superfield">superfield</a></em>.</p> <p>In order to make sense of this, some physics textbooks (e.g. <a class="existingWikiWord" href="/nlab/show/Bryce+DeWitt">de Witt</a> 92) posit a single “infinite dimensional Grassmann algebra” from which to draw the elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>θ</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\theta_i</annotation></semantics></math>. This approach has its pitfalls <a href="#Sachse08">Sachse 08, section 5.2</a>. The <a class="existingWikiWord" href="/nlab/show/functorial+geometry">functorial geometry</a> perspective (remark <a class="maruku-ref" href="#ASheafAsASpace"></a>) fixes this: the “arbitrary supply” of Grassmann variables is encoded by saying that</p> <ol> <li> <p>for each finite dimensional Grassmann algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mo>∧</mo> <mo>•</mo></msup><mo stretchy="false">⟨</mo><msub><mi>θ</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>θ</mi> <mi>q</mi></msub><mo stretchy="false">⟩</mo><mo>=</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mn>0</mn><mo stretchy="false">|</mo><mi>q</mi></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\wedge^\bullet \langle \theta_1, \cdots, \theta_q\rangle = C^\infty(\mathbb{R}^{0\vert q})</annotation></semantics></math> superfields have an expansion in terms of the generators <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>θ</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\theta_i</annotation></semantics></math>;</p> </li> <li> <p>these expressions are <em>covariant</em> with respect to change of Grassmann coordinates <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><mi>q</mi></mrow></msup><mo>→</mo><msup><mi>ℝ</mi> <mrow><mn>0</mn><mo stretchy="false">|</mo><mi>q</mi><mo>′</mo></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{0\vert q} \to \mathbb{R}^{0\vert q'}</annotation></semantics></math>.</p> </li> </ol> <p>There are of course the evident generalizations of the scalar valued superfields along the same lines. In general there is a (super-)<a class="existingWikiWord" href="/nlab/show/fiber+bundle">fiber bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mover><mo>→</mo><mi>π</mi></mover><mi>X</mi></mrow><annotation encoding="application/x-tex">E \overset{\pi}{\to} X</annotation></semantics></math> over (<a class="existingWikiWord" href="/nlab/show/super+spacetime">super</a>) <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> called the (super)-<a class="existingWikiWord" href="/nlab/show/field+bundle">field bundle</a> such that a field on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/section">section</a> of the field bundle (see also at <em><a class="existingWikiWord" href="/nlab/show/fiber+bundles+in+physics">fiber bundles in physics</a></em>). The super-space of sections of <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> is the following <a class="existingWikiWord" href="/nlab/show/fiber+product">fiber product</a> of the <a class="existingWikiWord" href="/nlab/show/mapping+space">mapping space</a></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>X</mi></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mi>E</mi><mo stretchy="false">]</mo><munder><mo>×</mo><mrow><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mrow></munder><mo stretchy="false">{</mo><msub><mi>id</mi> <mi>X</mi></msub><mo stretchy="false">}</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \Gamma_X(E) \;\coloneqq\; [X,E] \underset{[X,X]}{\times} \{id_X\} \,, </annotation></semantics></math></div> <p>i.e. the super-space with the <a class="existingWikiWord" href="/nlab/show/universal+property">universal property</a> that it makes the follwing <a class="existingWikiWord" href="/nlab/show/commuting+square">square commute</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><msub><mi>Γ</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mi>E</mi><mo stretchy="false">]</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>X</mi><mo>,</mo><mi>π</mi></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mo stretchy="false">{</mo><msub><mi>id</mi> <mi>X</mi></msub><mo stretchy="false">}</mo></mtd> <mtd><mo>↪</mo></mtd> <mtd><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \Gamma_X(E) &\longrightarrow& [X,E] \\ \downarrow && \downarrow^{\mathrlap{X,\pi}} \\ \{id_X\} &\hookrightarrow& [X,X] } \,. </annotation></semantics></math></div> <p>Then a <a class="existingWikiWord" href="/nlab/show/superfield">superfield</a> with values in <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> is a <a class="existingWikiWord" href="/nlab/show/generalized+element">generalized element</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Γ</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Gamma_X(E)</annotation></semantics></math>. By the <a class="existingWikiWord" href="/nlab/show/functorial+geometry">functorial geometry</a>this means that field is over every superpoint <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><mi>q</mi></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{0\vert q}</annotation></semantics></math> a homomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ψ</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>×</mo><msup><mi>ℝ</mi> <mrow><mn>0</mn><mo stretchy="false">|</mo><mi>q</mi></mrow></msup></mrow><annotation encoding="application/x-tex">\psi \colon X \times \mathbb{R}^{0\vert q}</annotation></semantics></math> such that</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></mtd> <mtd></mtd> <mtd><mi>E</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>ψ</mi></mpadded></msup><mo>↗</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>π</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>X</mi><mo>×</mo><msup><mi>ℝ</mi> <mrow><mn>0</mn><mo stretchy="false">|</mo><mi>q</mi></mrow></msup></mtd> <mtd><munder><mo>⟶</mo><mrow><msub><mi>pr</mi> <mn>1</mn></msub></mrow></munder></mtd> <mtd><mi>X</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ && E \\ & {}^{\mathllap{\psi}}\nearrow & \downarrow^{\mathrlap{\pi}} \\ X \times \mathbb{R}^{0 \vert q} &\underset{pr_1}{\longrightarrow}& X } \,. </annotation></semantics></math></div> <p>In a typical example <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is an ordinary <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> with <a class="existingWikiWord" href="/nlab/show/spin+structure">spin structure</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>=</mo><mi>Π</mi><mi>S</mi></mrow><annotation encoding="application/x-tex">E = \Pi S</annotation></semantics></math> is the odd version (according to example <a class="maruku-ref" href="#OddTangentBundle"></a>) of a <a class="existingWikiWord" href="/nlab/show/spinor+bundle">spinor bundle</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> <p>Then an element of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mi>Π</mi><mi>S</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[X,\Pi S]</annotation></semantics></math> is</p> <ol> <li> <p>over <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>0</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{0\vert 0}</annotation></semantics></math> no information;</p> </li> <li> <p>over <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">\mathbb{R}^{0 \vert 1}</annotation></semantics></math> an element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ψ</mi><mi>θ</mi></mrow><annotation encoding="application/x-tex">\psi \theta</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ψ</mi><mo>∈</mo><msub><mi>Γ</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\psi \in \Gamma_X(S)</annotation></semantics></math> an ordinary section of the <a class="existingWikiWord" href="/nlab/show/spinor+bundle">spinor bundle</a> (hence a <a class="existingWikiWord" href="/nlab/show/spinor+field">spinor field</a>).</p> </li> </ol> <p>and so on.</p> <p>This may be combined: For example if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mi>ℂ</mi><mo>̲</mo></munder><msub><mo>⊕</mo> <mi>X</mi></msub><mi>Π</mi></mrow><annotation encoding="application/x-tex">\underline{\mathbb{C}} \oplus_X \Pi </annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/direct+sum+of+vector+bundles">direct sum of vector bundles</a> of the <a class="existingWikiWord" href="/nlab/show/trivial+vector+bundle">trivial</a> <a class="existingWikiWord" href="/nlab/show/complex+line+bundle">complex line bundle</a> with an odd <a class="existingWikiWord" href="/nlab/show/spinor+bundle">spinor bundle</a>, then a <a class="existingWikiWord" href="/nlab/show/generalized+element">generalized element</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Γ</mi> <mi>X</mi></msub><mrow><mo>(</mo><munder><mi>ℂ</mi><mo>̲</mo></munder><msub><mo>⊕</mo> <mi>X</mi></msub><mi>Π</mi><mi>S</mi><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">\Gamma_X\left(\underline{\mathbb{C}} \oplus_X \Pi S \right)</annotation></semantics></math> is over <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">\mathbb{R}^{0\vert 1}</annotation></semantics></math> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>σ</mi><mo>=</mo><mi>ϕ</mi><mo>+</mo><mi>ψ</mi><mi>θ</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \sigma = \phi + \psi \theta \,, </annotation></semantics></math></div> <p>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 complex-vaued <a class="existingWikiWord" href="/nlab/show/scalar+field">scalar field</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ψ</mi></mrow><annotation encoding="application/x-tex">\psi</annotation></semantics></math> again a section of the spinor bundle.</p> <p>This way <a class="existingWikiWord" href="/nlab/show/bosonic+fields">bosonic fields</a> and <a class="existingWikiWord" href="/nlab/show/fermionic+fields">fermionic fields</a> may be combined into a singe <a class="existingWikiWord" href="/nlab/show/superfield">superfield</a> (see also at <em><a class="existingWikiWord" href="/nlab/show/super+multiplet">super multiplet</a></em>).</p> </div> <h2 id="Supermanifolds">Supermanifolds</h2> <p>We now define and then discuss the analog of <a class="existingWikiWord" href="/nlab/show/smooth+manifolds">smooth manifolds</a> in <a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a> – <a class="existingWikiWord" href="/nlab/show/supermanifolds">supermanifolds</a>. In the spirit of the entire presentation, we do so by applying a <a class="existingWikiWord" href="/nlab/show/general+abstract">general abstract</a> definition of “<a class="existingWikiWord" href="/nlab/show/V-manifold">V-manifold</a>” or “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-scheme” locally modeled on any given kind of model spaces to the special case where the local model spaces are <a class="existingWikiWord" href="/nlab/show/super+Cartesian+spaces">super Cartesian spaces</a> as discussed above. This general method is also discussed at <em><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+manifolds+and+orbifolds">geometry of physics – manifolds and orbifolds</a></em>, but we recall the relevant points below.</p> <p>Recall the <a class="existingWikiWord" href="/nlab/show/adjoint+pair">adjoint pair</a> of <a class="existingWikiWord" href="/nlab/show/endofunctors">endofunctors</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><mi>ℜ</mi><mo>⊣</mo><mi>ℑ</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>SuperFormalSmoothSet</mi><mo>⟶</mo><mi>SuperFormalSmoothSet</mi></mrow><annotation encoding="application/x-tex"> (\Re \dashv \Im) \;\colon\; SuperFormalSmoothSet \longrightarrow SuperFormalSmoothSet </annotation></semantics></math></div> <p>from Prop. <a class="maruku-ref" href="#ProgressionOfIdempotentEndofunctors"></a>. By prop. <a class="maruku-ref" href="#SuperSmoothSetsSystemOfAdjunctions"></a> and prop. <a class="maruku-ref" href="#CartSpCoreflectiveInclusion"></a> we know that <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">\Re</annotation></semantics></math> sends a (formal) <a class="existingWikiWord" href="/nlab/show/super+Cartesian+space">super Cartesian space</a> to its underlying reduced ordinary <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a>. For instance</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ℜ</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mi>p</mi><mo stretchy="false">|</mo><mi>q</mi></mrow></msup><mo stretchy="false">)</mo><mo>≃</mo><msup><mi>ℝ</mi> <mi>p</mi></msup></mrow><annotation encoding="application/x-tex"> \Re(\mathbb{R}^{p\vert q}) \simeq \mathbb{R}^p </annotation></semantics></math></div> <p>and generally</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ℜ</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>×</mo><mi>𝔻</mi><mo stretchy="false">)</mo><mo>≃</mo><msup><mi>𝔻</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex"> \Re(\mathbb{R}^n \times \mathbb{D}) \simeq \mathbb{D}^n </annotation></semantics></math></div> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔻</mi></mrow><annotation encoding="application/x-tex">\mathbb{D}</annotation></semantics></math> any <a class="existingWikiWord" href="/nlab/show/infinitesimally+thickened+point">infinitesimally thickened superpoint</a>.</p> <p>This is enough to find what its <a class="existingWikiWord" href="/nlab/show/right+adjoint">right adjoint</a> operation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℑ</mi></mrow><annotation encoding="application/x-tex">\Im</annotation></semantics></math> is doing:</p> <div class="num_prop" id="ImAction"> <h6 id="proposition_10">Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>SuperFormalSmoothSet</mi></mrow><annotation encoding="application/x-tex">X \in SuperFormalSmoothSet</annotation></semantics></math> (def. <a class="maruku-ref" href="#FormalSmoothSets"></a>), then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℑ</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">\Im X</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/super+formal+smooth+set">super formal smooth set</a> whose <a class="existingWikiWord" href="/nlab/show/functor+of+points">functor of points</a> is given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ℑ</mi><mi>X</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>×</mo><mi>𝔻</mi><mo>↦</mo><mi>X</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \Im X \;\colon\; \mathbb{R}^n \times \mathbb{D} \mapsto X(\mathbb{R}^n) </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math> and for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔻</mi></mrow><annotation encoding="application/x-tex">\mathbb{D}</annotation></semantics></math> any <a class="existingWikiWord" href="/nlab/show/infinitesimally+thickened+point">infinitesimally thickened superpoint</a>.</p> </div> <div class="proof"> <h6 id="proof_9">Proof</h6> <p>By applying the <a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a>, the <a class="existingWikiWord" href="/nlab/show/natural+bijection">natural bijection</a> between <a class="existingWikiWord" href="/nlab/show/hom-sets">hom-sets</a> that characterizes the <a class="existingWikiWord" href="/nlab/show/adjoint+pair">adjoint pair</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℜ</mi><mo>⊣</mo><mi>ℑ</mi></mrow><annotation encoding="application/x-tex">\Re \dashv \Im</annotation></semantics></math> and using the characterization 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">\Re</annotation></semantics></math> we get the following sequence of <a class="existingWikiWord" href="/nlab/show/natural+isomorphisms">natural isomorphisms</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mo stretchy="false">(</mo><mi>ℑ</mi><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>×</mo><mi>𝔻</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>≃</mo><mi>Hom</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>×</mo><mi>𝔻</mi><mo>,</mo><mi>ℑ</mi><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><mi>Hom</mi><mo stretchy="false">(</mo><mi>ℜ</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>×</mo><mi>𝔻</mi><mo stretchy="false">)</mo><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><mi>Hom</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><mi>X</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} (\Im X)(\mathbb{R}^n \times \mathbb{D}) & \simeq Hom( \mathbb{R}^n \times \mathbb{D} , \Im X) \\ & \simeq Hom(\Re(\mathbb{R}^n \times \mathbb{D}), X) \\ & \simeq Hom(\mathbb{R}^n, X) \\ & \simeq X(\mathbb{R}^n) \end{aligned} \,. </annotation></semantics></math></div></div> <p>Proposition <a class="maruku-ref" href="#ImAction"></a> means that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> for instance an ordinary <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> (regarded as a <a class="existingWikiWord" href="/nlab/show/super+formal+smooth+set">super formal smooth set</a> via prop. <a class="maruku-ref" href="#SmoothSetsContainSmoothManifolds"></a>) then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℑ</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">\Im X</annotation></semantics></math> is a rather exotic kind of <a class="existingWikiWord" href="/nlab/show/generalized+smooth+space">generalized smooth space</a>: it has the same finite smooth curves and other finite smooth shapes as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> does, but every <em>infinitesimal</em> curve or shape inside it is necessarily constant. A good way to think about this (which is also the precise way to think about it, if we speak in the <a class="existingWikiWord" href="/nlab/show/internal+language">internal language</a> of the <a class="existingWikiWord" href="/nlab/show/sheaf+topos">sheaf topos</a>) is that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℑ</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">\Im X</annotation></semantics></math> is the result obtained from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> by <strong>identifying all infinitesimally close points</strong> with each other. In <a class="existingWikiWord" href="/nlab/show/algebraic+geometry">algebraic geometry</a> this construction is often known as forming the <em><a class="existingWikiWord" href="/nlab/show/de+Rham+shape">de Rham shape</a></em> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> (<a href="#de+Rham+space#Simpson96">Simpson 96</a>). Here we will say <strong><a class="existingWikiWord" href="/nlab/show/infinitesimal+shape">infinitesimal shape</a></strong>.</p> <p>Another useful perspective on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℑ</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">\Im X</annotation></semantics></math> is the following:</p> <div class="num_defn" id="DiskBundle"> <h6 id="definition_8">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/infinitesimal+disk+bundle">infinitesimal disk bundle</a>)</strong></p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>SuperFormalSmoothSet</mi></mrow><annotation encoding="application/x-tex">X \in SuperFormalSmoothSet</annotation></semantics></math> (def. <a class="maruku-ref" href="#FormalSmoothSets"></a>) then we say that its <a class="existingWikiWord" href="/nlab/show/infinitesimal+disk+bundle">infinitesimal disk bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>T</mi> <mn>∞</mn></msup><mi>X</mi></mrow><annotation encoding="application/x-tex">T^\infty X</annotation></semantics></math> is the left vertical morphism in the following <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> diagram</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>T</mi> <mn>∞</mn></msup><mi>X</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>p</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><mover><mrow></mrow><mrow><mo stretchy="false">(</mo><mi>pb</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>η</mi> <mi>X</mi></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><munder><mo>⟶</mo><mrow><msub><mi>η</mi> <mi>X</mi></msub></mrow></munder></mtd> <mtd><mi>ℑ</mi><mi>X</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ T^\infty X &\longrightarrow& X \\ {}^{\mathllap{p}}\downarrow &\stackrel{(pb)}{}& \downarrow^{\mathrlap{\eta_X}} \\ X &\underset{\eta_X}{\longrightarrow}& \Im X } \,. </annotation></semantics></math></div> <p>Moreover, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo lspace="verythinmathspace">:</mo><mo>*</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x \colon \ast \to X</annotation></semantics></math> any <a class="existingWikiWord" href="/nlab/show/global+point">global point</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, then we say that the <a class="existingWikiWord" href="/nlab/show/formal+disk">formal disk</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/fiber">fiber</a> of the infinitesimal disk bundle over that point</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>𝔻</mi> <mi>x</mi></msub></mtd> <mtd><mo>⟶</mo></mtd> <mtd><msup><mi>T</mi> <mn>∞</mn></msup><mi>X</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><mover><mrow></mrow><mrow><mo stretchy="false">(</mo><mi>pb</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>p</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><mover><mrow></mrow><mrow><mo stretchy="false">(</mo><mi>pb</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>η</mi> <mi>X</mi></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd><munder><mo>⟶</mo><mi>x</mi></munder></mtd> <mtd><mi>X</mi></mtd> <mtd><munder><mo>⟶</mo><mrow><msub><mi>η</mi> <mi>X</mi></msub></mrow></munder></mtd> <mtd><mi>ℑ</mi><mi>X</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \mathbb{D}_x &\longrightarrow& T^\infty X &\longrightarrow& X \\ \downarrow &\stackrel{(pb)}{}& {}^{\mathllap{p}}\downarrow &\stackrel{(pb)}{}& \downarrow^{\mathrlap{\eta_X}} \\ \ast &\underset{x}{\longrightarrow}& X &\underset{\eta_X}{\longrightarrow}& \Im X } \,. </annotation></semantics></math></div></div> <div class="num_defn" id="LocalDiffeomorphisms"> <h6 id="definition_9">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/formally+%C3%A9tale+morphism">formally étale morphism</a>)</strong></p> <p>Given <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>,</mo><mi>Y</mi><mo>∈</mo><mi>SuperFormalSmoothSet</mi></mrow><annotation encoding="application/x-tex">X,Y\in SuperFormalSmoothSet</annotation></semantics></math> then a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f \;\colon\; X\longrightarrow Y</annotation></semantics></math> is called a <em><a class="existingWikiWord" href="/nlab/show/formally+%C3%A9tale+morphism">formally étale morphism</a></em> if its <a class="existingWikiWord" href="/nlab/show/natural+transformation">naturality square</a> of the <a class="existingWikiWord" href="/nlab/show/infinitesimal+shape+modality">infinitesimal shape modality</a> (prop. <a class="maruku-ref" href="#ImAction"></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></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>ℑ</mi><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>f</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>ℑ</mi><mi>f</mi></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>Y</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>ℑ</mi><mi>Y</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ X &\longrightarrow& \Im X \\ \downarrow^{\mathrlap{f}} && \downarrow^{\mathrlap{\Im f}} \\ Y &\longrightarrow& \Im Y } </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> square.</p> <p>We often indicate that a morphism satisfies this condition by labeling it “et” , hence</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></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>X</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>ℑ</mi><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>f</mi></mpadded></msup><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>et</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><mo>⇔</mo></mtd> <mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>f</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><mover><mrow></mrow><mrow><mo stretchy="false">(</mo><mi>pb</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>ℑ</mi><mi>f</mi></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>Y</mi></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>Y</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>ℑ</mi><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ X &&&& X &\longrightarrow& \Im X \\ {}^{\mathllap{f}}\downarrow^{\mathrlap{et}} &&\Leftrightarrow&& {}^{\mathllap{f}}\downarrow &\stackrel{(pb)}{}& \downarrow^{\mathrlap{\Im f}} \\ Y &&&& Y &\longrightarrow& \Im Y } \,. </annotation></semantics></math></div></div> <p>We unwind definition <a class="maruku-ref" href="#LocalDiffeomorphisms"></a> a little:</p> <div class="num_remark" id="FormallyEtaleUnwinding"> <h6 id="remark_4">Remark</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔻</mi></mrow><annotation encoding="application/x-tex">\mathbb{D}</annotation></semantics></math> be an <a class="existingWikiWord" href="/nlab/show/infinitesimally+thickened+point">infinitesimally thickened point</a>. This means that its reduction is the actual point, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℜ</mi><mi>𝔻</mi><mo>≃</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">\Re \mathbb{D} \simeq \ast</annotation></semantics></math>. By the <a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℜ</mi><mo>⊣</mo><mi>ℑ</mi></mrow><annotation encoding="application/x-tex">\Re \dashv \Im</annotation></semantics></math> from Prop. <a class="maruku-ref" href="#ProgressionOfIdempotentEndofunctors"></a> it follows that the image of the naturality square in def. <a class="maruku-ref" href="#LocalDiffeomorphisms"></a> under forming the <a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom</a> (def. <a class="maruku-ref" href="#InternalHom"></a>) out of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔻</mi></mrow><annotation encoding="application/x-tex">\mathbb{D}</annotation></semantics></math> is</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>X</mi> <mi>𝔻</mi></msup></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msup><mi>f</mi> <mi>𝔻</mi></msup></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mi>f</mi></msup></mtd></mtr> <mtr><mtd><msup><mi>Y</mi> <mi>𝔻</mi></msup></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ X^{\mathbb{D}} &\longrightarrow& X \\ \downarrow^{\mathrlap{f^\mathbb{D}}} && \downarrow^f \\ Y^{\mathbb{D}} &\longrightarrow& Y } </annotation></semantics></math></div> <p>Since the internal hom preserves <a class="existingWikiWord" href="/nlab/show/limits">limits</a> in its second argument (being <a class="existingWikiWord" href="/nlab/show/right+adjoint">right adjoint</a> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔻</mi><mo>×</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{D} \times (-)</annotation></semantics></math>) this is a <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> square if <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 a <a class="existingWikiWord" href="/nlab/show/formally+%C3%A9tale+morphism">formally étale morphism</a> according to def. <a class="maruku-ref" href="#LocalDiffeomorphisms"></a>. In this form the condition appears in <a href="#Yetter88">Yetter 88, def. 3.3.1</a>.</p> <p>If here we specify <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔻</mi><mo>=</mo><mi>Spec</mi><mo stretchy="false">(</mo><mi>ℝ</mi><mo stretchy="false">[</mo><mi>ϵ</mi><mo stretchy="false">]</mo><mo stretchy="false">/</mo><msup><mi>ϵ</mi> <mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{D} = Spec(\mathbb{R}[\epsilon]/\epsilon^2)</annotation></semantics></math> to be the <a class="existingWikiWord" href="/nlab/show/formal+dual">formal dual</a> of the <a class="existingWikiWord" href="/nlab/show/ring+of+dual+numbers">ring of dual numbers</a>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>X</mi> <mi>𝔻</mi></msup></mrow><annotation encoding="application/x-tex">X^{\mathbb{D}}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Y</mi> <mi>𝔻</mi></msup></mrow><annotation encoding="application/x-tex">Y^{\mathbb{D}}</annotation></semantics></math> are the respective <a class="existingWikiWord" href="/nlab/show/tangent+bundles">tangent bundles</a> by example <a class="maruku-ref" href="#CorepresentingTangentSpace"></a>. Hence in this case the condition that <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 a <a class="existingWikiWord" href="/nlab/show/formally+%C3%A9tale+morphism">formally étale morphism</a> according to def. <a class="maruku-ref" href="#LocalDiffeomorphisms"></a> implies that the square</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>T</mi><mi>X</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>T</mi><mi>f</mi></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>f</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>T</mi><mi>Y</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ T X &\longrightarrow& X \\ \downarrow^{\mathrlap{T f}} && \downarrow^{\mathrlap{f}} \\ T Y &\longrightarrow& Y } </annotation></semantics></math></div> <p>is a pullback square. For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>,</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X,Y</annotation></semantics></math> ordinary smooth manifolds via prop. <a class="maruku-ref" href="#SmoothSetsContainSmoothManifolds"></a>, this condition is the traditional definition that <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> be a <a class="existingWikiWord" href="/nlab/show/local+diffeomorphism">local diffeomorphism</a>.</p> </div> <div class="num_prop"> <h6 id="proposition_11">Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>,</mo><mi>Y</mi><mo>∈</mo><mi>SuperFormalSmoothSet</mi></mrow><annotation encoding="application/x-tex">X,Y \in SuperFormalSmoothSet</annotation></semantics></math> two ordinary <a class="existingWikiWord" href="/nlab/show/smooth+manifolds">smooth manifolds</a> via prop. <a class="maruku-ref" href="#SmoothSetsContainSmoothManifolds"></a>, then a morphism between them is a <a class="existingWikiWord" href="/nlab/show/formally+%C3%A9tale+morphism">formally étale morphism</a> according to def. <a class="maruku-ref" href="#LocalDiffeomorphisms"></a> precisely if it is a <a class="existingWikiWord" href="/nlab/show/local+diffeomorphism">local diffeomorphism</a> in the traditional sense.</p> </div> <div class="proof"> <h6 id="proof_10">Proof</h6> <p>By remark <a class="maruku-ref" href="#FormallyEtaleUnwinding"></a> the condition of def. <a class="maruku-ref" href="#LocalDiffeomorphisms"></a> on morphisms between smooth manifolds is equivalent to the traditional condition of being locally diffeo already when seen just under the internal hom out of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spec</mi><mo stretchy="false">(</mo><mi>ℝ</mi><mo stretchy="false">[</mo><mi>ϵ</mi><mo stretchy="false">]</mo><mo stretchy="false">(</mo><msup><mi>ϵ</mi> <mn>2</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spec(\mathbb{R}[\epsilon](\epsilon^2))</annotation></semantics></math>.</p> <p>But, as the name suggests, a <a class="existingWikiWord" href="/nlab/show/local+diffeomorphism">local diffeomorphism</a> of smooth manifolds is in particular also a <a class="existingWikiWord" href="/nlab/show/local+homeomorphism">local homeomorphism</a>. This means that around each point of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> there is actually an <a class="existingWikiWord" href="/nlab/show/open+neighbourhood">open neighbourhood</a> such that <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> restricts to a <a class="existingWikiWord" href="/nlab/show/diffeomorphism">diffeomorphism</a> on that neighbourhood. This implies that the full condition in def. <a class="maruku-ref" href="#LocalDiffeomorphisms"></a> holds, by an argument as in example <a class="maruku-ref" href="#ProjectioonOutOfCoproductIsFormllyEtale"></a>.</p> </div> <div class="num_example" id="ProjectioonOutOfCoproductIsFormllyEtale"> <h6 id="example_8">Example</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>∈</mo><mi>SuperFormalSmoothSet</mi></mrow><annotation encoding="application/x-tex">V \in SuperFormalSmoothSet</annotation></semantics></math> be given (def. <a class="maruku-ref" href="#FormalSmoothSets"></a>) and for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> any set, then the canonical morphism out of the <a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a> (<a class="existingWikiWord" href="/nlab/show/disjoint+union">disjoint union</a>) of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> copies of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> to itself is a local diffeomorphism according to def. <a class="maruku-ref" href="#LocalDiffeomorphisms"></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><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><mi>V</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>et</mi></mpadded></msup><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo stretchy="false">(</mo><msub><mi>id</mi> <mi>i</mi></msub><msub><mo stretchy="false">)</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>V</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \underset{i \in I}{\coprod} V \\ {}^{\mathllap{et}}\downarrow^{\mathrlap{(id_i)_{i \in I}}} \\ V } </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_11">Proof</h6> <p>Since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℑ</mi></mrow><annotation encoding="application/x-tex">\Im</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> by prop. <a class="maruku-ref" href="#SuperSmoothSetsSystemOfAdjunctions"></a>, Prop. <a class="maruku-ref" href="#ProgressionOfIdempotentEndofunctors"></a>, it preserves all <a class="existingWikiWord" href="/nlab/show/colimits">colimits</a> and hence in particular <a class="existingWikiWord" href="/nlab/show/coproducts">coproducts</a>, hence the image of the morphism under <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℑ</mi></mrow><annotation encoding="application/x-tex">\Im</annotation></semantics></math> is</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><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><mi>ℑ</mi><mi>V</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>et</mi></mpadded></msup><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo stretchy="false">(</mo><msub><mi>id</mi> <mi>i</mi></msub><msub><mo stretchy="false">)</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>ℑ</mi><mi>V</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \underset{i \in I}{\coprod} \Im V \\ {}^{\mathllap{et}}\downarrow^{\mathrlap{(id_i)_{i \in I}}} \\ \Im V } \,. </annotation></semantics></math></div> <p>Moreover, in any <a class="existingWikiWord" href="/nlab/show/sheaf+topos">sheaf topos</a> <a class="existingWikiWord" href="/nlab/show/universal+colimits">colimits are universal</a>, which means that maps out of colimits are preserved under <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a>. Moreover, the pullback of an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> is an isomorphism, and so we deduce that we have a pullback diagram of the form</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><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>V</mi> <mi>i</mi></msub></mtd> <mtd><mo>⟶</mo></mtd> <mtd><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><mi>ℑ</mi><mi>V</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo stretchy="false">(</mo><msub><mi>id</mi> <mi>i</mi></msub><msub><mo stretchy="false">)</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>pb</mi><mo stretchy="false">)</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo stretchy="false">(</mo><msub><mi>id</mi> <mi>i</mi></msub><msub><mo stretchy="false">)</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>V</mi></mtd> <mtd><munder><mo>⟶</mo><mrow><msub><mi>η</mi> <mi>V</mi></msub></mrow></munder></mtd> <mtd><mi>ℑ</mi><mi>V</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \underset{i \in I}{\coprod}V_i &\longrightarrow& \underset{i \in I}{\coprod} \Im V \\ {}^{{\mathllap{(id_i)_{i \in I}}}}\downarrow &(pb)& \downarrow^{\mathrlap{(id_i)_{i \in I}}} \\ V &\underset{\eta_V}{\longrightarrow}& \Im V } \,. </annotation></semantics></math></div> <p>But, again because <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℑ</mi></mrow><annotation encoding="application/x-tex">\Im</annotation></semantics></math> preserves colimits, this is manifestly the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℑ</mi></mrow><annotation encoding="application/x-tex">\Im</annotation></semantics></math>-naturality square of the original morphism.</p> </div> <div class="num_defn" id="VManifold"> <h6 id="definition_10">Definition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>∈</mo><mi>SuperFormalSmoothSet</mi></mrow><annotation encoding="application/x-tex">V \in SuperFormalSmoothSet</annotation></semantics></math> be given (def. <a class="maruku-ref" href="#FormalSmoothSets"></a>), equipped with the structure of a <a class="existingWikiWord" href="/nlab/show/group+object">group object</a>.</p> <p>A <em><a class="existingWikiWord" href="/nlab/show/V-manifold">V-manifold</a></em> is an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>SuperFormalSmoothSet</mi></mrow><annotation encoding="application/x-tex">X \in SuperFormalSmoothSet</annotation></semantics></math> such that there exists a <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/atlas">atlas</a></em>, namely a <a class="existingWikiWord" href="/nlab/show/correspondence">correspondence</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><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>U</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>et</mi></mpadded></msup><mo>↙</mo></mtd> <mtd></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mrow><mi>et</mi><mo>,</mo><mi>epi</mi></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>V</mi></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>X</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && U \\ & {}^{\mathllap{et}}\swarrow && \searrow^{\mathrlap{et, epi}} \\ V && && X } </annotation></semantics></math></div> <p>with both morphisms being <a class="existingWikiWord" href="/nlab/show/local+diffeomorphisms">local diffeomorphisms</a>, def. <a class="maruku-ref" href="#LocalDiffeomorphisms"></a>, and the right one in addition being an <a class="existingWikiWord" href="/nlab/show/epimorphism">epimorphism</a>.</p> <p>By prop. <a class="maruku-ref" href="#ProjectioonOutOfCoproductIsFormllyEtale"></a> this means that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is in particular a <a class="existingWikiWord" href="/nlab/show/V-manifold">V-manifold</a> if there exists a <a class="existingWikiWord" href="/nlab/show/set">set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> and a morphism out of the <a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> copies of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> which is a locally diffeomorphic epimorphism:</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><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><mi>V</mi></mtd></mtr> <mtr><mtd><msubsup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>epi</mi></mpadded> <mpadded width="0"><mi>et</mi></mpadded></msubsup></mtd></mtr> <mtr><mtd><mi>X</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \underset{i \in I}{\coprod} V \\ \downarrow^{\mathrlap{et}}_{\mathrlap{epi}} \\ X } </annotation></semantics></math></div> <p>Specifically if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>=</mo><msup><mi>ℝ</mi> <mrow><mi>p</mi><mo stretchy="false">|</mo><mi>q</mi></mrow></msup></mrow><annotation encoding="application/x-tex">V = \mathbb{R}^{p\vert q}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/super+Cartesian+space">super Cartesian space</a> regarded as a <a class="existingWikiWord" href="/nlab/show/group+object">group object</a> under a <a class="existingWikiWord" href="/nlab/show/translation+supergroup">translation supergroup</a> action. Then we say that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a <strong><a class="existingWikiWord" href="/nlab/show/supermanifold">supermanifold</a></strong> of <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">|</mo><mi>q</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(p\vert q)</annotation></semantics></math> if there exists</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><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msup><mi>ℝ</mi> <mrow><mi>p</mi><mo stretchy="false">|</mo><mi>q</mi></mrow></msup></mtd></mtr> <mtr><mtd><msubsup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>epi</mi></mpadded> <mpadded width="0"><mi>et</mi></mpadded></msubsup></mtd></mtr> <mtr><mtd><mi>X</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \underset{i \in I}{\coprod} \mathbb{R}^{p\vert q} \\ \downarrow^{\mathrlap{et}}_{\mathrlap{epi}} \\ X } </annotation></semantics></math></div></div> <div class="num_remark"> <h6 id="remark_5">Remark</h6> <p>There are in general several <a class="existingWikiWord" href="/nlab/show/translation+supergroup">translation supergroup</a> structures carried by a super Cartesian space. For instance in the discussion at <em><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+supersymmetry">geometry of physics – supersymmetry</a></em> we will consider <a class="existingWikiWord" href="/nlab/show/super+Minkowski+spacetime">super Minkowski</a> supergroup structure which exists for special pairs <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(p,q)</annotation></semantics></math>. Just the existence of a supermanifold structure in def. <a class="maruku-ref" href="#VManifold"></a> does not depend on the choice of group structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> (and could be ignored for much of the present purpose). But later on the choice affects for instance the concept of <a class="existingWikiWord" href="/nlab/show/torsion+of+a+G-structure">torsion of a G-structure</a> on the <a class="existingWikiWord" href="/nlab/show/V-manifold">V-manifold</a>.</p> </div> <div class="num_example" id="OddTangentBundle"> <h6 id="example_9">Example</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/odd+tangent+bundle">odd tangent bundle</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> of <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> <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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">E \to X</annotation></semantics></math> a smooth <a class="existingWikiWord" href="/nlab/show/vector+bundle">vector bundle</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/rank">rank</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">k \in \mathbb{N}</annotation></semantics></math>. Then there is a <a class="existingWikiWord" href="/nlab/show/supermanifold">supermanifold</a> (def. <a class="maruku-ref" href="#VManifold"></a>) of dimension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">|</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n \vert k)</annotation></semantics></math>, denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi><mi>E</mi></mrow><annotation encoding="application/x-tex">\Pi E</annotation></semantics></math>, as follows.</p> <p>Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝒪</mi><mo stretchy="false">(</mo><mi>Π</mi><mi>E</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><msubsup><mo>∧</mo> <mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow> <mo>•</mo></msubsup><msub><mi>Γ</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>⊕</mo><mspace width="thickmathspace"></mspace><msub><mi>Γ</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><msup><mi>E</mi> <mo>*</mo></msup><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>⊕</mo><mspace width="thickmathspace"></mspace><mrow><mo>(</mo><msub><mi>Γ</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><msup><mi>E</mi> <mo>*</mo></msup><mo stretchy="false">)</mo><msub><mo>∧</mo> <mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></msub><msub><mi>Γ</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><msup><mi>E</mi> <mo>*</mo></msup><mo stretchy="false">)</mo><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mo>⊕</mo><mspace width="thickmathspace"></mspace><mi>⋯</mi></mrow><annotation encoding="application/x-tex"> \mathcal{O}(\Pi E) \;\coloneqq\; \wedge^\bullet_{C^\infty(X)} \Gamma_X(E) = C^\infty(X) \;\oplus\; \Gamma_X(E^\ast) \;\oplus\; \left(\Gamma_X(E^\ast) \wedge_{C^\infty(X)} \Gamma_X(E^\ast)\right) \;\oplus\; \cdots </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/supercommutative+superalgebra">supercommutative superalgebra</a> which is the <a class="existingWikiWord" href="/nlab/show/exterior+algebra">exterior algebra</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^\infty(x)</annotation></semantics></math> of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^\infty(X)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/module">module</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Γ</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><msup><mi>E</mi> <mo>*</mo></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Gamma_X(E^\ast)</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/smooth+sections">smooth sections</a> (as in the <a class="existingWikiWord" href="/nlab/show/smooth+Serre-Swan+theorem">smooth Serre-Swan theorem</a>, Prop. <a class="maruku-ref" href="#FirstTwoMagicPropertiesOfAlgebrasOfSmoothFunctions"></a>) of the <a class="existingWikiWord" href="/nlab/show/dual+vector+bundle">dual vector bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">E^\ast</annotation></semantics></math>.</p> <p>The underlying <a class="existingWikiWord" href="/nlab/show/functor+of+points">functor of points</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi><mi>E</mi></mrow><annotation encoding="application/x-tex">\Pi E</annotation></semantics></math> (remark <a class="maruku-ref" href="#ASheafAsASpace"></a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Π</mi><mi>E</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>SuperCartSp</mi> <mi>op</mi></msup><mo>⟶</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex"> \Pi E \;\colon\; SuperCartSp^{op} \longrightarrow Set </annotation></semantics></math></div> <p>is the one that is <a class="existingWikiWord" href="/nlab/show/representable+functor">represented</a> by this algebra:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Π</mi><mi>E</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>ℝ</mi> <mrow><mi>p</mi><mo stretchy="false">|</mo><mi>q</mi></mrow></msup><mo>↦</mo><msub><mi>Hom</mi> <mrow><msub><mi>sCalg</mi> <mi>ℝ</mi></msub></mrow></msub><mo stretchy="false">(</mo><msubsup><mo>∧</mo> <mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow> <mo>•</mo></msubsup><msub><mi>Γ</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo>,</mo><mspace width="thickmathspace"></mspace><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mi>ℝ</mi></msub><msubsup><mo>∧</mo> <mi>ℝ</mi> <mo>•</mo></msubsup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>q</mi></msup><msup><mo stretchy="false">)</mo> <mo>*</mo></msup><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Pi E \;\colon\; \mathbb{R}^{p \vert q} \mapsto Hom_{sCalg_{\mathbb{R}}}( \wedge^\bullet_{C^\infty(X)} \Gamma_X(E) , \; C^\infty(\mathbb{R}^n) \otimes_{\mathbb{R}} \wedge^\bullet_{\mathbb{R}} (\mathbb{R}^q)^\ast ) \,. </annotation></semantics></math></div> <p>For instance if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>=</mo><mi>T</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">E = T X</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, so that the dual bundle is the <a class="existingWikiWord" href="/nlab/show/cotangent+bundle">cotangent bundle</a>, then</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝒪</mi><mo stretchy="false">(</mo><mi>Π</mi><mi>T</mi><mi>X</mi><mo stretchy="false">)</mo><mo>≃</mo><msup><mi>Ω</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≔</mo><msubsup><mo>∧</mo> <mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow> <mo>•</mo></msubsup><msub><mi>Γ</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><msup><mi>T</mi> <mo>*</mo></msup><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathcal{O}(\Pi T X) \simeq \Omega^\bullet(X) \coloneqq \wedge^\bullet_{C^\infty(X)}\Gamma_X(T^\ast X) </annotation></semantics></math></div> <p>is the superalgebra of smooth <a class="existingWikiWord" href="/nlab/show/differential+forms">differential forms</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> with respect to the wedge product, with any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>-form regarded as being in degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mspace width="thinmathspace"></mspace><mi>mod</mi><mspace width="thinmathspace"></mspace><mn>2</mn><mo>∈</mo><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">p \,mod\, 2 \in \mathbb{Z}/2</annotation></semantics></math>.</p> <p>This <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi><mi>T</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">\Pi T X</annotation></semantics></math> is often called the <strong><a class="existingWikiWord" href="/nlab/show/odd+tangent+bundle">odd tangent bundle</a></strong>.</p> <p>Notice that generally we may think of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi><mi>E</mi></mrow><annotation encoding="application/x-tex">\Pi E</annotation></semantics></math> as being the <a class="existingWikiWord" href="/nlab/show/superspace">superspace</a> which is obtained from the base manifold <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> by adding an (odd-graded) infinitesimal thickening where an “infinitesimal step” away from some point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x \in X</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/vector">vector</a> in the <a class="existingWikiWord" href="/nlab/show/fiber">fiber</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mi>x</mi></msub></mrow><annotation encoding="application/x-tex">E_x</annotation></semantics></math>. This is particularly suggestive in the case that <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> is the tangent bundle, because tangent vectors precisely want to be thought of as the infinitesimal paths in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> <p>Below we see that this is not a coincidence, and discuss the formal proof that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi><mi>T</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">\Pi T X</annotation></semantics></math> is the superspace of (odd-graded) infinitesimal paths in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> </div> <div class="num_prop" id="BosonicModalityPreservesLocalDiffeomorphism"> <h6 id="proposition_12">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/bosonic+modality">bosonic modality</a> preserves <a class="existingWikiWord" href="/nlab/show/formally+%C3%A9tale+morphisms">formally étale morphisms</a>)</strong></p> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f \;\colon\; X \longrightarrow Y</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/formally+%C3%A9tale+morphism">formally étale morphism</a>, def. <a class="maruku-ref" href="#LocalDiffeomorphisms"></a>, then so is its image <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>f</mi><mo>⇝</mo></mover><mo lspace="verythinmathspace">:</mo><mover><mi>X</mi><mo>⇝</mo></mover><mo>⟶</mo><mover><mi>Y</mi><mo>⇝</mo></mover></mrow><annotation encoding="application/x-tex">\stackrel{\rightsquigarrow}{f}\colon \stackrel{\rightsquigarrow}{X} \longrightarrow \stackrel{\rightsquigarrow}{Y}</annotation></semantics></math> under the <a class="existingWikiWord" href="/nlab/show/bosonic+modality">bosonic modality</a>.</p> </div> <div class="proof"> <h6 id="proof_12">Proof</h6> <p>By Prop. <a class="maruku-ref" href="#SuperSmoothSetsSystemOfAdjunctions"></a> we have equivalences</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mrow><mi>ℑ</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><mo>⇝</mo></mover><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mi>ℑ</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mi>ℑ</mi><mover><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><mo>⇝</mo></mover><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \overset{\rightsquigarrow}{\Im(-)} \;\simeq\; \Im(-) \;\simeq\; \Im \overset{\rightsquigarrow}{(-)} \,. </annotation></semantics></math></div> <p>The first of these equivalences is the “left <a class="existingWikiWord" href="/nlab/show/Aufhebung">Aufhebung</a>” which is explicit in Prop. <a class="maruku-ref" href="#SuperSmoothSetsSystemOfAdjunctions"></a>. The second equivalence is under the <a class="existingWikiWord" href="/nlab/show/Yoneda+embedding">Yoneda embedding</a>, equivalent to the “<a class="existingWikiWord" href="/nlab/show/adjunct">adjunct</a> equivalence”</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mrow><mi>ℜ</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><mo>⇉</mo></mover><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mi>ℜ</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \overset{\rightrightarrows}{\Re(-)} \;\simeq\; \Re(-) </annotation></semantics></math></div> <p>which follows from the progression in Prop. <a class="maruku-ref" href="#SuperSmoothSetsSystemOfAdjunctions"></a> since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⇉</mo></mrow><annotation encoding="application/x-tex">\rightrightarrows</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⇝</mo></mrow><annotation encoding="application/x-tex">\rightsquigarrow</annotation></semantics></math> have the same <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> of <a class="existingWikiWord" href="/nlab/show/modal+objects">modal objects</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>FormalSmoothset</mi><mover><mo>↪</mo><mrow><msub><mi>ι</mi> <mi>sup</mi></msub></mrow></mover><mi>SuperFormalSmoothSet</mi></mrow><annotation encoding="application/x-tex">FormalSmoothset \overset{\iota_{sup}}{\hookrightarrow} SuperFormalSmoothSet</annotation></semantics></math>, and since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⇝</mo><mo>></mo><mi>ℜ</mi></mrow><annotation encoding="application/x-tex">\rightsquigarrow \gt \Re</annotation></semantics></math>, by <a class="maruku-eqref" href="#eq:ProgressionOfModalitiesOnSuperFormalSmoothSet">(1)</a>.</p> <p>Moreover, since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⇝</mo></mrow><annotation encoding="application/x-tex">\rightsquigarrow</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/preserved+limit">preserves</a> <a class="existingWikiWord" href="/nlab/show/pullbacks">pullbacks</a> (being a <a class="existingWikiWord" href="/nlab/show/right+adjoint">right adjoint</a>, by <a href="geometry+of+physics+--+categories+and+toposes#AdjointsPreserveCoLimits">this Prop.</a>). Hence hitting a pullback diagram which exhibits a <a class="existingWikiWord" href="/nlab/show/formally+%C3%A9tale+morphism">formally étale morphism</a> <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> (Def. <a class="maruku-ref" href="#LocalDiffeomorphisms"></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></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>ℑ</mi><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>f</mi></mpadded></msup></mtd> <mtd><mo stretchy="false">(</mo><mi>pb</mi><mo stretchy="false">)</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>ℑ</mi><mi>f</mi></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>Y</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>ℑ</mi><mi>Y</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ X &\longrightarrow& \Im X \\ \downarrow^{\mathrlap{f}} &(pb)& \downarrow^{\mathrlap{\Im f}} \\ Y &\longrightarrow& \Im Y } </annotation></semantics></math></div> <p>with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⇝</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\rightsquigarrow\;\;</annotation></semantics></math> yields a pullback diagram</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><mover><mi>X</mi><mo>⇝</mo></mover></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>ℑ</mi><mover><mi>X</mi><mo>⇝</mo></mover></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mover><mi>f</mi><mo>⇝</mo></mover></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>ℑ</mi><mover><mi>f</mi><mo>⇝</mo></mover></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mover><mi>Y</mi><mo>⇝</mo></mover></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>ℑ</mi><mover><mi>Y</mi><mo>⇝</mo></mover></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \stackrel{\rightsquigarrow}{X} &\longrightarrow& \Im \stackrel{\rightsquigarrow}{X} \\ \downarrow^{\mathrlap{\stackrel{\rightsquigarrow}{f}}} && \downarrow^{\mathrlap{\Im \stackrel{\rightsquigarrow}{f}}} \\ \stackrel{\rightsquigarrow}{Y} &\longrightarrow& \Im \stackrel{\rightsquigarrow}{Y} } </annotation></semantics></math></div> <p>that witnesses <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mo lspace="0em" rspace="thinmathspace">f</mo><mo>⇝</mo></mover></mrow><annotation encoding="application/x-tex">\overset{\rightsquigarrow}{\f}</annotation></semantics></math> as being <a class="existingWikiWord" href="/nlab/show/formally+%C3%A9tale+morphisms">formally étale morphisms</a> formally étale.</p> </div> <div class="num_cor"> <h6 id="corollary">Corollary</h6> <p>The bosonic space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>X</mi><mo>⇝</mo></mover></mrow><annotation encoding="application/x-tex">\stackrel{\rightsquigarrow}{X}</annotation></semantics></math> underlying a <a class="existingWikiWord" href="/nlab/show/V-manifold">V-manifold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, def. <a class="maruku-ref" href="#VManifold"></a>, is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>V</mi><mo>⇝</mo></mover></mrow><annotation encoding="application/x-tex">\stackrel{\rightsquigarrow}{V}</annotation></semantics></math>-manifold.</p> </div> <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> <h2 id="DeRhamComplexOfSuperDifferentialForms">Super differential forms</h2> <p>We discuss the super-geometric analog of <a class="existingWikiWord" href="/nlab/show/differential+forms">differential forms</a> on <a class="existingWikiWord" href="/nlab/show/supermanifolds">supermanifolds</a>, first with <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a> in <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">\mathbb{R}</annotation></semantics></math>, then with coefficients in a <a class="existingWikiWord" href="/nlab/show/super+Lie+algebra">super Lie algebra</a>.</p> <p>Recall from def. <a class="maruku-ref" href="#SuperCartesianSpace"></a>:</p> <p>A <a class="existingWikiWord" href="/nlab/show/super+Cartesian+space">super Cartesian space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mi>p</mi><mo stretchy="false">|</mo><mi>q</mi></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{p|q}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/formal+dual">formal dual</a> of the <a class="existingWikiWord" href="/nlab/show/commutative+superalgebra">commutative superalgebra</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mi>p</mi><mo stretchy="false">|</mo><mi>q</mi></mrow></msup><mo stretchy="false">)</mo><mo>≔</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>p</mi></msup><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mi>ℝ</mi></msub><msup><mo>∧</mo> <mo>•</mo></msup><msup><mi>ℝ</mi> <mi>q</mi></msup></mrow><annotation encoding="application/x-tex"> C^\infty(\mathbb{R}^{p|q}) \coloneqq C^\infty(\mathbb{R}^p)\otimes_{\mathbb{R}}\wedge^\bullet \mathbb{R}^q </annotation></semantics></math></div> <p>in that a smooth function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><msub><mi>p</mi> <mn>1</mn></msub><mo stretchy="false">|</mo><msub><mi>q</mi> <mn>1</mn></msub></mrow></msup><mo>⟶</mo><msup><mi>ℝ</mi> <mrow><msub><mi>p</mi> <mn>2</mn></msub><mo stretchy="false">|</mo><msub><mi>q</mi> <mn>2</mn></msub></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{p_1|q_1}\longrightarrow \mathbb{R}^{p_2|q_2}</annotation></semantics></math> is equivalently (by definition!) a superalgebra homomorphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><msub><mi>p</mi> <mn>1</mn></msub><mo stretchy="false">|</mo><msub><mi>q</mi> <mn>1</mn></msub></mrow></msup><mo stretchy="false">)</mo><mo>⟵</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><msub><mi>p</mi> <mn>2</mn></msub><mo stretchy="false">|</mo><msub><mi>q</mi> <mn>2</mn></msub></mrow></msup><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> C^\infty(\mathbb{R}^{p_1|q_1}) \longleftarrow C^\infty(\mathbb{R}^{p_2|q_2}) \,. </annotation></semantics></math></div> <p>Notice then that from knowledge of an <a class="existingWikiWord" href="/nlab/show/algebra+of+functions">algebra of functions</a> one obtains the corresponding <a class="existingWikiWord" href="/nlab/show/de+Rham+complex">de Rham complex</a> by the idea of <a class="existingWikiWord" href="/nlab/show/K%C3%A4hler+differentials">Kähler differentials</a>. As discussed there, this statement requires a little care in the smooth context, but the result is still immediate:</p> <p>For <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> a <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a>, then its <a class="existingWikiWord" href="/nlab/show/de+Rham+complex">de Rham complex</a> is the <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 <a class="existingWikiWord" href="/nlab/show/dg-algebra">dg-algebra</a> whose underlying <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+vector+space">graded vector space</a> is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>p</mi></msup><mo stretchy="false">)</mo><mo>=</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>p</mi></msup><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mi>ℝ</mi></msub><msup><mo>∧</mo> <mo>•</mo></msup><mo stretchy="false">⟨</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>x</mi> <mn>1</mn></msup><mo>,</mo><mi>⋯</mi><mo>,</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>x</mi> <mi>p</mi></msup><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex"> \Omega^\bullet(\mathbb{R}^p) = C^\infty(\mathbb{R}^p) \otimes_{\mathbb{R}} \wedge^\bullet \langle \mathbf{d}x^1, \cdots, \mathbf{d}x^p\rangle </annotation></semantics></math></div> <p>and whose <a class="existingWikiWord" href="/nlab/show/differential">differential</a> is defined in degree-0 by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>d</mi></mstyle><mi>f</mi><mo>≔</mo><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow> <mi>p</mi></munderover><mfrac><mrow><mo>∂</mo><mi>f</mi></mrow><mrow><mo>∂</mo><msup><mi>x</mi> <mi>i</mi></msup></mrow></mfrac><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>x</mi> <mi>i</mi></msup></mrow><annotation encoding="application/x-tex"> \mathbf{d} f \coloneqq \sum_{i = 1}^p \frac{\partial f}{\partial x^i} \mathbf{d}x^i </annotation></semantics></math></div> <p>and extended from there to all degrees by the graded Leibniz rule.</p> <p>It is immediate to generalize this to <a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a>, one just needs to be sure to apply the <a class="existingWikiWord" href="/nlab/show/signs+in+supergeometry">sign rule</a> throughout.</p> <div class="num_defn" id="SuperDeRhamComplex"> <h6 id="definition_11">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/de+Rham+complex">de Rham complex</a> of <a class="existingWikiWord" href="/nlab/show/super+differential+forms">super differential forms</a>)</strong></p> <p>The <a class="existingWikiWord" href="/nlab/show/de+Rham+complex">de Rham complex</a> of <a class="existingWikiWord" href="/nlab/show/super+differential+forms">super differential forms</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mi>p</mi><mo stretchy="false">|</mo><mi>q</mi></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Omega^\bullet(\mathbb{R}^{p|q})</annotation></semantics></math> on a <a class="existingWikiWord" href="/nlab/show/super+Cartesian+space">super Cartesian space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mi>p</mi><mo stretchy="false">|</mo><mi>q</mi></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{p|q}</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><mi>ℤ</mi><mo>,</mo><msub><mi>ℤ</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathbb{Z},\mathbb{Z}_2)</annotation></semantics></math>-bigraded commutative algebra</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></mtd> <mtd><msup><mi>Ω</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mi>p</mi><mo stretchy="false">|</mo><mi>q</mi></mrow></msup><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo></mtd> <mtd><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mi>p</mi><mo stretchy="false">|</mo><mi>q</mi></mrow></msup><mo stretchy="false">)</mo></mtd> <mtd><msub><mo>⊗</mo> <mi>ℝ</mi></msub></mtd> <mtd><msup><mo>∧</mo> <mo>•</mo></msup><mo stretchy="false">⟨</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>x</mi> <mn>1</mn></msup><mo>,</mo><mi>⋯</mi><mo>,</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>x</mi> <mi>p</mi></msup><mo>,</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>θ</mi> <mn>1</mn></msup><mo>,</mo><mi>⋯</mi><mo>,</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>θ</mi> <mi>q</mi></msup></mtd> <mtd><mo stretchy="false">⟩</mo></mtd></mtr> <mtr><mtd><mtext>bidegree:</mtext></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mi>even</mi><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mi>even</mi><mo stretchy="false">)</mo></mtd> <mtd><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mi>odd</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ & \Omega^\bullet(\mathbb{R}^{p|q}) &=& C^\infty(\mathbb{R}^{p|q}) & \otimes_{\mathbb{R}} & \wedge^\bullet \langle & \mathbf{d}x^1, \cdots, \mathbf{d}x^p, & \mathbf{d}\theta^1, \cdots, \mathbf{d}\theta^q & \rangle \\ \text{bidegree:} & && (0,even) &&& (1,even) & (1,odd) } </annotation></semantics></math></div> <p>whose <a class="existingWikiWord" href="/nlab/show/differential">differential</a> is defined in degree-0 by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>d</mi></mstyle><mi>f</mi><mo>≔</mo><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow> <mi>p</mi></munderover><mfrac><mrow><mo>∂</mo><mi>f</mi></mrow><mrow><mo>∂</mo><msup><mi>x</mi> <mi>i</mi></msup></mrow></mfrac><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>x</mi> <mi>i</mi></msup></mrow><annotation encoding="application/x-tex"> \mathbf{d} f \coloneqq \sum_{i = 1}^p \frac{\partial f}{\partial x^i} \mathbf{d}x^i </annotation></semantics></math></div> <p>and extended from there to all degree by the graded Leibniz rule.</p> </div> <div class="num_remark"> <h6 id="remark_6">Remark</h6> <p>We may write</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><mi>n</mi><mo>,</mo><mi>σ</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>ℤ</mi><mo>×</mo><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex"> (n,\sigma)\in \mathbb{Z} \times \mathbb{Z}_2 </annotation></semantics></math></div> <p>for elements in this bigrading group.</p> <p>In this notation the grading of the elements in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mi>p</mi><mo stretchy="false">|</mo><mi>q</mi></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Omega^\bullet(\mathbb{R}^{p|q})</annotation></semantics></math> is all induced by the fact that the de Rham differential <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>d</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{d}</annotation></semantics></math> itself is a <a class="existingWikiWord" href="/nlab/show/derivation">derivation</a> of degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mi>even</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(1,even)</annotation></semantics></math>.</p> <table><thead><tr><th>generator</th><th>bi-degree</th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>x</mi> <mi>a</mi></msup></mrow><annotation encoding="application/x-tex">x^a</annotation></semantics></math></td><td style="text-align: left;">(0,even)</td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>θ</mi> <mi>α</mi></msup></mrow><annotation encoding="application/x-tex">\theta^\alpha</annotation></semantics></math></td><td style="text-align: left;">(0,odd)</td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>d</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{d}</annotation></semantics></math></td><td style="text-align: left;">(1,even)</td></tr> </tbody></table> <p>Here the last line means that we have</p> <table><thead><tr><th>generator</th><th>bi-degree</th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>x</mi> <mi>a</mi></msup></mrow><annotation encoding="application/x-tex">x^a</annotation></semantics></math></td><td style="text-align: left;">(0,even)</td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>θ</mi> <mi>α</mi></msup></mrow><annotation encoding="application/x-tex">\theta^\alpha</annotation></semantics></math></td><td style="text-align: left;">(0,odd)</td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>x</mi> <mi>a</mi></msup></mrow><annotation encoding="application/x-tex">\mathbf{d}x^a</annotation></semantics></math></td><td style="text-align: left;">(1,even)</td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>θ</mi> <mi>α</mi></msup></mrow><annotation encoding="application/x-tex">\mathbf{d}\theta^\alpha</annotation></semantics></math></td><td style="text-align: left;">(1,odd)</td></tr> </tbody></table> <p>The formula for the “cohomologically- and super-graded commutativity” in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mi>p</mi><mo stretchy="false">|</mo><mi>q</mi></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Omega^\bullet(\mathbb{R}^{p|q})</annotation></semantics></math> is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>∧</mo><mi>β</mi><mo>=</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><msub><mi>n</mi> <mi>α</mi></msub><msub><mi>n</mi> <mi>β</mi></msub><mo>+</mo><msub><mi>σ</mi> <mi>α</mi></msub><msub><mi>σ</mi> <mi>β</mi></msub></mrow></msup><mspace width="thickmathspace"></mspace><mi>β</mi><mo>∧</mo><mi>α</mi></mrow><annotation encoding="application/x-tex"> \alpha \wedge \beta = \; (- 1)^{n_\alpha n_\beta + \sigma_\alpha \sigma_\beta} \; \beta \wedge \alpha </annotation></semantics></math></div> <p>for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>,</mo><mi>β</mi><mo>∈</mo><msup><mi>Ω</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mi>p</mi><mo stretchy="false">|</mo><mi>q</mi></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\alpha, \beta \in \Omega^\bullet(\mathbb{R}^{p|q})</annotation></semantics></math> of homogeneous <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo>×</mo><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}\times \mathbb{Z}_2</annotation></semantics></math>-degree. Hence there are <em>two</em> contributions to the sign picked up when exchanging two super-differential forms in the wedge product:</p> <ol> <li> <p>there is a “cohomological sign” which for commuting a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">n_1</annotation></semantics></math>-forms past an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">n_2</annotation></semantics></math>-form is <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> </li> <li> <p>in addition there is a “super-grading” which for commuting a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>σ</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\sigma_1</annotation></semantics></math>-graded coordinate function past a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>σ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\sigma_2</annotation></semantics></math>-graded coordinate function (possibly under the de Rham differential) is <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>σ</mi> <mn>1</mn></msub><msub><mi>σ</mi> <mn>2</mn></msub></mrow></msup></mrow><annotation encoding="application/x-tex">(-1)^{\sigma_1 \sigma_2}</annotation></semantics></math>.</p> </li> </ol> </div> <div class="num_example"> <h6 id="example_10">Example</h6> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>x</mi> <mrow><msub><mi>a</mi> <mn>1</mn></msub></mrow></msup><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>x</mi> <mrow><msub><mi>a</mi> <mn>2</mn></msub></mrow></msup><mo stretchy="false">)</mo><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>x</mi> <mrow><msub><mi>a</mi> <mn>2</mn></msub></mrow></msup><mo stretchy="false">)</mo><msup><mi>x</mi> <mrow><msub><mi>a</mi> <mn>1</mn></msub></mrow></msup></mrow><annotation encoding="application/x-tex"> x^{a_1} (\mathbf{d}x^{a_2}) = + (\mathbf{d}x^{a_2}) x^{a_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><msup><mi>θ</mi> <mi>α</mi></msup><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>x</mi> <mi>a</mi></msup><mo stretchy="false">)</mo><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>x</mi> <mi>a</mi></msup><mo stretchy="false">)</mo><msup><mi>θ</mi> <mi>α</mi></msup></mrow><annotation encoding="application/x-tex"> \theta^\alpha (\mathbf{d}x^a) = + (\mathbf{d}x^a) \theta^\alpha </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>θ</mi> <mrow><msub><mi>α</mi> <mn>1</mn></msub></mrow></msup><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>θ</mi> <mrow><msub><mi>α</mi> <mn>2</mn></msub></mrow></msup><mo stretchy="false">)</mo><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>θ</mi> <mrow><msub><mi>α</mi> <mn>2</mn></msub></mrow></msup><mo stretchy="false">)</mo><msup><mi>θ</mi> <mrow><msub><mi>α</mi> <mn>1</mn></msub></mrow></msup></mrow><annotation encoding="application/x-tex"> \theta^{\alpha_1} (\mathbf{d}\theta^{\alpha_2}) = - (\mathbf{d}\theta^{\alpha_2}) \theta^{\alpha_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><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>x</mi> <mrow><msub><mi>a</mi> <mn>1</mn></msub></mrow></msup><mo>∧</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>x</mi> <mrow><msub><mi>a</mi> <mn>2</mn></msub></mrow></msup><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>x</mi> <mrow><msub><mi>a</mi> <mn>2</mn></msub></mrow></msup><mo>∧</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>x</mi> <mrow><msub><mi>a</mi> <mn>1</mn></msub></mrow></msup></mrow><annotation encoding="application/x-tex"> \mathbf{d}x^{a_1} \wedge \mathbf{d} x^{a_2} = - \mathbf{d} x^{a_2} \wedge \mathbf{d} x^{a_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><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>x</mi> <mi>a</mi></msup><mo>∧</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>θ</mi> <mi>α</mi></msup><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>θ</mi> <mi>α</mi></msup><mo>∧</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>x</mi> <mi>a</mi></msup></mrow><annotation encoding="application/x-tex"> \mathbf{d}x^a \wedge \mathbf{d} \theta^{\alpha} = - \mathbf{d}\theta^{\alpha} \wedge \mathbf{d} x^a </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>θ</mi> <mrow><msub><mi>α</mi> <mn>1</mn></msub></mrow></msup><mo>∧</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>θ</mi> <mrow><msub><mi>α</mi> <mn>2</mn></msub></mrow></msup><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>θ</mi> <mrow><msub><mi>α</mi> <mn>2</mn></msub></mrow></msup><mo>∧</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>θ</mi> <mrow><msub><mi>α</mi> <mn>1</mn></msub></mrow></msup></mrow><annotation encoding="application/x-tex"> \mathbf{d}\theta^{\alpha_1} \wedge \mathbf{d} \theta^{\alpha_2} = + \mathbf{d}\theta^{\alpha_2} \wedge \mathbf{d} \theta^{\alpha_1} </annotation></semantics></math></div></div> <p>See at <em><a class="existingWikiWord" href="/nlab/show/signs+in+supergeometry">signs in supergeometry</a></em> for further discussion, for literature, and for mentioning of <em>another</em> popular sign convention, which is different but in the end yields the same cohomology.</p> <p>We want to discuss the generalization of the concept of <a class="existingWikiWord" href="/nlab/show/Lie+algebra+valued+differential+forms">Lie algebra valued differential forms</a> from ordinary <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a> to <a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a>. To that end, we first recall the following neat formulation of ordinary Lie algebra valued differential forms due to Cartan. This will lend itself in fact not only to the generalization to <a class="existingWikiWord" href="/nlab/show/super+Lie+algebras">super Lie algebras</a> but further to <em><a class="existingWikiWord" href="/nlab/show/super+L-%E2%88%9E+algebras">super L-∞ algebras</a></em>, which is what is needed for the description of higher dimensional <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a>.</p> <div class="num_defn" id="CEAlgebra"> <h6 id="definition_12">Definition</h6> <p>The <em><a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebra">Chevalley-Eilenberg algebra</a></em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CE</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">CE(\mathfrak{g})</annotation></semantics></math> of a finite dimensional <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/semifree+dga">semifree</a> graded-commutative <a class="existingWikiWord" href="/nlab/show/dg-algebra">dg-algebra</a> whose underlying graded algebra is the <a class="existingWikiWord" href="/nlab/show/Grassmann+algebra">Grassmann algebra</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mo>∧</mo> <mo>•</mo></msup><msup><mi>𝔤</mi> <mo>*</mo></msup><mo>=</mo><mi>k</mi><mo>⊕</mo><msup><mi>𝔤</mi> <mo>*</mo></msup><mo>⊕</mo><mo stretchy="false">(</mo><msup><mi>𝔤</mi> <mo>*</mo></msup><mo>∧</mo><msup><mi>𝔤</mi> <mo>*</mo></msup><mo stretchy="false">)</mo><mo>⊕</mo><mi>⋯</mi></mrow><annotation encoding="application/x-tex"> \wedge^\bullet \mathfrak{g}^* = k \oplus \mathfrak{g}^* \oplus (\mathfrak{g}^* \wedge \mathfrak{g}^* ) \oplus \cdots </annotation></semantics></math></div> <p>(with 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 skew-symmetrized power in degree <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>)</p> <p>and whose <a class="existingWikiWord" href="/nlab/show/differential">differential</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math> (of degree +1) is on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝔤</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">\mathfrak{g}^*</annotation></semantics></math> the dual of the Lie bracket</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>d</mi><msub><mo stretchy="false">|</mo> <mrow><msup><mi>𝔤</mi> <mo>*</mo></msup></mrow></msub><mo>:</mo><mo>=</mo><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mo stretchy="false">]</mo> <mo>*</mo></msup><mo>:</mo><msup><mi>𝔤</mi> <mo>*</mo></msup><mo>→</mo><msup><mi>𝔤</mi> <mo>*</mo></msup><mo>∧</mo><msup><mi>𝔤</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex"> d|_{\mathfrak{g}^*} := [-,-]^* : \mathfrak{g}^* \to \mathfrak{g}^* \wedge \mathfrak{g}^* </annotation></semantics></math></div> <p>extended uniquely as a graded <a class="existingWikiWord" href="/nlab/show/derivation">derivation</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mo>∧</mo> <mo>•</mo></msup><msup><mi>𝔤</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">\wedge^\bullet \mathfrak{g}^*</annotation></semantics></math>.</p> <p>That this differential indeed squares to 0, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>∘</mo><mi>d</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">d \circ d = 0</annotation></semantics></math>, is precisely the fact that the Lie bracket satisfies the <a class="existingWikiWord" href="/nlab/show/Jacobi+identity">Jacobi identity</a>.</p> </div> <div class="num_remark" id="CEAlgebraInTermsOfBasis"> <h6 id="remark_7">Remark</h6> <p>If in the situation of prop. <a class="maruku-ref" href="#CEAlgebra"></a> we choose a dual basis <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msup><mi>t</mi> <mi>a</mi></msup><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{t^a\}</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝔤</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">\mathfrak{g}^*</annotation></semantics></math> and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msup><mi>C</mi> <mi>a</mi></msup><msub><mrow></mrow> <mrow><mi>b</mi><mi>c</mi></mrow></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{C^a{}_{b c}\}</annotation></semantics></math> be the structure constants of the Lie bracket in that basis, then the action of the differential on the basis generators is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>d</mi><msup><mi>t</mi> <mi>a</mi></msup><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>C</mi> <mi>a</mi></msup><msub><mrow></mrow> <mrow><mi>b</mi><mi>c</mi></mrow></msub><msup><mi>t</mi> <mi>b</mi></msup><mo>∧</mo><msup><mi>t</mi> <mi>c</mi></msup><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> d t^a = - \frac{1}{2} C^a{}_{b c} t^b \wedge t^c \,, </annotation></semantics></math></div> <p>where here and in the following a sum over repeated indices is implicit.</p> </div> <div class="num_prop" id="CEIfFullyFaithful"> <h6 id="proposition_13">Proposition</h6> <p>The construction of Chevalley-Eilenberg algebras in def. <a class="maruku-ref" href="#CEAlgebra"></a> yields a <a class="existingWikiWord" href="/nlab/show/fully+faithful+functor">fully faithful functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>CE</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><mi>LieAlg</mi><mo>⟶</mo><msup><mi>dgAlg</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex"> CE(-) \colon LieAlg \longrightarrow dgAlg^{op} </annotation></semantics></math></div> <p>embedding <a class="existingWikiWord" href="/nlab/show/Lie+algebras">Lie algebras</a> into <a class="existingWikiWord" href="/nlab/show/formal+duals">formal duals</a> of <a class="existingWikiWord" href="/nlab/show/differential+graded+algebras">differential graded algebras</a>. Its image consists of precisely of the <a class="existingWikiWord" href="/nlab/show/semifree+dg-algebras">semifree dg-algebras</a>, those whose underlying <a class="existingWikiWord" href="/nlab/show/graded+algebra">graded algebra</a> (forgetting the <a class="existingWikiWord" href="/nlab/show/differential">differential</a>) is a <a class="existingWikiWord" href="/nlab/show/Grassmann+algebra">Grassmann algebra</a> generated on a <a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a>.</p> </div> <div class="num_defn" id="WeilForLInfinitityAlgebra"> <h6 id="definition_13">Definition</h6> <p>Given a <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math>, its <strong><a class="existingWikiWord" href="/nlab/show/Weil+algebra">Weil algebra</a></strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">W(\mathfrak{g})</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/semi-free+dga">semi-free dga</a> whose underlying graded-commutative algebra is the <a class="existingWikiWord" href="/nlab/show/exterior+algebra">exterior algebra</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mo>∧</mo> <mo>•</mo></msup><mo stretchy="false">(</mo><msup><mi>𝔤</mi> <mo>*</mo></msup><mo>⊕</mo><msup><mi>𝔤</mi> <mo>*</mo></msup><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \wedge^\bullet (\mathfrak{g}^* \oplus \mathfrak{g}^*[1]) </annotation></semantics></math></div> <p>on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝔤</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">\mathfrak{g}^*</annotation></semantics></math> and a shifted copy of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝔤</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">\mathfrak{g}^*</annotation></semantics></math>, and whose <a class="existingWikiWord" href="/nlab/show/differential">differential</a> is the sum</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mrow><mi>W</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo></mrow></msub><mo>=</mo><msub><mi>d</mi> <mrow><mi>CE</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo></mrow></msub><mo>+</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mrow><annotation encoding="application/x-tex"> d_{W(\mathfrak{g})} = d_{CE(\mathfrak{g})} + \mathbf{d} </annotation></semantics></math></div> <p>of two graded <a class="existingWikiWord" href="/nlab/show/derivations">derivations</a> of degree +1 defined by</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>d</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{d}</annotation></semantics></math> acts by degree shift <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝔤</mi> <mo>*</mo></msup><mo>→</mo><msup><mi>𝔤</mi> <mo>*</mo></msup><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathfrak{g}^* \to \mathfrak{g}^*[1]</annotation></semantics></math> on elements in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝔤</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">\mathfrak{g}^*</annotation></semantics></math> and by 0 on elements of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝔤</mi> <mo>*</mo></msup><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathfrak{g}^*[1]</annotation></semantics></math>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mrow><mi>CE</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">d_{CE(\mathfrak{g})}</annotation></semantics></math> acts on unshifted elements in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝔤</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">\mathfrak{g}^*</annotation></semantics></math> as the differential of the <a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebra">Chevalley-Eilenberg algebra</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math> and is extended uniquely to shifted generators by graded-commutattivity</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>d</mi> <mrow><mi>CE</mi><mo stretchy="false">(</mo><mi>𝔤</mi></mrow></msub><mo>,</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><mo stretchy="false">]</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> [d_{CE(\mathfrak{g}}, \mathbf{d}] = 0 </annotation></semantics></math></div> <p>with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>d</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{d}</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mrow><mi>CE</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo></mrow></msub><mstyle mathvariant="bold"><mi>d</mi></mstyle><mi>ω</mi><mo>:</mo><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><msub><mi>d</mi> <mrow><mi>CE</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo></mrow></msub><mi>ω</mi></mrow><annotation encoding="application/x-tex"> d_{CE(\mathfrak{g})} \mathbf{d} \omega := - \mathbf{d} d_{CE(\mathfrak{g})} \omega </annotation></semantics></math></div> <p>for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo>∈</mo><msup><mo>∧</mo> <mn>1</mn></msup><msup><mi>𝔤</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">\omega \in \wedge^1 \mathfrak{g}^*</annotation></semantics></math>.</p> </li> </ul> </div> <div class="num_prop" id="LieAlgValuedFormsViaDgAlgHoms"> <h6 id="proposition_14">Proposition</h6> <p>Given a <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math>, then a <a class="existingWikiWord" href="/nlab/show/Lie+algebra+valued+differential+form">Lie algebra valued differential form</a> on, say, a <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>, is equivalently a dg-algebra homomorphims</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>p</mi></msup><mo stretchy="false">)</mo><mo>⟵</mo><mi>W</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><mi>A</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \Omega^\bullet(\mathbb{R}^p) \longleftarrow W(\mathfrak{g}) \colon A \,, </annotation></semantics></math></div> <p>hence there is a <a class="existingWikiWord" href="/nlab/show/natural+bijection">natural bijection</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>p</mi></msup><mo>,</mo><mi>𝔤</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>Hom</mi> <mi>dgAlg</mi></msub><mo stretchy="false">(</mo><mi>W</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo><mo>,</mo><msup><mi>Ω</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>p</mi></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Omega^1(\mathbb{R}^p, \mathfrak{g}) \simeq Hom_{dgAlg}(W(\mathfrak{g}), \Omega^\bullet(\mathbb{R}^p)) \,. </annotation></semantics></math></div> <p>The form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is <em>flat</em> in that its <a class="existingWikiWord" href="/nlab/show/curvature">curvature</a> <a class="existingWikiWord" href="/nlab/show/differential+2-form">differential 2-form</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">F_A</annotation></semantics></math> vanishes, precisely if this morphism factors through the CE-algebra.</p> </div> <div class="num_remark"> <h6 id="remark_8">Remark</h6> <p>With a choice of basis as in remark <a class="maruku-ref" href="#CEAlgebraInTermsOfBasis"></a>, then the content of prop. <a class="maruku-ref" href="#LieAlgValuedFormsViaDgAlgHoms"></a> is seen in components as follows:</p> <p>a dg-algebra homomorphism is first of all a homomorphism of <a class="existingWikiWord" href="/nlab/show/graded+algebras">graded algebras</a>, and since the domain <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">W(\mathfrak{g})</annotation></semantics></math> is free as a graded algebra, such is entirely determined by what it does to the generators</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>t</mi> <mi>a</mi></msup><mo>,</mo></mtd> <mtd><mo>↦</mo></mtd> <mtd><msup><mi>A</mi> <mi>a</mi></msup></mtd> <mtd><mo>∈</mo><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mi>r</mi> <mi>a</mi></msup></mtd> <mtd><mo>↦</mo></mtd> <mtd><msup><mi>F</mi> <mi>a</mi></msup></mtd> <mtd><mo>∈</mo><msup><mi>Ω</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ t^a, &\mapsto& A^a & \in \Omega^1(\mathbb{R}^n) \\ r^a &\mapsto& F^a & \in \Omega^2(\mathbb{R}^n) } \,. </annotation></semantics></math></div> <p>But being a dg-algebra homomorphism, this assignment needs to respect the differentials on both sides. For the original generators this gives</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>t</mi> <mi>a</mi></msup></mtd> <mtd><mo>↦</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mi>A</mi> <mi>a</mi></msup></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>d</mi> <mrow><mi>W</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo></mrow></msub></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mstyle mathvariant="bold"><mi>d</mi></mstyle> <mi>dR</mi></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mo lspace="verythinmathspace" rspace="0em">−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>C</mi> <mi>a</mi></msup><msub><mrow></mrow> <mrow><mi>b</mi><mi>c</mi></mrow></msub><msup><mi>t</mi> <mi>b</mi></msup><mo>∧</mo><msup><mi>t</mi> <mi>c</mi></msup><mo>+</mo><msup><mi>r</mi> <mi>a</mi></msup></mtd> <mtd><mo>↦</mo></mtd> <mtd><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>C</mi> <mi>a</mi></msup><msub><mrow></mrow> <mrow><mi>b</mi><mi>c</mi></mrow></msub><msup><mi>A</mi> <mi>b</mi></msup><mo>∧</mo><msup><mi>A</mi> <mi>c</mi></msup><mo>+</mo><msup><mi>F</mi> <mi>a</mi></msup><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo></mtd> <mtd><msub><mstyle mathvariant="bold"><mi>d</mi></mstyle> <mi>dR</mi></msub><msup><mi>A</mi> <mi>a</mi></msup></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ t^a &\mapsto&&& A^a \\ \downarrow^{\mathrlap{d_{W(\mathfrak{g})}}} &&&& \downarrow^{\mathrlap{\mathbf{d}_{dR}}} \\ - \frac{1}{2} C^a{}_{b c} t^b \wedge t^c + r^a &\mapsto& (- \frac{1}{2} C^a{}_{b c} A^b \wedge A^c + F^a) &=& \mathbf{d}_{dR} A^a } \,. </annotation></semantics></math></div> <p>With this satisfied, then, by the very nature of the <a class="existingWikiWord" href="/nlab/show/Weil+algebra">Weil algebra</a>, the differential is automatically respected also on the shifted generators. This statement is the <em><a class="existingWikiWord" href="/nlab/show/Bianchi+identity">Bianchi identity</a></em>.</p> </div> <p>Now to pass this to <a class="existingWikiWord" href="/nlab/show/superalgebra">superalgebra</a>.</p> <div class="num_defn" id="SuperGrassmannAlgebra"> <h6 id="definition_14">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>=</mo><msub><mi>V</mi> <mi>even</mi></msub><mo>⊕</mo><msub><mi>V</mi> <mi>odd</mi></msub></mrow><annotation encoding="application/x-tex">V = V_{even} \oplus V_{odd}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/super+vector+space">super vector space</a>, then its <a class="existingWikiWord" href="/nlab/show/Grassmann+algebra">Grassmann algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mo>∧</mo> <mo>•</mo></msup><mi>V</mi></mrow><annotation encoding="application/x-tex">\wedge^\bullet V</annotation></semantics></math> is the free <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><msub><mi>ℤ</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathbb{Z},\mathbb{Z}_2)</annotation></semantics></math>-bigraded commutative algebra subject to</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mn>1</mn></msub><mo>∧</mo><msub><mi>v</mi> <mn>2</mn></msub><mo>=</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><msub><mi>σ</mi> <mn>1</mn></msub><msub><mi>σ</mi> <mn>2</mn></msub></mrow></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> v_1 \wedge v_2 = (-1) (-1)^{\sigma_1 \sigma_2} \,. </annotation></semantics></math></div></div> <p>In the spirit of prop. <a class="maruku-ref" href="#CEIfFullyFaithful"></a> we may then simply say that:</p> <div class="num_defn" id="SuperLieAlgebraViaCE"> <h6 id="definition_15">Definition</h6> <p>A <a class="existingWikiWord" href="/nlab/show/super+Lie+algebra">super Lie algebra</a> structure on a <a class="existingWikiWord" href="/nlab/show/super+vector+space">super vector space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/formal+dual">formal dual</a> of a <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><msub><mi>ℤ</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathbb{Z},\mathbb{Z}_2)</annotation></semantics></math>-bigraded commutative differential algebra</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>CE</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>(</mo><msup><mo>∧</mo> <mo>•</mo></msup><msup><mi>V</mi> <mo>*</mo></msup><mo>,</mo><mspace width="thickmathspace"></mspace><mi>d</mi><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> CE(\mathfrak{g}) = \left( \wedge^\bullet V^\ast, \; d \right) </annotation></semantics></math></div> <p>(with differential <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math> of degree (1,even)) such that the underlying graded algebra is the super Grassmann algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mo>∧</mo> <mo>•</mo></msup><msup><mi>𝔤</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">\wedge^\bullet \mathfrak{g}^\ast</annotation></semantics></math> via def. <a class="maruku-ref" href="#SuperGrassmannAlgebra"></a>.</p> <p>We call this again the <a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebra">Chevalley-Eilenberg algebra</a> of the super Lie algebra dually defined thereby.</p> <p>Similarly, the <a class="existingWikiWord" href="/nlab/show/Weil+algebra">Weil algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">W(\mathfrak{g})</annotation></semantics></math> is obtained from this by adding a generator in degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mi>σ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(2,\sigma)</annotation></semantics></math> for each previous generator in degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mi>σ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(1,\sigma)</annotation></semantics></math> and extending the differential as in def. <a class="maruku-ref" href="#WeilForLInfinitityAlgebra"></a>.</p> </div> <p>Unwinding what this means, one finds that it is equivalent to the following more traditional definition:</p> <div class="num_prop" id="SuperLieAlgebraTraditional"> <h6 id="proposition_15">Proposition</h6> <p>A <a class="existingWikiWord" href="/nlab/show/super+Lie+algebra">super Lie algebra</a> is equivalently</p> <ol> <li> <p>a <a class="existingWikiWord" href="/nlab/show/super+vector+space">super vector space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi><mo>=</mo><msub><mi>𝔤</mi> <mi>even</mi></msub><mo>⊕</mo><msub><mi>𝔤</mi> <mi>odd</mi></msub></mrow><annotation encoding="application/x-tex">\mathfrak{g} = \mathfrak{g}_{even} \oplus \mathfrak{g}_{odd}</annotation></semantics></math>;</p> </li> <li> <p>equipped with a bilinear bracket</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><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo><mo>:</mo><mi>𝔤</mi><mo>⊗</mo><mi>𝔤</mi><mo>→</mo><mi>𝔤</mi></mrow><annotation encoding="application/x-tex"> [-,-] : \mathfrak{g}\otimes \mathfrak{g} \to \mathfrak{g} </annotation></semantics></math></div> <p>which is <em>graded</em> skew-symmetric: for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">x,y \in \mathfrak{g}</annotation></semantics></math> two elements of homogeneous degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>σ</mi> <mi>x</mi></msub></mrow><annotation encoding="application/x-tex">\sigma_x</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>σ</mi> <mi>y</mi></msub></mrow><annotation encoding="application/x-tex">\sigma_y</annotation></semantics></math>, respectively, then</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><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">]</mo><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><msub><mi>σ</mi> <mi>x</mi></msub><msub><mi>σ</mi> <mi>y</mi></msub></mrow></msup><mo stretchy="false">[</mo><mi>y</mi><mo>,</mo><mi>x</mi><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> [x,y] = -(-1)^{\sigma_x \sigma_y} [y,x] \,, </annotation></semantics></math></div></li> <li> <p>that satisfies the <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/Jacobi+identity">Jacobi identity</a> in that for any three elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>∈</mo><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">x,y,z \in \mathfrak{g}</annotation></semantics></math> of homogeneous super-degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>σ</mi> <mi>x</mi></msub><mo>,</mo><msub><mi>σ</mi> <mi>y</mi></msub><mo>,</mo><msub><mi>σ</mi> <mi>z</mi></msub><mo>∈</mo><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\sigma_x,\sigma_y,\sigma_z\in \mathbb{Z}_2</annotation></semantics></math> then</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><mi>x</mi><mo>,</mo><mo stretchy="false">[</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo><mo>=</mo><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">]</mo><mo>,</mo><mi>z</mi><mo stretchy="false">]</mo><mo>+</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><msub><mi>σ</mi> <mi>x</mi></msub><mo>⋅</mo><msub><mi>σ</mi> <mi>y</mi></msub></mrow></msup><mo stretchy="false">[</mo><mi>y</mi><mo>,</mo><mo stretchy="false">[</mo><mi>x</mi><mo>,</mo><mi>z</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [x, [y, z]] = [[x,y],z] + (-1)^{\sigma_x \cdot \sigma_y} [y, [x,z]] \,. </annotation></semantics></math></div></li> </ol> </div> <p>But with def. <a class="maruku-ref" href="#SuperLieAlgebraViaCE"></a> we immediately known, in view of prop. <a class="maruku-ref" href="#LieAlgValuedFormsViaDgAlgHoms"></a>, what <em>super Lie algebra valued super differential forms</em> should be:</p> <div class="num_defn" id="SuperLieAlgValuedDiffForms"> <h6 id="definition_16">Definition</h6> <p>Given a <a class="existingWikiWord" href="/nlab/show/super+Lie+algebra">super Lie algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math>, def. <a class="maruku-ref" href="#SuperLieAlgebraViaCE"></a>, prop. <a class="maruku-ref" href="#SuperLieAlgebraTraditional"></a>, then a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math>-valued super-differential form on the <a class="existingWikiWord" href="/nlab/show/super+Cartesian+space">super Cartesian space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mi>p</mi><mo stretchy="false">|</mo><mi>q</mi></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{p|q}</annotation></semantics></math> is a <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><msub><mi>ℤ</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathbb{Z},\mathbb{Z}_2)</annotation></semantics></math>-graded dg-algebra homomorphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mi>p</mi><mo stretchy="false">|</mo><mi>q</mi></mrow></msup><mo stretchy="false">)</mo><mo>⟵</mo><mi>W</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>A</mi></mrow><annotation encoding="application/x-tex"> \Omega^\bullet(\mathbb{R}^{p|q}) \longleftarrow W(\mathfrak{g}) \;\colon\; A </annotation></semantics></math></div> <p>from the <a class="existingWikiWord" href="/nlab/show/Weil+algebra">Weil algebra</a> according to def. <a class="maruku-ref" href="#SuperLieAlgebraViaCE"></a>, to the super de Rham complex of def. <a class="maruku-ref" href="#SuperDeRhamComplex"></a>.</p> <p>Accordingly we write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mi>p</mi><mo stretchy="false">|</mo><mi>q</mi></mrow></msup><mo>,</mo><mi>𝔤</mi><mo stretchy="false">)</mo><mo>≔</mo><msub><mi>Hom</mi> <mi>dgAlg</mi></msub><mo stretchy="false">(</mo><mi>W</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo><mo>,</mo><msup><mi>Ω</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mi>p</mi><mo stretchy="false">|</mo><mi>q</mi></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Omega^1(\mathbb{R}^{p|q}, \mathfrak{g}) \coloneqq Hom_{dgAlg}(W(\mathfrak{g}), \Omega^\bullet(\mathbb{R}^{p|q})) \,. </annotation></semantics></math></div></div> <div class="num_example"> <h6 id="example_11">Example</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi><mo>≔</mo><msup><mi>ℝ</mi> <mrow><mn>1</mn><mo stretchy="false">|</mo><mn>0</mn></mrow></msup><mo>=</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g} \coloneqq \mathbb{R}^{1|0} = \mathbb{R}</annotation></semantics></math> be the ordinary abelian <a class="existingWikiWord" href="/nlab/show/line+Lie+algebra">line Lie algebra</a>. Then</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mi>p</mi><mo stretchy="false">|</mo><mi>q</mi></mrow></msup><mo>,</mo><msup><mi>ℝ</mi> <mrow><mn>1</mn><mo stretchy="false">|</mo><mn>0</mn></mrow></msup><mo stretchy="false">)</mo><mo>≃</mo><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mi>p</mi><mo stretchy="false">|</mo><mi>q</mi></mrow></msup><msub><mo stretchy="false">)</mo> <mi>even</mi></msub></mrow><annotation encoding="application/x-tex"> \Omega^1(\mathbb{R}^{p|q}, \mathbb{R}^{1|0}) \simeq \Omega^1(\mathbb{R}^{p|q})_{even} </annotation></semantics></math></div> <p>is the set of super-differential forms in degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mi>even</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(1,even)</annotation></semantics></math>.</p> <p>Similarly with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi><mo>=</mo><msup><mi>ℝ</mi> <mrow><mn>0</mn><mo stretchy="false">|</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\mathfrak{g} = \mathbb{R}^{0|1}</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/odd+line">odd line</a> regarded as an abelian super Lie algebra, then</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mi>p</mi><mo stretchy="false">|</mo><mi>q</mi></mrow></msup><mo>,</mo><msup><mi>ℝ</mi> <mrow><mn>0</mn><mo stretchy="false">|</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mo>≃</mo><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mi>p</mi><mo stretchy="false">|</mo><mi>q</mi></mrow></msup><msub><mo stretchy="false">)</mo> <mi>odd</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Omega^1(\mathbb{R}^{p|q}, \mathbb{R}^{0|1}) \simeq \Omega^1(\mathbb{R}^{p|q})_{odd} \,. </annotation></semantics></math></div> <p>So generally for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math> an ordinary Lie algebra regarded as a super Lie algebra, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mi>p</mi><mo stretchy="false">|</mo><mi>q</mi></mrow></msup><mo>,</mo><mi>𝔤</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Omega^1(\mathbb{R}^{p|q}, \mathfrak{g})</annotation></semantics></math> is bigger than <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>p</mi></msup><mo>,</mo><mi>𝔤</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Omega^1(\mathbb{R}^p,\mathfrak{g})</annotation></semantics></math>.</p> <p>This is an issue to be dealt with when describing <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a> in terms of Cartan fields on <a class="existingWikiWord" href="/nlab/show/supermanifolds">supermanifolds</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, because the actual <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> manifold one cares about is just the <a class="existingWikiWord" href="/nlab/show/bosonic+modality">bosonic part</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>X</mi><mo>⇝</mo></mover></mrow><annotation encoding="application/x-tex">\stackrel{\rightsquigarrow}{X}</annotation></semantics></math>. This issue is dealt with by the concept of <a href="D'Auria-Fre+formulation+of+supergravity#Rheonomy">rheonomy</a>.</p> </div> <h2 id="references">References</h2> <h3 id="general_theory">General theory</h3> <p>Some historically influential remarks on <a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a> are due to</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Yuri+Manin">Yuri Manin</a>, <em><a class="existingWikiWord" href="/nlab/show/New+Dimensions+in+Geometry">New Dimensions in Geometry</a></em>, talk at Arbeitstagung, Bonn 1984</li> </ul> <p>Introductory lecture notes include</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Yuri+Manin">Yuri Manin</a>, chapter 4 of <em><a class="existingWikiWord" href="/nlab/show/Gauge+Field+Theory+and+Complex+Geometry">Gauge Field Theory and Complex Geometry</a></em>, Grundlehren der Mathematischen Wissenschaften 289, Springer 1988</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gennadi+Sardanashvily">Gennadi Sardanashvily</a>, <em>Lectures on supergeometry</em> (<a href="http://arxiv.org/abs/0910.0092">arXiv:0910.0092</a>)</p> </li> </ul> <p>Many texts discuss supergeometry only in the context of <a class="existingWikiWord" href="/nlab/show/supersymmetry">supersymmetry</a>, but notice that the former exists and is relevant even if the latter does not or is not.</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Daniel+Freed">Daniel Freed</a>, <em><a class="existingWikiWord" href="/nlab/show/Five+lectures+on+supersymmetry">Five lectures on supersymmetry</a></em></p> </li> <li> <p>L. Caston, R. Fioresi, <em>Mathematical Foundations of Supersymmetry</em> (<a href="http://arxiv.org/abs/0710.5742">arXiv:0710.5742</a>)</p> </li> </ul> <h3 id="for_classical_field_theories_with_fermions">For classical field theories with fermions</h3> <p>The <a class="existingWikiWord" href="/nlab/show/experiment">experimental</a> observation that <a class="existingWikiWord" href="/nlab/show/phase+spaces">phase spaces</a> of <a class="existingWikiWord" href="/nlab/show/classical+field+theories">classical field theories</a> with <a class="existingWikiWord" href="/nlab/show/fermions">fermions</a> (such as <a class="existingWikiWord" href="/nlab/show/electrons">electrons</a> or <a class="existingWikiWord" href="/nlab/show/quarks">quarks</a>) are <a class="existingWikiWord" href="/nlab/show/superspaces">superspaces</a> goes back to</p> <ul> <li id="Pauli25"> <p><a class="existingWikiWord" href="/nlab/show/Wolfgang+Pauli">Wolfgang Pauli</a>, <em>Über den Zusammenhang des Abschlusses der Elektronengruppen im Atom mit der Komplexstruktur der Spektren</em>, Zeitschrift für Physik, February 1925, Volume 31, Issue 1, pp 765-783</p> </li> <li id="Fierz39"> <p><a class="existingWikiWord" href="/nlab/show/Markus+Fierz">Markus Fierz</a>, <em>Über die relativistische Theorie kräftefreier Teilchen mit beliebigem Spin</em>. Helvetica Physica Acta. 12 (1): 3–37. (1939) <a href="https://dx.doi.org/10.5169%2Fseals-110930">doi:10.5169/seals-110930</a></p> </li> <li id="Pauli40"> <p><a class="existingWikiWord" href="/nlab/show/Wolfgang+Pauli">Wolfgang Pauli</a>, <em>The connection between spin and statistics</em>, Phys. Rev. 58, 716–722 (1940)</p> </li> </ul> <p>Discussion of <a class="existingWikiWord" href="/nlab/show/classical+field+theory">classical field theory</a> with <a class="existingWikiWord" href="/nlab/show/fermions">fermions</a> as taking place on <a class="existingWikiWord" href="/nlab/show/supermanifolds">supermanifolds</a> (supergometric <a class="existingWikiWord" href="/nlab/show/field+bundles">field bundles</a> and <a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a>) includes the following references:</p> <ul> <li id="DeligneFreed99"> <p><a class="existingWikiWord" href="/nlab/show/Pierre+Deligne">Pierre Deligne</a>, <a class="existingWikiWord" href="/nlab/show/Daniel+Freed">Daniel Freed</a>, <em>Classical field theory</em> (1999) (<a href="https://publications.ias.edu/sites/default/files/79_ClassicalFieldTheory.pdf">pdf</a>)</p> <p>this is a chapter in</p> <p><a class="existingWikiWord" href="/nlab/show/Pierre+Deligne">P. Deligne</a>, <a class="existingWikiWord" href="/nlab/show/Pavel+Etingof">P. Etingof</a>, <a class="existingWikiWord" href="/nlab/show/Dan+Freed">D.S. Freed</a>, L. Jeffrey, <a class="existingWikiWord" href="/nlab/show/David+Kazhdan">D. Kazhdan</a>, J. Morgan, D.R. Morrison, <a class="existingWikiWord" href="/nlab/show/Edward+Witten">E. Witten</a> (eds.) <em><a class="existingWikiWord" href="/nlab/show/Quantum+Fields+and+Strings">Quantum Fields and Strings</a>, A course for mathematicians</em>, 2 vols. Amer. Math. Soc. Providence 1999. (<a href="http://www.math.ias.edu/qft">web version</a>)</p> </li> <li id="Freed01"> <p><a class="existingWikiWord" href="/nlab/show/Daniel+Freed">Daniel Freed</a>, <em>Classical field theory and Supersymmetry</em>, IAS/Park City Mathematics Series Volume 11 (2001) (<a href="https://www.ma.utexas.edu/users/dafr/pcmi.pdf">pdf</a>)</p> </li> <li id="GiachettaMangiarottiSardanashvily09"> <p>Giovanni Giachetta, Luigi Mangiarotti, <a class="existingWikiWord" href="/nlab/show/Gennadi+Sardanashvily">Gennadi Sardanashvily</a>, chapter 3 of <em>Advanced classical field theory</em>, World Scientific (2009)</p> </li> <li id="Sardanashvily12"> <p><a class="existingWikiWord" href="/nlab/show/Gennadi+Sardanashvily">Gennadi Sardanashvily</a>, <em>Grassmann-graded Lagrangian theory of even and odd variables</em> (<a href="https://arxiv.org/abs/1206.2508">arXiv:1206.2508</a>)</p> </li> <li id="Sardanashvily16"> <p><a class="existingWikiWord" href="/nlab/show/Gennadi+Sardanashvily">Gennadi Sardanashvily</a>, <em>Noether’s Theorems: Applications in Mechanics and Field Theory</em>, Studies in Variational Geometry, 2016</p> </li> </ul> <h3 id="ReferencesOverSuperpoints">In the topos over superpoints</h3> <p>The observation that the study of super-structures in mathematics is usefully regarded as taking place over the <a class="existingWikiWord" href="/nlab/show/base+topos">base topos</a> on the <a class="existingWikiWord" href="/nlab/show/site">site</a> of <a class="existingWikiWord" href="/nlab/show/super+points">super points</a> has been made around 1984 in</p> <ul> <li id="Schwarz84"> <p><a class="existingWikiWord" href="/nlab/show/Albert+Schwarz">Albert Schwarz</a>, <em>On the definition of superspace</em>, Teoret. Mat. Fiz. (1984) Volume 60, Number 1, Pages 37–42, (<a href="http://www.mathnet.ru/links/b12306f831b8c37d32d5ba8511d60c93/tmf5111.pdf">russian original pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Alexander+Voronov">Alexander Voronov</a>, <em>Maps of supermanifolds</em> , Teoret. Mat. Fiz. (1984) Volume 60, Number 1, Pages 43–48</p> </li> </ul> <p>and in</p> <ul> <li id="Molotkov84">V. Molotkov., <em>Infinite-dimensional <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>ℤ</mi> <mn>2</mn> <mi>k</mi></msubsup></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2^k</annotation></semantics></math>-supermanifolds</em> , ICTP preprints, IC/84/183, 1984.</li> </ul> <p>A summary/review is in the appendix of</p> <ul> <li id="KonechnySchwarz97"> <p>Anatoly Konechny and <a class="existingWikiWord" href="/nlab/show/Albert+Schwarz">Albert Schwarz</a>,</p> <p><em>On <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>k</mi><mo>⊕</mo><mi>l</mi><mo stretchy="false">|</mo><mi>q</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(k \oplus l|q)</annotation></semantics></math>-dimensional supermanifolds</em> in <a class="existingWikiWord" href="/nlab/show/Julius+Wess">Julius Wess</a>, V. Akulov (eds.) <em>Supersymmetry and Quantum Field Theory</em> (D. Volkov memorial volume) Springer-Verlag, 1998, Lecture Notes in Physics, 509 , (<a href="http://arxiv.org/abs/hep-th/9706003">arXiv:hep-th/9706003</a>)</p> <p><em>Theory of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>k</mi><mo>⊕</mo><mi>l</mi><mo stretchy="false">|</mo><mi>q</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(k \oplus l|q)</annotation></semantics></math>-dimensional supermanifolds</em> Sel. math., New ser. 6 (2000) 471 - 486</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Albert+Schwarz">Albert Schwarz</a>, I- Shapiro, <em>Supergeometry and Arithmetic Geometry</em> (<a href="http://arxiv.org/abs/hep-th/0605119">arXiv:hep-th/0605119</a>)</p> </li> </ul> <p>A review of all this as geometry in the <a class="existingWikiWord" href="/nlab/show/topos">topos</a> over the category of <a class="existingWikiWord" href="/nlab/show/superpoints">superpoints</a> is in</p> <ul> <li id="Sachse08"><a class="existingWikiWord" href="/nlab/show/Christoph+Sachse">Christoph Sachse</a>, <em>A Categorical Formulation of Superalgebra and Supergeometry</em> (<a href="http://arxiv.org/abs/0802.4067">arXiv:0802.4067</a>)</li> </ul> <h3 id="in_the_topos_over_super_cartesian_spaces">In the topos over super Cartesian spaces</h3> <p>The <a href="#ReferencesOverSuperpoints">above</a> perspective of supergeometry in the topos over superpoints is a restriction of the perspective in the topos over <a class="existingWikiWord" href="/nlab/show/super+Cartesian+spaces">super Cartesian spaces</a> which we use here. This in turn is essentially just the specification to <a class="existingWikiWord" href="/nlab/show/supercommutative+superalgebra">supercommutative superalgebra</a> of <a class="existingWikiWord" href="/nlab/show/Alexander+Grothendieck">Alexander Grothendieck</a>‘s concept of “<a class="existingWikiWord" href="/nlab/show/functorial+geometry">functorial geometry</a>” as laid out in</p> <ul> <li id="Grothendieck65"><a class="existingWikiWord" href="/nlab/show/Alexander+Grothendieck">Alexander Grothendieck</a>, <em>Introduction au langage fonctoriel</em>, course in Algiers in November 1965, lecture notes by <a class="existingWikiWord" href="/nlab/show/Max+Karoubi">Max Karoubi</a>, <a href="http://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/GrothAlgiers.pdf">pdf scan</a>.</li> </ul> <p>Grothendieck amplified that this functorial perspective is superior to the perspective on schemes as <a class="existingWikiWord" href="/nlab/show/locally+ringed+spaces">locally ringed spaces</a> in the lecture</p> <ul> <li id="Grothendieck73"><a class="existingWikiWord" href="/nlab/show/Alexander+Grothendieck">Alexander Grothendieck</a>, <em>Introduction to functorial algebraic geometry, part 1: affine algebraic geometry</em>, summer school in Buffalo, 1973, lecture notes by Federico Gaeta (<a href="http://matematicas.unex.es/~navarro/res/ifag.pdf">pdf scan</a>)</li> </ul> <p>Further amplification of Grothendieck’s amplification may be found in the short text</p> <ul> <li id="Lawvere03"><a class="existingWikiWord" href="/nlab/show/William+Lawvere">William Lawvere</a>, <em>Grothendieck’s 1973 Buffalo Colloquium</em>, posting to the mailing list <em>categories@mta.ca</em>, March 2003 (<a href="http://permalink.gmane.org/gmane.science.mathematics.categories/2228">gmane archive</a>)</li> </ul> <p>The application of this perspective to supergeometry is sometimes known as <a class="existingWikiWord" href="/nlab/show/synthetic+differential+supergeometry">synthetic differential supergeometry</a>:</p> <ul> <li id="Yetter88"><a class="existingWikiWord" href="/nlab/show/David+Yetter">David Yetter</a>, <em>Models for synthetic supergeometry</em>, Cahiers, 29, 2 (1988) (<a href="http://www.numdam.org/item?id=CTGDC_1988__29_2_87_0">NUMDAM</a>)</li> </ul> <p>The formulation via the axioms of <a class="existingWikiWord" href="/nlab/show/differential+cohesion">differential cohesion</a> that we use here follows</p> <ul> <li id="Schreiber"> <p><a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/schreiber/show/differential+cohomology+in+a+cohesive+topos">differential cohomology in a cohesive topos</a></em></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, Sec. 3.1.3 of: <em><a class="existingWikiWord" href="/schreiber/show/Proper+Orbifold+Cohomology">Proper Orbifold Cohomology</a></em> (<a href="https://arxiv.org/abs/2008.01101">arXiv:2008.01101</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Igor+Khavkine">Igor Khavkine</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/schreiber/show/Synthetic+variational+calculus">Synthetic geometry of differential equations Part I – Jets and comonad structure</a></em> (<a href="https://arxiv.org/abs/1701.06238">arXiv:1701.06238</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Felix+Wellen">Felix Wellen</a>, <em><a class="existingWikiWord" href="/schreiber/show/thesis+Wellen">Formalizing Cartan Geometry in Modal Homotopy Type Theory</a></em>, 2017 (<a href="https://arxiv.org/abs/1806.05966">arXiv:1806.05966</a>, <a href="http://www.math.kit.edu/iag3/~wellen/media/diss.pdf">pdf</a>)</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on October 12, 2023 at 14:10:45. 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