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class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Lie theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+theory">∞-Lie theory</a></strong> (<a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a>)</p> <p><strong>Background</strong></p> <p><em>Smooth structure</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+smooth+space">generalized smooth space</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/diffeological+space">diffeological space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fr%C3%B6licher+space">Frölicher space</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+topos">smooth topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Cahiers+topos">Cahiers topos</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth ∞-groupoid</a>, <a class="existingWikiWord" href="/nlab/show/concrete+smooth+%E2%88%9E-groupoid">concrete smooth ∞-groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/synthetic+differential+%E2%88%9E-groupoid">synthetic differential ∞-groupoid</a></p> </li> </ul> <p><em>Higher groupoids</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-groupoid">2-groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/strict+%E2%88%9E-groupoid">strict ∞-groupoid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/crossed+complex">crossed complex</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/simplicial+group">simplicial group</a></li> </ul> </li> </ul> <p><em>Lie theory</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+theory">Lie theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integration</a>, <a class="existingWikiWord" href="/nlab/show/Lie+differentiation">Lie differentiation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie%27s+three+theorems">Lie's three theorems</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+theory+for+stacky+Lie+groupoids">Lie theory for stacky Lie groupoids</a></p> </li> </ul> </li> </ul> <p><strong>∞-Lie groupoids</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+groupoid">∞-Lie groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+groupoid">strict ∞-Lie groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+groupoid">Lie groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differentiable+stack">differentiable stack</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orbifold">orbifold</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+group">∞-Lie group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/simple+Lie+group">simple Lie group</a>, <a class="existingWikiWord" href="/nlab/show/semisimple+Lie+group">semisimple Lie group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+2-group">Lie 2-group</a></p> </li> </ul> </li> </ul> <p><strong>∞-Lie algebroids</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebroid">∞-Lie algebroid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Lie+algebroid">Lie algebroid</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+%E2%88%9E-algebroid+representation">Lie ∞-algebroid representation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E-algebra">L-∞-algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+L-%E2%88%9E+algebras">model structure for L-∞ algebras</a>: <a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-Lie+algebras">on dg-Lie algebras</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-coalgebras">on dg-coalgebras</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+Lie+algebras">on simplicial Lie algebras</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/semisimple+Lie+algebra">semisimple Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/compact+Lie+algebra">compact Lie algebra</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+2-algebra">Lie 2-algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/strict+Lie+2-algebra">strict Lie 2-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+crossed+module">differential crossed module</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+3-algebra">Lie 3-algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/differential+2-crossed+module">differential 2-crossed module</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dg-Lie+algebra">dg-Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/simplicial+Lie+algebra">simplicial Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+L-%E2%88%9E+algebra">super L-∞ algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/super+Lie+algebra">super Lie algebra</a></li> </ul> </li> </ul> <p><strong>Formal Lie groupoids</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/formal+group">formal group</a>, <a class="existingWikiWord" href="/nlab/show/formal+groupoid">formal groupoid</a></li> </ul> <p><strong>Cohomology</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+algebra+cohomology">Lie algebra cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebra">Chevalley-Eilenberg algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Weil+algebra">Weil algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/invariant+polynomial">invariant polynomial</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Killing+form">Killing form</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nonabelian+Lie+algebra+cohomology">nonabelian Lie algebra cohomology</a></p> </li> </ul> <p><strong>Homotopy</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/homotopy+groups+of+a+Lie+groupoid">homotopy groups of a Lie groupoid</a></li> </ul> <p><strong>Related topics</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Weil+theory">∞-Chern-Weil theory</a></li> </ul> <p><strong>Examples</strong></p> <p><em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Lie groupoids</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+groupoid">Atiyah Lie groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid">fundamental ∞-groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/path+groupoid">path groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/path+n-groupoid">path n-groupoid</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">smooth principal ∞-bundle</a></p> </li> </ul> <p><em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Lie groups</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/orthogonal+group">orthogonal group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/special+orthogonal+group">special orthogonal group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spin+group">spin group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+2-group">string 2-group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fivebrane+6-group">fivebrane 6-group</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/unitary+group">unitary group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/special+unitary+group">special unitary group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+groupoid">circle Lie n-group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/circle+group">circle group</a></li> </ul> </li> </ul> <p><em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Lie algebroids</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/tangent+Lie+algebroid">tangent Lie algebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/action+Lie+algebroid">action Lie algebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+algebroid">Atiyah Lie algebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+Lie+n-algebroid">symplectic Lie n-algebroid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Poisson+Lie+algebroid">Poisson Lie algebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Courant+Lie+algebroid">Courant Lie algebroid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/generalized+complex+geometry">generalized complex geometry</a></li> </ul> </li> </ul> </li> </ul> <p><em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Lie algebras</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/general+linear+Lie+algebra">general linear Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orthogonal+Lie+algebra">orthogonal Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/special+orthogonal+Lie+algebra">special orthogonal Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/endomorphism+L-%E2%88%9E+algebra">endomorphism L-∞ algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/automorphism+%E2%88%9E-Lie+algebra">automorphism ∞-Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+Lie+2-algebra">string Lie 2-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fivebrane+Lie+6-algebra">fivebrane Lie 6-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity+Lie+3-algebra">supergravity Lie 3-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity+Lie+6-algebra">supergravity Lie 6-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/line+Lie+n-algebra">line Lie n-algebra</a></p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <ul> <li><a href='#PresentationByCE'>Presentation by dg-algebras and simplicial presheaves</a></li> <li><a href='#GeneralAbstractDefinition'>General abstract definition</a></li> </ul> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#ModelsForTheAbstractAxioms'>Models for the abstract axioms</a></li> <li><a href='#Cohomology'>Cohomology of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Lie algebroids</a></li> </ul> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#classes_of_examples'>Classes of examples</a></li> <li><a href='#LieAlgebroidsAsInfinLieAlgebroids'>Lie algebroids regarded as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Lie algebroids</a></li> <ul> <li><a href='#SmoothLociOfInfinitesimalSimplices'>Smooth loci of infinitesimal simplices</a></li> <li><a href='#TangentLieAlgebroid'>Tangent Lie algebroid</a></li> <li><a href='#LieAlgebra'>Lie algebra</a></li> </ul> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>An <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Lie algebroid</em> is a <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth ∞-groupoid</a> (or rather a <a class="existingWikiWord" href="/nlab/show/synthetic-differential+%E2%88%9E-groupoid">synthetic-differential ∞-groupoid</a>) all whose <a class="existingWikiWord" href="/nlab/show/k-morphism">k-morphism</a>s for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> have <em><a class="existingWikiWord" href="/nlab/show/infinitesimal+space">infinitesimal</a> extension</em> (are infinitesimal neighbours of an identity <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-morphism).</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Lie algebroids are to <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+groupoids">∞-Lie groupoids</a> as <a class="existingWikiWord" href="/nlab/show/Lie+algebras">Lie algebras</a> are to <a class="existingWikiWord" href="/nlab/show/Lie+groups">Lie groups</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a> - <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+groupoid">Lie groupoid</a> - <a class="existingWikiWord" href="/nlab/show/Lie+algebroid">Lie algebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-Lie group</a> - <a class="existingWikiWord" href="/nlab/show/L-infinity-algebra">∞-Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+groupoid">∞-Lie groupoid</a> - <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Lie algebroid</strong> .</p> </li> </ul> <h2 id="definition">Definition</h2> <p>We discuss <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Lie algebroids in the <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>th</mi></msub><mo>:</mo><mo>=</mo></mrow><annotation encoding="application/x-tex">\mathbf{H}_{th} := </annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/SynthDiff%E2%88%9EGrpd">SynthDiff∞Grpd</a> of <a class="existingWikiWord" href="/nlab/show/synthetic+differential+%E2%88%9E-groupoid">synthetic differential ∞-groupoid</a>s. This is an <a href="http://ncatlab.org/nlab/show/cohesive+(infinity%2C1)-topos#InfinitesimalCohesion">infinitesimal cohesive neigbourhood</a> of the cohesive <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-topos <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>:</mo><mo>=</mo></mrow><annotation encoding="application/x-tex">\mathbf{H} :=</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Smooth%E2%88%9EGrpd">Smooth∞Grpd</a> of <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth ∞-groupoid</a>s, which is exhibited by the <a href="http://ncatlab.org/nlab/show/cohesive+(infinity%2C1)-topos#InfinitesimalPaths">infinitesimal path</a> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-geometric+morphism">(∞,1)-geometric morphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>Π</mi> <mi>inf</mi></msub><mo>⊣</mo><msub><mi>Disc</mi> <mi>inf</mi></msub><mo>⊣</mo><msub><mi>Γ</mi> <mi>inf</mi></msub><mo stretchy="false">)</mo><mo>:</mo><mi>SynthDiff</mi><mn>∞</mn><mi>Grpd</mi><mo>→</mo><mi>Smooth</mi><mn>∞</mn><mi>Grpd</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (\Pi_{inf} \dashv Disc_{inf} \dashv \Gamma_{inf}) : SynthDiff\infty Grpd \to Smooth\infty Grpd \,. </annotation></semantics></math></div> <h3 id="PresentationByCE">Presentation by dg-algebras and simplicial presheaves</h3> <p>We consider <em>presentations</em> of the general abstract definition <a class="maruku-ref" href="#TheGeneralAbstractDefinition"></a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Lie algebroids by constructing in the standard <a class="existingWikiWord" href="/nlab/show/model+structure">model structure</a>-<a class="existingWikiWord" href="/nlab/show/presentable+%28%E2%88%9E%2C1%29-category">presentation</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SynthDiff</mi><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">SynthDiff\infty Grpd</annotation></semantics></math> by <a class="existingWikiWord" href="/nlab/show/simplicial+presheaves">simplicial presheaves</a> on <a class="existingWikiWord" href="/nlab/show/CartSp">CartSp</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mrow></mrow> <mi>synthdiff</mi></msub></mrow><annotation encoding="application/x-tex">{}_{synthdiff}</annotation></semantics></math> certain classes of simplicial presheaves in the image of <a class="existingWikiWord" href="/nlab/show/semifree+dga">semi-free</a> <a class="existingWikiWord" href="/nlab/show/differential+graded+algebra">differential graded algebra</a>s under the <a class="existingWikiWord" href="/nlab/show/monoidal+Dold-Kan+correspondence">monoidal Dold-Kan correspondence</a>. This amounts to identifying the traditional description of of <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a>s, <a class="existingWikiWord" href="/nlab/show/Lie+algebroid">Lie algebroid</a>s and <a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebra">L-∞ algebra</a>s by their <a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebra">Chevalley-Eilenberg algebra</a>s as a convenient characterization of the corresponding <a class="existingWikiWord" href="/nlab/show/cosimplicial+algebra">cosimplicial algebra</a>s whose formal dual simplicial presheaves are manifest presentations of infinitesimal <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth ∞-groupoid</a>s.</p> <div class="num_defn" id="LInftyGlgebroid"> <h6 id="definition_2">Definition</h6> <p>Let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub><mi>Algd</mi><mo>↪</mo><msubsup><mi>dgCAlg</mi> <mi>ℝ</mi> <mi>op</mi></msubsup></mrow><annotation encoding="application/x-tex"> L_\infty Algd \hookrightarrow dgCAlg_{\mathbb{R}}^{op} </annotation></semantics></math></div> <p>be the <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> on the <a class="existingWikiWord" href="/nlab/show/opposite+category">opposite category</a> of cochain <a class="existingWikiWord" href="/nlab/show/dg-algebra">dg-algebra</a>s 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 those dg-algebras that are</p> <ul> <li> <p>graded-commutative;</p> </li> <li> <p>concentrated in non-negative degree (the <a class="existingWikiWord" href="/nlab/show/differential">differential</a> being of degree +1 );</p> </li> <li> <p>in degree 0 of the form <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> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo></mrow><annotation encoding="application/x-tex">X \in</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/SmoothMfd">SmoothMfd</a>;</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/semifree+dga">semifree</a>: their underlying <a class="existingWikiWord" href="/nlab/show/graded+algebra">graded algebra</a> is <a class="existingWikiWord" href="/nlab/show/isomorphic">isomorphic</a> to an <a class="existingWikiWord" href="/nlab/show/exterior+algebra">exterior algebra</a> on 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">\mathbb{N}</annotation></semantics></math>-graded locally free <a class="existingWikiWord" href="/nlab/show/projective+object">projective</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>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></p> </li> <li> <p>of finite <a class="existingWikiWord" href="/nlab/show/rank">rank</a>;</p> </li> </ul> </div> <p>We call this the category of <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebroids</strong>.</p> <p>More in detail, an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔞</mi><mo>∈</mo><msub><mi>L</mi> <mn>∞</mn></msub><mi>Algd</mi></mrow><annotation encoding="application/x-tex">\mathfrak{a} \in L_\infty Algd</annotation></semantics></math> may be identified (non-canonically) with a pair <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>CE</mi><mo stretchy="false">(</mo><mi>𝔞</mi><mo stretchy="false">)</mo><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(CE(\mathfrak{a}), X)</annotation></semantics></math>, where</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>SmoothMfd</mi></mrow><annotation encoding="application/x-tex">X \in SmoothMfd</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> – called the <strong>base space</strong> of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebroid ;</p> </li> <li> <p><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">\mathfrak{a}</annotation></semantics></math> is the module of smooth <a class="existingWikiWord" href="/nlab/show/section">section</a>s of an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">\mathbb{N}</annotation></semantics></math>-graded <a class="existingWikiWord" href="/nlab/show/vector+bundle">vector bundle</a> of degreewise finite <a class="existingWikiWord" href="/nlab/show/rank">rank</a>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CE</mi><mo stretchy="false">(</mo><mi>𝔞</mi><mo stretchy="false">)</mo><mo>=</mo><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><msup><mi>𝔞</mi> <mo>*</mo></msup><mo>,</mo><msub><mi>d</mi> <mi>𝔞</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">CE(\mathfrak{a}) = (\wedge^\bullet_{C^\infty(X)} \mathfrak{a}^*, d_{\mathfrak{a}})</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/semifree+dga">semifree dga</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝔞</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">\mathfrak{a}^*</annotation></semantics></math> – a <a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebra">Chevalley-Eilenberg algebra</a> – where</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><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><msup><mi>𝔞</mi> <mo>*</mo></msup><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><msubsup><mi>𝔞</mi> <mn>0</mn> <mo>*</mo></msubsup><mspace width="thickmathspace"></mspace><mo>⊕</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><msubsup><mi>𝔞</mi> <mn>0</mn> <mo>*</mo></msubsup><msub><mo>∧</mo> <mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></msub><msubsup><mi>𝔞</mi> <mn>0</mn> <mo>*</mo></msubsup><mo>⊕</mo><msubsup><mi>𝔞</mi> <mn>1</mn> <mo>*</mo></msubsup><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>⊕</mo><mspace width="thickmathspace"></mspace><mi>⋯</mi></mrow><annotation encoding="application/x-tex"> \wedge^\bullet_{C^\infty(X)}\mathfrak{a}^* = C^\infty(X) \; \oplus \; \mathfrak{a}^*_0 \; \oplus \; ( \mathfrak{a}^*_0 \wedge_{C^\infty(X)} \mathfrak{a}^*_0 \oplus \mathfrak{a}^*_1 ) \; \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>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>th summand on the right being in degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>.</p> </li> </ul> <div class="num_defn" id="LInfinityAlgebras"> <h6 id="definition_3">Definition</h6> <p>An <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebroid with base space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>=</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">X = *</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/point">point</a> is an <a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebra">L-∞ 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>, or rather is the <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> of an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebra. We write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">b \mathfrak{g}</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebroids over the point. They form the full <a class="existingWikiWord" href="/nlab/show/subcategory">subcategory</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub><mi>Alg</mi><mo>↪</mo><msub><mi>L</mi> <mn>∞</mn></msub><mi>Algd</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> L_\infty Alg \hookrightarrow L_\infty Algd \,. </annotation></semantics></math></div></div> <p>We now construct an embedding of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub><mi>Algs</mi></mrow><annotation encoding="application/x-tex">L_\infty Algs</annotation></semantics></math> into <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SynthDiff</mi><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">SynthDiff\infty Grpd</annotation></semantics></math>.</p> <p>The functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Ξ</mi><mo>:</mo><msubsup><mi>Ch</mi> <mo>+</mo> <mo>•</mo></msubsup><mo stretchy="false">(</mo><mi>ℝ</mi><mo stretchy="false">)</mo><mo>→</mo><msubsup><mi>Vect</mi> <mi>ℝ</mi> <mi>Δ</mi></msubsup></mrow><annotation encoding="application/x-tex"> \Xi : Ch^\bullet_+(\mathbb{R}) \to Vect_{\mathbb{R}}^{\Delta} </annotation></semantics></math></div> <p>of the <a class="existingWikiWord" href="/nlab/show/Dold-Kan+correspondence">Dold-Kan correspondence</a> from non-negatively graded <a class="existingWikiWord" href="/nlab/show/cochain+complex">cochain complex</a>es of <a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a>s to <a class="existingWikiWord" href="/nlab/show/cosimplicial+object">cosimplicial</a> vector spaces is a <a class="existingWikiWord" href="/nlab/show/lax+monoidal+functor">lax monoidal functor</a> and hence induces (see <a class="existingWikiWord" href="/nlab/show/monoidal+Dold-Kan+correspondence">monoidal Dold-Kan correspondence</a>) a functor (which we shall denote by the same symbol)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Ξ</mi><mo>:</mo><msubsup><mi>dgAlg</mi> <mi>ℝ</mi> <mo>+</mo></msubsup><mo>→</mo><msubsup><mi>Alg</mi> <mi>ℝ</mi> <mi>Δ</mi></msubsup></mrow><annotation encoding="application/x-tex"> \Xi : dgAlg_{\mathbb{R}}^+ \to Alg_{\mathbb{R}}^{\Delta} </annotation></semantics></math></div> <p>from non-negatively graded cochain <a class="existingWikiWord" href="/nlab/show/dg-algebra">dg-algebra</a>s to <a class="existingWikiWord" href="/nlab/show/cosimplicial+algebra">cosimplicial algebra</a>s (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>).</p> <div class="num_defn" id="PresentationByMonoidalDoldKan"> <h6 id="definition_4">Definition</h6> <p>Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Ξ</mi><mo>:</mo><msub><mi>L</mi> <mn>∞</mn></msub><mi>Algd</mi><mo>→</mo><mo stretchy="false">(</mo><msubsup><mi>CAlg</mi> <mi>ℝ</mi> <mi>Δ</mi></msubsup><msup><mo stretchy="false">)</mo> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex"> \Xi : L_\infty Algd \to (CAlg_{\mathbb{R}}^\Delta)^{op} </annotation></semantics></math></div> <p>for the restriction of the above <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ξ</mi></mrow><annotation encoding="application/x-tex">\Xi</annotation></semantics></math> along the inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub><mi>Algd</mi><mo>↪</mo><msubsup><mi>dgAlg</mi> <mi>ℝ</mi> <mi>op</mi></msubsup></mrow><annotation encoding="application/x-tex">L_\infty Algd \hookrightarrow dgAlg^{op}_{\mathbb{R}}</annotation></semantics></math>:</p> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔞</mi><mo>∈</mo><msub><mi>L</mi> <mn>∞</mn></msub><mi>Algd</mi></mrow><annotation encoding="application/x-tex">\mathfrak{a} \in L_\infty Algd</annotation></semantics></math> the underlying cosimplicial vector space of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ξ</mi><mi>𝔞</mi></mrow><annotation encoding="application/x-tex">\Xi \mathfrak{a}</annotation></semantics></math> 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>Ξ</mi><mi>𝔞</mi><mo>:</mo><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>↦</mo><munderover><mo lspace="thinmathspace" rspace="thinmathspace">⨁</mo> <mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow> <mi>n</mi></munderover><mi>CE</mi><mo stretchy="false">(</mo><mi>𝔞</mi><msub><mo stretchy="false">)</mo> <mi>i</mi></msub><mo>⊗</mo><msup><mo>∧</mo> <mi>i</mi></msup><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex"> \Xi \mathfrak{a} : [n] \mapsto \bigoplus_{i = 0}^{n} CE(\mathfrak{a})_i \otimes \wedge^i \mathbb{R}^n </annotation></semantics></math></div> <p>and the product of 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 structure on the right is given on homogeneous elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>ω</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">(</mo><mi>λ</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>CE</mi><mo stretchy="false">(</mo><mi>𝔞</mi><msub><mo stretchy="false">)</mo> <mi>i</mi></msub><mo>⊗</mo><msup><mo>∧</mo> <mi>i</mi></msup><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">(\omega,x), (\lambda,y) \in CE(\mathfrak{a})_i \otimes \wedge^i \mathbb{R}^n</annotation></semantics></math> in the <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a> by</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>ω</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><mi>λ</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>ω</mi><mo>∧</mo><mi>λ</mi><mo>,</mo><mi>x</mi><mo>∧</mo><mi>y</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (\omega , x)\cdot (\lambda ,y) = (\omega \wedge \lambda , x \wedge y) \,. </annotation></semantics></math></div> <p>(Notice that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ξ</mi><mi>𝔞</mi></mrow><annotation encoding="application/x-tex">\Xi \mathfrak{a}</annotation></semantics></math> is indeed a <em>commutative</em> cosimplicial algebra, since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math> and <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 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>ω</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\omega,x)</annotation></semantics></math> are by definition in the same degree.)</p> <p>To define the <a class="existingWikiWord" href="/nlab/show/cosimplicial+object">cosimplicial structure</a>, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>e</mi> <mi>j</mi></msub><msubsup><mo stretchy="false">}</mo> <mrow><mi>j</mi><mo>=</mo><mn>0</mn></mrow> <mi>n</mi></msubsup></mrow><annotation encoding="application/x-tex">\{e_j\}_{j = 0}^n</annotation></semantics></math> be the canonical <a class="existingWikiWord" href="/nlab/show/basis">basis</a> 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> and consider also the basis <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>v</mi> <mi>j</mi></msub><msubsup><mo stretchy="false">}</mo> <mrow><mi>j</mi><mo>=</mo><mn>0</mn></mrow> <mi>n</mi></msubsup></mrow><annotation encoding="application/x-tex">\{v_j\}_{j = 0}^n</annotation></semantics></math> 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>v</mi> <mi>j</mi></msub><mo>:</mo><mo>=</mo><msub><mi>e</mi> <mi>j</mi></msub><mo>−</mo><msub><mi>e</mi> <mn>0</mn></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> v_{j} := e_j - e_{0} \,. </annotation></semantics></math></div> <p>Then for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>:</mo><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo><mo>→</mo><mo stretchy="false">[</mo><mi>l</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\alpha : [k] \to [l]</annotation></semantics></math> a morphism in the <a class="existingWikiWord" href="/nlab/show/simplex">simplex</a> category, set</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>α</mi><msub><mi>v</mi> <mi>j</mi></msub><mo>:</mo><mo>=</mo><msub><mi>v</mi> <mrow><mi>α</mi><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo></mrow></msub><mo>−</mo><msub><mi>v</mi> <mrow><mi>α</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex"> \alpha v_j := v_{\alpha(j)} - v_{\alpha(0)} </annotation></semantics></math></div> <p>and extend this skew-multilinearly to a map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>:</mo><msup><mo>∧</mo> <mo>•</mo></msup><msup><mi>ℝ</mi> <mi>k</mi></msup><mo>→</mo><msup><mo>∧</mo> <mo>•</mo></msup><msup><mi>ℝ</mi> <mi>l</mi></msup></mrow><annotation encoding="application/x-tex">\alpha : \wedge^\bullet \mathbb{R}^k \to \wedge^\bullet \mathbb{R}^l</annotation></semantics></math>. In terms of all this the action of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math> on homogeneous elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>ω</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\omega,x)</annotation></semantics></math> in the cosimplicial algebra is defined by</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><mo stretchy="false">(</mo><mi>ω</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo>↦</mo><mo stretchy="false">(</mo><mi>ω</mi><mo>,</mo><mi>α</mi><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><msub><mi>d</mi> <mi>𝔞</mi></msub><mi>ω</mi><mo>,</mo><msub><mi>v</mi> <mrow><mi>α</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msub><mo>∧</mo><mi>α</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \alpha : (\omega, x) \mapsto (\omega, \alpha x) + (d_\mathfrak{a} \omega , v_{\alpha(0)}\wedge \alpha(x)) </annotation></semantics></math></div></div> <p>This is due to (<a href="#CastiglioniCortinas">CastiglioniCortinas, (1), (2), (20), (22)</a>).</p> <p>We shall refine the image of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ξ</mi></mrow><annotation encoding="application/x-tex">\Xi</annotation></semantics></math> to cosimplicial <a class="existingWikiWord" href="/nlab/show/smooth+algebra">smooth algebra</a>s. Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo>:</mo><mo>=</mo></mrow><annotation encoding="application/x-tex">T := </annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/CartSp">CartSp</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mrow></mrow> <mi>smooth</mi></msub></mrow><annotation encoding="application/x-tex">{}_{smooth}</annotation></semantics></math> be the category of <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a>s and <a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</a>s between them, regarded as a <a class="existingWikiWord" href="/nlab/show/Lawvere+theory">Lawvere theory</a>. Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>SmoothAlg</mi><mo>:</mo><mo>=</mo><mi>T</mi><mi>Alg</mi></mrow><annotation encoding="application/x-tex"> SmoothAlg := T Alg </annotation></semantics></math></div> <p>for its category of <a class="existingWikiWord" href="/nlab/show/algebra+over+a+Lawvere+theory">algebras</a>: these are the <a class="existingWikiWord" href="/nlab/show/smooth+algebra">smooth algebra</a>s.</p> <p>Notice that there is a canonical <a class="existingWikiWord" href="/nlab/show/forgetful+functor">forgetful functor</a></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><mi>SmoothAlg</mi><mo>→</mo><msub><mi>CAlg</mi> <mi>ℝ</mi></msub></mrow><annotation encoding="application/x-tex"> U : SmoothAlg \to CAlg_{\mathbb{R}} </annotation></semantics></math></div> <p>to the category of <a class="existingWikiWord" href="/nlab/show/associative+algebra">comutative associative algebras</a> over the <a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a>.</p> <div class="num_prop" id="SmoothMonoidalDoldKan"> <h6 id="proposition">Proposition</h6> <p>There is a unique factorization of the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ξ</mi><mo>:</mo><msub><mi>L</mi> <mn>∞</mn></msub><mi>Algd</mi><mo>→</mo><mo stretchy="false">(</mo><msubsup><mi>CAlg</mi> <mi>ℝ</mi> <mi>Δ</mi></msubsup><msup><mo stretchy="false">)</mo> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">\Xi : L_\infty Algd \to (CAlg_{\mathbb{R}}^\Delta)^{op}</annotation></semantics></math> from def. <a class="maruku-ref" href="#PresentationByMonoidalDoldKan"></a> through the <a class="existingWikiWord" href="/nlab/show/forgetful+functor">forgetful functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msubsup><mi>SmoothAlg</mi> <mi>ℝ</mi> <mi>Δ</mi></msubsup><msup><mo stretchy="false">)</mo> <mi>op</mi></msup><mo>→</mo><mo stretchy="false">(</mo><msubsup><mi>CAlg</mi> <mi>ℝ</mi> <mi>Δ</mi></msubsup><msup><mo stretchy="false">)</mo> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">(SmoothAlg_{\mathbb{R}}^\Delta)^{op} \to (CAlg_{\mathbb{R}}^\Delta)^{op}</annotation></semantics></math> such that for any <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">\mathfrak{a}</annotation></semantics></math> over base space <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> the degree-0 algebra of smooth functions <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> lifts to its canonical structure as a <a class="existingWikiWord" href="/nlab/show/smooth+algebra">smooth algebra</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></mtd> <mtd></mtd> <mtd><mo stretchy="false">(</mo><msup><mi>SmoothAlg</mi> <mi>Δ</mi></msup><msup><mo stretchy="false">)</mo> <mi>op</mi></msup></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↗</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>U</mi></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>L</mi> <mn>∞</mn></msub><mi>Algd</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mo stretchy="false">(</mo><msubsup><mi>CAlg</mi> <mi>ℝ</mi> <mi>Δ</mi></msubsup><msup><mo stretchy="false">)</mo> <mi>op</mi></msup></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ && (SmoothAlg^{\Delta})^{op} \\ & \nearrow & \downarrow^{\mathrlap{U}} \\ L_\infty Algd &\to& (CAlg_{\mathbb{R}}^\Delta)^{op} } \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof">Proof</h6> <p>Observe that for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> the algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Ξ</mi><mi>𝔞</mi><msub><mo stretchy="false">)</mo> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">(\Xi \mathfrak{a})_n</annotation></semantics></math> is a finite nilpotent extension 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></mrow><annotation encoding="application/x-tex">C^\infty(X)</annotation></semantics></math>. The claim then follows with using <a class="existingWikiWord" href="/nlab/show/Hadamard%27s+lemma">Hadamard's lemma</a> to write every smooth function of sums as a finite Taylor expansion with a smooth rest term. See the examples at <a class="existingWikiWord" href="/nlab/show/smooth+algebra">smooth algebra</a> for more details on this kind of argument.</p> </div> <div class="num_defn" id="EmbeddingOfThePresentation"> <h6 id="definition_5">Definition</h6> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>:</mo><msub><mi>L</mi> <mn>∞</mn></msub><mi>Algd</mi><mo>→</mo><mi>SynthDiff</mi><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">i : L_\infty Algd \to SynthDiff\infty Grpd</annotation></semantics></math> for the composite <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub><mi>Algd</mi><mover><mo>→</mo><mi>Ξ</mi></mover><mo stretchy="false">(</mo><msup><mi>SmoothAlg</mi> <mi>Δ</mi></msup><msup><mo stretchy="false">)</mo> <mi>op</mi></msup><mover><mo>→</mo><mi>j</mi></mover><mo stretchy="false">[</mo><msubsup><mi>CartSp</mi> <mi>synthdiff</mi> <mi>op</mi></msubsup><mo>,</mo><mi>sSet</mi><mo stretchy="false">]</mo><mover><mo>→</mo><mrow><mi>P</mi><mi>Q</mi></mrow></mover><mo stretchy="false">(</mo><mo stretchy="false">[</mo><msubsup><mi>CartSp</mi> <mi>synthdiff</mi> <mi>op</mi></msubsup><mo>,</mo><mi>sSet</mi><msub><mo stretchy="false">]</mo> <mi>loc</mi></msub><msup><mo stretchy="false">)</mo> <mo>∘</mo></msup><mo>≃</mo><mi>SynthDiff</mi><mn>∞</mn><mi>Grpd</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> L_\infty Algd \stackrel{\Xi}{\to} (SmoothAlg^{\Delta})^{op} \stackrel{j}{\to} [CartSp_{synthdiff}^{op}, sSet] \stackrel{P Q}{\to} ([CartSp_{synthdiff}^{op}, sSet]_{loc})^\circ \simeq SynthDiff\infty Grpd \,, </annotation></semantics></math></div> <p>where the first morphism is the <a class="existingWikiWord" href="/nlab/show/monoidal+Dold-Kan+correspondence">monoidal Dold-Kan correspondence</a> as in prop. <a class="maruku-ref" href="#SmoothMonoidalDoldKan"></a>, the second is the external degreewise <a class="existingWikiWord" href="/nlab/show/Yoneda+embedding">Yoneda embedding</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mi>Q</mi></mrow><annotation encoding="application/x-tex">P Q</annotation></semantics></math> is any fibrant-cofibrant <a class="existingWikiWord" href="/nlab/show/resolution">resolution</a> functor in the local <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">model structure on simplicial presheaves</a>. The last equivalence holds as discussed there and at <a class="existingWikiWord" href="/nlab/show/models+for+%E2%88%9E-stack+%28%E2%88%9E%2C1%29-toposes">models for ∞-stack (∞,1)-toposes</a>.</p> </div> <div class="num_remark" id="RemarkOnModelStructure"> <h6 id="remark">Remark</h6> <p>We do not consider the standard <a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-algebras">model structure on dg-algebras</a> and do not consider <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub><mi>Algd</mi></mrow><annotation encoding="application/x-tex">L_\infty Algd</annotation></semantics></math> itself as a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> and do not consider an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> spanned by it. Instead, the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>:</mo><msub><mi>L</mi> <mn>∞</mn></msub><mi>Algd</mi><mo>→</mo><mi>SynthDiff</mi><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">i : L_\infty Algd \to SynthDiff\infty Grpd</annotation></semantics></math> only serves to exhibit a class of objects in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SynthDiff</mi><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">SynthDiff\infty Grpd</annotation></semantics></math>, which below in the section <a href="#ModelsForTheAbstractAxioms">Models for the abstract axioms</a> we show are indeed <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Lie algebroids by the general abstract definition, <a class="maruku-ref" href="#TheGeneralAbstractDefinition"></a>. All the <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> of objects in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub><mi>Algd</mi></mrow><annotation encoding="application/x-tex">L_\infty Algd</annotation></semantics></math> is that of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SynthDiff</mi><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">SynthDiff\infty Grpd</annotation></semantics></math> after this embedding.</p> </div> <h3 id="GeneralAbstractDefinition">General abstract definition</h3> <p>We may abstractly formalize this in an <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-topos">(infinity,1)-topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> with <a class="existingWikiWord" href="/nlab/show/differential+cohesion">differential cohesion</a> as follows.</p> <p>Recall that a <a class="existingWikiWord" href="/nlab/show/groupoid+object+in+an+%28infinity%2C1%29-category">groupoid object in an (infinity,1)-category</a> is equivalently an <a class="existingWikiWord" href="/nlab/show/1-epimorphism">1-epimorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>⟶</mo><mi>𝒢</mi></mrow><annotation encoding="application/x-tex">X \longrightarrow \mathcal{G}</annotation></semantics></math>, thought of as exhibiting an <a class="existingWikiWord" href="/nlab/show/atlas">atlas</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> for the groupoid <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">\mathcal{G}</annotation></semantics></math>.</p> <p>Now an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Lie algebroid is supposed to be an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoid which is only infinitesimally extended over its base space <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>. Hence:</p> <p>A groupoid object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>⟶</mo><mi>𝒢</mi></mrow><annotation encoding="application/x-tex">p \colon X \longrightarrow \mathcal{G}</annotation></semantics></math> is <em>infinitesimal</em> if under the <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><mi>ℜ</mi></mrow><annotation encoding="application/x-tex">\Re</annotation></semantics></math> (equivalently under the <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><mi>ℑ</mi></mrow><annotation encoding="application/x-tex">\Im</annotation></semantics></math>) the <a class="existingWikiWord" href="/nlab/show/atlas">atlas</a> becomes an <a class="existingWikiWord" href="/nlab/show/equivalence+in+an+%28infinity%2C1%29-category">equivalence</a>: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℜ</mi><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo><mo>,</mo><mi>ℑ</mi><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>Equiv</mi></mrow><annotation encoding="application/x-tex">\Re(p), \Im(p) \in Equiv</annotation></semantics></math>.</p> <p>For example the tangent <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Lie algebroid <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> of any <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/unit+of+a+monad">unit</a> of the <a class="existingWikiWord" href="/nlab/show/infinitesimal+shape+modality">infinitesimal shape modality</a>.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>η</mi> <mi>X</mi> <mi>ℑ</mi></msubsup><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mover><mo>⟶</mo><mrow></mrow></mover><mi>ℑ</mi><mi>X</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \eta^{\Im}_X \;\colon\; X \stackrel{}{\longrightarrow} \Im X \,. </annotation></semantics></math></div> <p>It follows that every such <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Lie algebroid <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mi>𝒢</mi></mrow><annotation encoding="application/x-tex">X \to \mathcal{G}</annotation></semantics></math> canonically maps to the tangent <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Lie algebroid 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> – the <em>anchor map</em>. The naturality square of the unit <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>η</mi> <mi>p</mi> <mi>ℑ</mi></msubsup></mrow><annotation encoding="application/x-tex">\eta^{\Im}_{p}</annotation></semantics></math> exhibits the morphism:</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><mover><mo>⟶</mo><mi>id</mi></mover></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>p</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msubsup><mi>η</mi> <mi>X</mi> <mi>ℑ</mi></msubsup></mrow></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>ℑ</mi><mi>X</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msubsup><mo stretchy="false">↓</mo> <mo>≃</mo> <mpadded width="0"><mrow><mi>ℑ</mi><mi>p</mi></mrow></mpadded></msubsup></mtd></mtr> <mtr><mtd><mi>𝒢</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><msubsup><mi>η</mi> <mi>𝒢</mi> <mi>ℑ</mi></msubsup></mrow></mover></mtd> <mtd><mi>ℑ</mi><mi>𝒢</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ X & \stackrel{id}{\longrightarrow} & X \\ \downarrow^{\mathrlap{p}} && \downarrow^{\mathrlap{\eta^\Im_X}} \\ && \Im X \\ \downarrow && \downarrow^{\mathrlap{\Im p}}_\simeq \\ \mathcal{G} &\stackrel{\eta^{\Im}_{\mathcal{G}}}{\longrightarrow}& \Im \mathcal{G} } </annotation></semantics></math></div> <h2 id="properties">Properties</h2> <h3 id="general">General</h3> <div class="num_prop" id="prop"> <h6 id="proposition_2">Proposition</h6> <p>The full subcategory category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub><mi>Alg</mi><mo>↪</mo><msub><mi>L</mi> <mn>∞</mn></msub><mi>Algd</mi></mrow><annotation encoding="application/x-tex">L_\infty Alg \hookrightarrow L_\infty Algd</annotation></semantics></math> from def. <a class="maruku-ref" href="#LInfinityAlgebras"></a> is equivalent to the traditional definition of the category of <a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebra">L-∞ algebra</a>s and “weak morphisms” / “sh-maps” between them.</p> <p>The full subcategory <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>LieAlgd</mi><mo>↪</mo><msub><mi>L</mi> <mn>∞</mn></msub><mi>Algd</mi></mrow><annotation encoding="application/x-tex">LieAlgd \hookrightarrow L_\infty Algd</annotation></semantics></math> on the 1-truncated objects is equivalent to the traditional category of <a class="existingWikiWord" href="/nlab/show/Lie+algebroid">Lie algebroid</a>s (over <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a>s).</p> <p>In particular the joint intersection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Lie</mi><mi>Alg</mi><mo>↪</mo><msub><mi>L</mi> <mn>∞</mn></msub><mi>Alg</mi></mrow><annotation encoding="application/x-tex">Lie Alg \hookrightarrow L_\infty Alg</annotation></semantics></math> on the 1-truncated <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebras is equivalent to the category of ordinary <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a>s.</p> </div> <p>This is discussed in detail at <a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebra">L-∞ algebra</a> and <a class="existingWikiWord" href="/nlab/show/Lie+algebroid">Lie algebroid</a>.</p> <h3 id="ModelsForTheAbstractAxioms">Models for the abstract axioms</h3> <p>Above we have given a general abstract definition, def. <a class="maruku-ref" href="#TheGeneralAbstractDefinition"></a>, of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Lie algebroids, and then a concrete construction in terms of <a class="existingWikiWord" href="/nlab/show/dg-algebra">dg-algebra</a>s, def. <a class="maruku-ref" href="#PresentationByMonoidalDoldKan"></a>. Here we discuss that this concrete construction is indeed a presentation for objects satisfying the abstract axioms.</p> <p>As in the discussion at <a class="existingWikiWord" href="/nlab/show/SynthDiff%E2%88%9EGrpd">SynthDiff∞Grpd</a> we now present this <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a> by the <a class="existingWikiWord" href="/nlab/show/hypercompletion">hypercompletion</a> of the <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">model structure on simplicial presheaves</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>FSmoothDiff</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy="false">]</mo> <mrow><mi>proj</mi><mo>,</mo><mi>loc</mi></mrow></msub></mrow><annotation encoding="application/x-tex">[FSmoothDiff^{op}, sSet]_{proj,loc}</annotation></semantics></math> of formal smooth manifolds.</p> <div class="num_lemma" id="CofibrantResolutionOfLinfinityAlgebroid"> <h6 id="lemma">Lemma</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔞</mi><mo>∈</mo><msub><mi>L</mi> <mn>∞</mn></msub><mi>Algd</mi></mrow><annotation encoding="application/x-tex">\mathfrak{a} \in L_\infty Algd</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo stretchy="false">(</mo><mi>𝔞</mi><mo stretchy="false">)</mo><mo>∈</mo><mo stretchy="false">[</mo><msup><mi>FSmoothMfd</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy="false">]</mo> <mrow><mi>proj</mi><mo>,</mo><mi>loc</mi></mrow></msub></mrow><annotation encoding="application/x-tex">i(\mathfrak{a}) \in [FSmoothMfd^{op}, sSet]_{proj,loc}</annotation></semantics></math> its image in the standard presentation for <a class="existingWikiWord" href="/nlab/show/SynthDiff%E2%88%9EGrpd">SynthDiff∞Grpd</a>, we have that</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><mo>∫</mo> <mrow><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo><mo>∈</mo><mi>Δ</mi></mrow></msup><mstyle mathvariant="bold"><mi>Δ</mi></mstyle><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo><mo>⋅</mo><mi>i</mi><mo stretchy="false">(</mo><mi>𝔞</mi><msub><mo stretchy="false">)</mo> <mi>k</mi></msub><mo>)</mo></mrow><mover><mo>→</mo><mo>≃</mo></mover><mi>i</mi><mo stretchy="false">(</mo><mi>𝔞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \left( \int^{[k]\in \Delta} \mathbf{\Delta}[k] \cdot i(\mathfrak{a})_k \right) \stackrel{\simeq}{\to} i(\mathfrak{a}) </annotation></semantics></math></div> <p>is a cofibrant <a class="existingWikiWord" href="/nlab/show/resolution">resolution</a>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Δ</mi></mstyle><mo>:</mo><mi>Δ</mi><mo>→</mo><mi>sSet</mi></mrow><annotation encoding="application/x-tex">\mathbf{\Delta} : \Delta \to sSet</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/fat+simplex">fat simplex</a>.</p> </div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>We have</p> <ol> <li> <p>The <a class="existingWikiWord" href="/nlab/show/fat+simplex">fat simplex</a> is cofibrant in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>Δ</mi><mo>,</mo><msub><mi>sSet</mi> <mi>Quillen</mi></msub><msub><mo stretchy="false">]</mo> <mi>proj</mi></msub></mrow><annotation encoding="application/x-tex">[\Delta, sSet_{Quillen}]_{proj}</annotation></semantics></math>.</p> </li> <li> <p>The canonical morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Δ</mi></mstyle><mo>→</mo><mi>Δ</mi></mrow><annotation encoding="application/x-tex">\mathbf{\Delta} \to \Delta</annotation></semantics></math> is a weak equivalence between cofibrant objects in the <a class="existingWikiWord" href="/nlab/show/Reedy+model+structure">Reedy model structure</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>Δ</mi><mo>,</mo><msub><mi>sSet</mi> <mi>Quillen</mi></msub><msub><mo stretchy="false">]</mo> <mi>Reedy</mi></msub></mrow><annotation encoding="application/x-tex">[\Delta, sSet_{Quillen}]_{Reedy}</annotation></semantics></math>.</p> </li> <li> <p>Because every representable <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>FSmoothMfd</mi><mo>↪</mo><mo stretchy="false">[</mo><msup><mi>FSmoothMfd</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy="false">]</mo> <mrow><mi>proj</mi><mo>,</mo><mi>loc</mi></mrow></msub></mrow><annotation encoding="application/x-tex">FSmoothMfd \hookrightarrow [FSmoothMfd^{op}, sSet]_{proj,loc}</annotation></semantics></math> is cofibrant, the object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo stretchy="false">(</mo><mi>𝔞</mi><msub><mo stretchy="false">)</mo> <mo>•</mo></msub><mo>∈</mo><mo stretchy="false">[</mo><msup><mi>Δ</mi> <mi>op</mi></msup><mo>,</mo><mo stretchy="false">[</mo><msup><mi>FSmoothMfd</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy="false">]</mo> <mrow><mi>proj</mi><mo>,</mo><mi>loc</mi></mrow></msub><msub><mo stretchy="false">]</mo> <mi>inj</mi></msub></mrow><annotation encoding="application/x-tex">i(\mathfrak{a})_\bullet \in [\Delta^{op}, [FSmoothMfd^{op}, sSet]_{proj,loc} ]_{inj}</annotation></semantics></math> is cofibrant.</p> </li> <li> <p>Every <a class="existingWikiWord" href="/nlab/show/simplicial+presheaf">simplicial presheaf</a> is cofibrant regarded as an object <a class="existingWikiWord" href="/nlab/show/Reedy+model+structure">Reedy model structure</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>Δ</mi> <mi>op</mi></msup><mo>,</mo><mo stretchy="false">[</mo><msup><mi>FSmoothMfd</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy="false">]</mo> <mi>inj</mi></msub><msub><mo stretchy="false">]</mo> <mi>Reedy</mi></msub></mrow><annotation encoding="application/x-tex">[\Delta^{op}, [FSmoothMfd^{op}, sSet]_{inj}]_{Reedy}</annotation></semantics></math>.</p> </li> </ol> <p>Now the <a class="existingWikiWord" href="/nlab/show/coend">coend</a> over the <a class="existingWikiWord" href="/nlab/show/tensoring">tensoring</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> <mrow><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo><mo>∈</mo><mi>Δ</mi></mrow></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>:</mo><mo stretchy="false">[</mo><mi>Δ</mi><mo>,</mo><msub><mi>sSet</mi> <mi>Quillen</mi></msub><msub><mo stretchy="false">]</mo> <mi>proj</mi></msub><mo>×</mo><mo stretchy="false">[</mo><msup><mi>Δ</mi> <mi>op</mi></msup><mo>,</mo><mo stretchy="false">[</mo><msup><mi>FSmoothMfd</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy="false">]</mo> <mrow><mi>proj</mi><mo>,</mo><mi>loc</mi></mrow></msub><msub><mo stretchy="false">]</mo> <mi>inj</mi></msub><mo>→</mo><mo stretchy="false">[</mo><msup><mi>FSmoothMfd</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy="false">]</mo> <mrow><mi>proj</mi><mo>,</mo><mi>loc</mi></mrow></msub></mrow><annotation encoding="application/x-tex"> \int^{[k] \in \Delta} (-)\cdot (-) : [\Delta, sSet_{Quillen}]_{proj} \times [\Delta^{op}, [FSmoothMfd^{op}, sSet]_{proj,loc} ]_{inj} \to [FSmoothMfd^{op}, sSet]_{proj,loc} </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/Quillen+bifunctor">Quillen bifunctor</a> (as discussed there) for the projective and injective <a class="existingWikiWord" href="/nlab/show/global+model+structure+on+functors">global model structure on functors</a> on the <a class="existingWikiWord" href="/nlab/show/simplex+category">simplex category</a> and its opposite as indicated. This implies the cofibrancy.</p> <p>It is also a <a class="existingWikiWord" href="/nlab/show/Quillen+bifunctor">Quillen bifunctor</a> (as discussed there) for the <a class="existingWikiWord" href="/nlab/show/Reedy+model+structure">Reedy model structure</a>s</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mo>∫</mo> <mrow><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo><mo>∈</mo><mi>Δ</mi></mrow></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>:</mo><mo stretchy="false">[</mo><mi>Δ</mi><mo>,</mo><msub><mi>sSet</mi> <mi>Quillen</mi></msub><msub><mo stretchy="false">]</mo> <mi>Reedy</mi></msub><mo>×</mo><mo stretchy="false">[</mo><msup><mi>Δ</mi> <mi>op</mi></msup><mo>,</mo><mo stretchy="false">[</mo><msup><mi>FSmoothMfd</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy="false">]</mo> <mi>inj</mi></msub><msub><mo stretchy="false">]</mo> <mi>Reedy</mi></msub><mo>→</mo><mo stretchy="false">[</mo><msup><mi>FSmoothMfd</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy="false">]</mo> <mi>inj</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \int^{[k] \in \Delta} (-)\cdot (-) : [\Delta, sSet_{Quillen}]_{Reedy} \times [\Delta^{op}, [FSmoothMfd^{op}, sSet]_{inj} ]_{Reedy} \to [FSmoothMfd^{op}, sSet]_{inj} \,. </annotation></semantics></math></div> <p>Using the <a class="existingWikiWord" href="/nlab/show/factorization+lemma">factorization lemma</a> this implies the weak equivalence (this is the argument of the <a class="existingWikiWord" href="/nlab/show/Bousfield-Kan+map">Bousfield-Kan map</a>).</p> </div> <div class="num_prop" id="LInftyAlgIsInfinitesimalObject"> <h6 id="proposition_3">Proposition</h6> <p>Let <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> be an <a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebra">L-∞ algebra</a>, regarded as an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebroid <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mi>𝔤</mi><mo>∈</mo><msub><mi>L</mi> <mn>∞</mn></msub><mi>Algd</mi></mrow><annotation encoding="application/x-tex">b \mathfrak{g} \in L_\infty Algd</annotation></semantics></math> over the point by the embedding of def. <a class="maruku-ref" href="#LInfinityAlgebras"></a>.</p> <p>Then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo stretchy="false">(</mo><mi>b</mi><mi>𝔤</mi><mo stretchy="false">)</mo><mo>∈</mo></mrow><annotation encoding="application/x-tex">i(b \mathfrak{g}) \in </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/SynthDiff%E2%88%9EGrpd">SynthDiff∞Grpd</a> is an <a href="http://nlab.mathforge.org/nlab/show/cohesive+%28infinity%2C1%29-topos#LieAlgebras">infinitesimal cohesive object</a>, in that it is <a href="http://nlab.mathforge.org/nlab/show/cohesive+%28infinity%2C1%29-topos#Homotopy">geometrically contractible</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>b</mi><mi>𝔤</mi><mo>≃</mo><mo>*</mo></mrow><annotation encoding="application/x-tex"> \Pi b \mathfrak{g} \simeq * </annotation></semantics></math></div> <p>and has as underlying <a class="existingWikiWord" href="/nlab/show/discrete+%E2%88%9E-groupoid">discrete ∞-groupoid</a> the point</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Γ</mi><mi>b</mi><mi>𝔤</mi><mo>≃</mo><mo>*</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Gamma b \mathfrak{g} \simeq * \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_3">Proof</h6> <p>We present now <a class="existingWikiWord" href="/nlab/show/SynthDiff%E2%88%9EGrpd">SynthDiff∞Grpd</a> by the <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">model structure on simplicial presheaves</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msubsup><mi>CartSp</mi> <mi>synthdiff</mi> <mi>op</mi></msubsup><mo>,</mo><mi>sSet</mi><msub><mo stretchy="false">]</mo> <mrow><mi>proj</mi><mo>,</mo><mi>loc</mi></mrow></msub></mrow><annotation encoding="application/x-tex">[CartSp_{synthdiff}^{op}, sSet]_{proj,loc}</annotation></semantics></math>. Since <a class="existingWikiWord" href="/nlab/show/CartSp">CartSp</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mrow></mrow> <mi>synthdiff</mi></msub></mrow><annotation encoding="application/x-tex">{}_{synthdiff}</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-cohesive+site">∞-cohesive site</a> we have by the discussion there that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi></mrow><annotation encoding="application/x-tex">\Pi</annotation></semantics></math> is presented by the left <a class="existingWikiWord" href="/nlab/show/derived+functor">derived functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝕃</mi><mi>lim</mi><mo>→</mo></mrow><annotation encoding="application/x-tex">\mathbb{L} \lim\to</annotation></semantics></math> of the degreewise <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi></mrow><annotation encoding="application/x-tex">\Gamma</annotation></semantics></math> is presented by the left <a class="existingWikiWord" href="/nlab/show/derived+functor">derived functor</a> of evaluation on the point.</p> <p>With lemma <a class="maruku-ref" href="#CofibrantResolutionOfLinfinityAlgebroid"></a> we can evaluate</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>𝕃</mi><munder><mi>lim</mi> <mo>→</mo></munder><mo stretchy="false">)</mo><mi>i</mi><mo stretchy="false">(</mo><mi>b</mi><mi>𝔤</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>≃</mo><munder><mi>lim</mi> <mo>→</mo></munder><msup><mo>∫</mo> <mrow><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo><mo>∈</mo><mi>Δ</mi></mrow></msup><mstyle mathvariant="bold"><mi>Δ</mi></mstyle><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo><mo>⋅</mo><mo stretchy="false">(</mo><mi>b</mi><mi>𝔤</mi><msub><mo stretchy="false">)</mo> <mi>k</mi></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><msup><mo>∫</mo> <mrow><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo><mo>∈</mo><mi>Δ</mi></mrow></msup><mstyle mathvariant="bold"><mi>Δ</mi></mstyle><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo><mo>⋅</mo><munder><mi>lim</mi> <mo>→</mo></munder><mo stretchy="false">(</mo><mi>b</mi><mi>𝔤</mi><msub><mo stretchy="false">)</mo> <mi>k</mi></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><msup><mo>∫</mo> <mrow><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo><mo>∈</mo><mi>Δ</mi></mrow></msup><mstyle mathvariant="bold"><mi>Δ</mi></mstyle><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo><mo>⋅</mo><mo>*</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} (\mathbb{L} \lim_\to) i(b\mathfrak{g}) & \simeq \lim_\to \int^{[k] \in \Delta} \mathbf{\Delta}[k] \cdot (b \mathfrak{g})_{k} \\ & \simeq \int^{[k] \in \Delta} \mathbf{\Delta}[k] \cdot \lim_\to (b \mathfrak{g})_{k} \\ & = \int^{[k] \in \Delta} \mathbf{\Delta}[k] \cdot * \end{aligned} \,, </annotation></semantics></math></div> <p>because each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>b</mi><mi>𝔤</mi><msub><mo stretchy="false">)</mo> <mi>n</mi></msub><mo>∈</mo><mi>InfPoint</mi><mo>↪</mo><msub><mi>CartSp</mi> <mi>smooth</mi></msub></mrow><annotation encoding="application/x-tex">(b \mathfrak{g})_n \in InfPoint \hookrightarrow CartSp_{smooth}</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/infinitesimally+thickened+point">infinitesimally thickened point</a>, hence representable and hence sent to the point by the colimit functor.</p> <p>That this is equivalent to the point follows from the fact that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>∅</mi><mo>→</mo><mstyle mathvariant="bold"><mi>Δ</mi></mstyle></mrow><annotation encoding="application/x-tex">\emptyset \to \mathbf{\Delta}</annotation></semantics></math> is an acylic cofibration in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>Δ</mi><mo>,</mo><msub><mi>sSet</mi> <mi>Quillen</mi></msub><msub><mo stretchy="false">]</mo> <mi>proj</mi></msub></mrow><annotation encoding="application/x-tex">[\Delta, sSet_{Quillen}]_{proj}</annotation></semantics></math>, and that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mo>∫</mo> <mrow><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo><mo>∈</mo><mi>Δ</mi></mrow></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>:</mo><mo stretchy="false">[</mo><mi>Δ</mi><mo>,</mo><msub><mi>sSet</mi> <mi>Quillen</mi></msub><msub><mo stretchy="false">]</mo> <mi>proj</mi></msub><mo>×</mo><mo stretchy="false">[</mo><msup><mi>Δ</mi> <mi>op</mi></msup><mo>,</mo><msub><mi>sSet</mi> <mi>Qillen</mi></msub><msub><mo stretchy="false">]</mo> <mi>inj</mi></msub><mo>→</mo><msub><mi>sSet</mi> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex"> \int^{[k] \in \Delta} (-)\times (-) : [\Delta, sSet_{Quillen}]_{proj} \times [\Delta^{op}, sSet_{Qillen}]_{inj} \to sSet_{Quillen} </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/Quillen+bifunctor">Quillen bifunctor</a>, using that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo><mo>∈</mo><mo stretchy="false">[</mo><msup><mi>Δ</mi> <mi>op</mi></msup><mo>,</mo><msub><mi>sSet</mi> <mi>Quillen</mi></msub><msub><mo stretchy="false">]</mo> <mi>inj</mi></msub></mrow><annotation encoding="application/x-tex">* \in [\Delta^{op}, sSet_{Quillen}]_{inj}</annotation></semantics></math> is cofibrant.</p> <p>Similarily, we have degreewise that</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><mo>*</mo><mo>,</mo><mo stretchy="false">(</mo><mi>b</mi><mi>𝔤</mi><msub><mo stretchy="false">)</mo> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mo>*</mo></mrow><annotation encoding="application/x-tex"> Hom(*, (b \mathfrak{g})_n) = * </annotation></semantics></math></div> <p>by the fact that an <a class="existingWikiWord" href="/nlab/show/infinitesimally+thickened+point">infinitesimally thickened point</a> has a single global point. Therefore the claim for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi></mrow><annotation encoding="application/x-tex">\Gamma</annotation></semantics></math> follows analogously.</p> </div> <div class="num_prop"> <h6 id="proposition_4">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝔞</mi><mo>→</mo><mi>T</mi><mi>X</mi><mo stretchy="false">)</mo><mo>∈</mo><msub><mi>L</mi> <mn>∞</mn></msub><mi>Algd</mi><mo>↪</mo><mo stretchy="false">[</mo><msub><mi>CartSp</mi> <mi>synthdiff</mi></msub><mo>,</mo><mi>sSet</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">(\mathfrak{a} \to T X) \in L_\infty Algd \hookrightarrow [CartSp_{synthdiff}, sSet]</annotation></semantics></math> be an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebroid, def. <a class="maruku-ref" href="#LInftyGlgebroid"></a>, over a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth 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>, regarded as a <a class="existingWikiWord" href="/nlab/show/simplicial+presheaf">simplicial presheaf</a> and hence as a presentation for an object in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SynthDiff</mi><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">SynthDiff \infty Grpd</annotation></semantics></math> according to def. <a class="maruku-ref" href="#EmbeddingOfThePresentation"></a>.</p> <p>We have an equivalence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>Π</mi></mstyle> <mi>inf</mi></msub><mo stretchy="false">(</mo><mi>𝔞</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mstyle mathvariant="bold"><mi>Π</mi></mstyle> <mi>inf</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{\Pi}_{inf}(\mathfrak{a}) \simeq \mathbf{\Pi}_{inf}(X) \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_4">Proof</h6> <p>Let first <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><msub><mi>CartSp</mi> <mi>synthdiff</mi></msub></mrow><annotation encoding="application/x-tex">X = U \in CartSp_{synthdiff}</annotation></semantics></math> be a representable. Then according to prop. <a class="maruku-ref" href="#CofibrantResolutionOfLinfinityAlgebroid"></a> we have that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mi>𝔞</mi><mo stretchy="false">^</mo></mover><mo>:</mo><mo>=</mo><mrow><mo>(</mo><msup><mo>∫</mo> <mrow><mi>k</mi><mo>∈</mo><mi>Δ</mi></mrow></msup><mstyle mathvariant="bold"><mi>Δ</mi></mstyle><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo><mo>⋅</mo><msub><mi>𝔞</mi> <mi>k</mi></msub><mo>)</mo></mrow><mo>≃</mo><mi>𝔞</mi></mrow><annotation encoding="application/x-tex"> \hat \mathfrak{a} := \left( \int^{k \in \Delta} \mathbf{\Delta}[k] \cdot \mathfrak{a}_k \right) \simeq \mathfrak{a} </annotation></semantics></math></div> <p>is cofibrant in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msubsup><mi>CartSp</mi> <mi>synthdiff</mi> <mi>op</mi></msubsup><mo>,</mo><mi>sSet</mi><msub><mo stretchy="false">]</mo> <mi>proj</mi></msub></mrow><annotation encoding="application/x-tex">[CartSp_{synthdiff}^{op}, sSet]_{proj}</annotation></semantics></math> . Therefore by <a href="http://ncatlab.org/nlab/show/cohesive+%28infinity%2C1%29-topos+--+infinitesimal+cohesion#InfinitesimalNeighbourhoodFromInfinitesimalSite">this proposition</a> on the presentation of infinitesimal neighbourhoods by <a class="existingWikiWord" href="/nlab/show/simplicial+presheaves">simplicial presheaves</a> over <em>infinitesimal neighbourhood sites</em> we compute the <a class="existingWikiWord" href="/nlab/show/derived+functor">derived functor</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><msub><mstyle mathvariant="bold"><mi>Π</mi></mstyle> <mi>inf</mi></msub><mo stretchy="false">(</mo><mi>𝔞</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>≃</mo><msub><mi>i</mi> <mo>*</mo></msub><msup><mi>i</mi> <mo>*</mo></msup><mi>𝔞</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><mi>𝕃</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>∘</mo><mi>p</mi><mo stretchy="false">)</mo><mi>𝕃</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>∘</mo><mi>i</mi><mo stretchy="false">)</mo><mi>𝔞</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>∘</mo><mi>i</mi><mi>p</mi><mo stretchy="false">)</mo><mover><mi>𝔞</mi><mo stretchy="false">^</mo></mover></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \mathbf{\Pi}_{inf}(\mathfrak{a}) & \simeq i_* i^* \mathfrak{a} \\ & \simeq \mathbb{L} ((-) \circ p) \mathbb{L} ((-) \circ i) \mathfrak{a} \\ & \simeq ((-) \circ i p ) \hat \mathfrak{a} \end{aligned} </annotation></semantics></math></div> <p>with the notation as used there.</p> <p>In view of def. <a class="maruku-ref" href="#PresentationByMonoidalDoldKan"></a> we have for all <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> that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔞</mi> <mi>k</mi></msub><mo>=</mo><mi>X</mi><mo>×</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">\mathfrak{a}_k = X \times D</annotation></semantics></math> where <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> is an <a class="existingWikiWord" href="/nlab/show/infinitesimally+thickened+point">infinitesimally thickened point</a>. Therefore <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>∘</mo><mi>i</mi><mi>p</mi><mo stretchy="false">)</mo><msub><mi>𝔞</mi> <mi>k</mi></msub><mo>=</mo><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>∘</mo><mi>i</mi><mi>p</mi><mo stretchy="false">)</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">((-) \circ i p ) \mathfrak{a}_k = ((-) \circ i p ) X</annotation></semantics></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> and hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>∘</mo><mi>i</mi><mi>p</mi><mo stretchy="false">)</mo><mover><mi>𝔞</mi><mo stretchy="false">^</mo></mover><mo>≃</mo><msub><mstyle mathvariant="bold"><mi>Π</mi></mstyle> <mi>inf</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">((-) \circ i p ) \hat \mathfrak{a} \simeq \mathbf{\Pi}_{inf}(X)</annotation></semantics></math>.</p> <p>For general <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> choose first a cofibrant <a class="existingWikiWord" href="/nlab/show/resolution">resolution</a> by a <a class="existingWikiWord" href="/nlab/show/split+hypercover">split hypercover</a> that is degreewise a coproduct of representables (which always exists, by the discussion at <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">model structure on simplicial presheaves</a>), then pull back the above discussion to these covers.</p> </div> <div class="num_cor" id="LInfinityAlgebrboidsAreFormalInfinityGroupoids"> <h6 id="corollary">Corollary</h6> <p>Every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebroid in the sense of def. <a class="maruku-ref" href="#LInftyGlgebroid"></a> under the embedding of def. <a class="maruku-ref" href="#EmbeddingOfThePresentation"></a> is indeed a <a class="existingWikiWord" href="/nlab/show/formal+cohesive+%E2%88%9E-groupoid">formal cohesive ∞-groupoid</a> in the sense of def. <a class="maruku-ref" href="#TheGeneralAbstractDefinition"></a>.</p> </div> <h3 id="Cohomology">Cohomology of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Lie algebroids</h3> <p>We discuss the relation between the intrinsic <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebroids when regarded as objects of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SynthDiff</mi><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">SynthDiff\infty Grpd</annotation></semantics></math>, and the ordinary cohomology of their <a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebra">Chevalley-Eilenberg algebra</a>s. For more on this see <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebroid+cohomology">∞-Lie algebroid cohomology</a>.</p> <div class="num_prop" id="IntrinsicRealCohomologyByCECohomology"> <h6 id="proposition_5">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔞</mi><mo>∈</mo><msub><mi>L</mi> <mn>∞</mn></msub><mi>Algd</mi></mrow><annotation encoding="application/x-tex">\mathfrak{a} \in L_\infty Algd</annotation></semantics></math> be an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebroid. Then its <a href="">intrinsic real cohomoloogy</a> in <a class="existingWikiWord" href="/nlab/show/SynthDiff%E2%88%9EGrpd">SynthDiff∞Grpd</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>𝔞</mi><mo>,</mo><mi>ℝ</mi><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><msub><mi>π</mi> <mn>0</mn></msub><mi>SynthDiff</mi><mn>∞</mn><mi>Grpd</mi><mo stretchy="false">(</mo><mi>𝔞</mi><mo>,</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>ℝ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> H^n(\mathfrak{a}, \mathbb{R}) := \pi_0 SynthDiff\infty Grpd(\mathfrak{a}, \mathbf{B}^n \mathbb{R}) </annotation></semantics></math></div> <p>coincides with its ordinary <a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebra+cohomology">L-∞ algebra cohomology</a>: the <a class="existingWikiWord" href="/nlab/show/cochain+cohomology">cochain cohomology</a> of its <a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebra">Chevalley-Eilenberg 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>H</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>𝔞</mi><mo>,</mo><mi>ℝ</mi><mo stretchy="false">)</mo><mo>≃</mo><msup><mi>H</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>CE</mi><mo stretchy="false">(</mo><mi>𝔞</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> H^n(\mathfrak{a}, \mathbb{R}) \simeq H^n(CE(\mathfrak{a})) \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_5">Proof</h6> <p>By <a href="http://nlab.mathforge.org/nlab/show/synthetic+differential+infinity-groupoid#StrucCohomology">this discussion</a> at <a class="existingWikiWord" href="/nlab/show/SynthDiff%E2%88%9EGrpd">SynthDiff∞Grpd</a> we have that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>𝔞</mi><mo>,</mo><mi>ℝ</mi><mo stretchy="false">)</mo><mo>≃</mo><msup><mi>H</mi> <mi>n</mi></msup><msup><mi>N</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>𝕃</mi><mi>𝒪</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">(</mo><mi>𝔞</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> H^n(\mathfrak{a}, \mathbb{R}) \simeq H^n N^\bullet(\mathbb{L}\mathcal{O})(i(\mathfrak{a})) \,. </annotation></semantics></math></div> <p>By lemma <a class="maruku-ref" href="#CofibrantResolutionOfLinfinityAlgebroid"></a> this is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>⋯</mi><mo>≃</mo><msup><mi>H</mi> <mi>n</mi></msup><msup><mi>N</mi> <mo>•</mo></msup><mrow><mo>(</mo><msup><mo>∫</mo> <mrow><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo><mo>∈</mo><mi>Δ</mi></mrow></msup><mstyle mathvariant="bold"><mi>Δ</mi></mstyle><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo><mo>⋅</mo><mi>𝒪</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">(</mo><mi>𝔞</mi><msub><mo stretchy="false">)</mo> <mi>k</mi></msub><mo stretchy="false">)</mo><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \cdots \simeq H^n N^\bullet \left( \int^{[k] \in \Delta} \mathbf{\Delta}[k] \cdot \mathcal{O}(i(\mathfrak{a})_k) \right) \,. </annotation></semantics></math></div> <p>Observe that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒪</mi><mo stretchy="false">(</mo><mi>𝔞</mi><msub><mo stretchy="false">)</mo> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">\mathcal{O}(\mathfrak{a})_\bullet</annotation></semantics></math> is cofibrant in the <a class="existingWikiWord" href="/nlab/show/Reedy+model+structure">Reedy model structure</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>Δ</mi> <mi>op</mi></msup><mo>,</mo><mo stretchy="false">(</mo><msup><mi>SmoothAlg</mi> <mrow><msub><mi>Δ</mi> <mi>proj</mi></msub></mrow></msup><msup><mo stretchy="false">)</mo> <mi>op</mi></msup><msub><mo stretchy="false">]</mo> <mi>Reedy</mi></msub></mrow><annotation encoding="application/x-tex">[\Delta^{op}, (SmoothAlg^{\Delta_{proj}})^{op}]_{Reedy}</annotation></semantics></math> relative to the opposite of the projective <a class="existingWikiWord" href="/nlab/show/model+structure+on+cosimplicial+algebras">model structure on cosimplicial algebras</a>: the map from the latching object 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> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>SmoothAlg</mi> <mi>Δ</mi></msup><msup><mo stretchy="false">)</mo> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">SmoothAlg^\Delta)^{op}</annotation></semantics></math> is dually in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SmoothAlg</mi><mo>↪</mo><msup><mi>SmoothAlg</mi> <mi>Δ</mi></msup></mrow><annotation encoding="application/x-tex">SmoothAlg \hookrightarrow SmoothAlg^\Delta</annotation></semantics></math> the projection</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mo>⊕</mo> <mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow> <mi>n</mi></msubsup><mi>CE</mi><mo stretchy="false">(</mo><mi>𝔞</mi><msub><mo stretchy="false">)</mo> <mi>i</mi></msub><mo>⊗</mo><msup><mo>∧</mo> <mi>i</mi></msup><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>→</mo><msubsup><mo>⊕</mo> <mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msubsup><mi>CE</mi><mo stretchy="false">(</mo><mi>𝔞</mi><msub><mo stretchy="false">)</mo> <mi>i</mi></msub><mo>⊗</mo><msup><mo>∧</mo> <mi>i</mi></msup><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex"> \oplus_{i = 0}^n CE(\mathfrak{a})_i \otimes \wedge^i \mathbb{R}^n \to \oplus_{i = 0}^{n-1} CE(\mathfrak{a})_i \otimes \wedge^i \mathbb{R}^n </annotation></semantics></math></div> <p>hence is a surjection, hence a fibration in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>SmoothAlg</mi> <mi>proj</mi> <mi>Δ</mi></msubsup></mrow><annotation encoding="application/x-tex">SmoothAlg^\Delta_{proj}</annotation></semantics></math> and therefore indeed a cofibration in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msubsup><mi>SmoothAlg</mi> <mi>proj</mi> <mi>Δ</mi></msubsup><msup><mo stretchy="false">)</mo> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">(SmoothAlg^\Delta_{proj})^{op}</annotation></semantics></math>.</p> <p>Therefore using the <a class="existingWikiWord" href="/nlab/show/Quillen+bifunctor">Quillen bifunctor</a> property of the coend over the tensoring in reverse to lemma <a class="maruku-ref" href="#CofibrantResolutionOfLinfinityAlgebroid"></a> the above is equivalent to</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>⋯</mi><mo>≃</mo><msup><mi>H</mi> <mi>n</mi></msup><msup><mi>N</mi> <mo>•</mo></msup><mrow><mo>(</mo><msup><mo>∫</mo> <mrow><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo><mo>∈</mo><mi>Δ</mi></mrow></msup><mi>Δ</mi><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo><mo>⋅</mo><mi>𝒪</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">(</mo><mi>𝔞</mi><msub><mo stretchy="false">)</mo> <mi>k</mi></msub><mo stretchy="false">)</mo><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> \cdots \simeq H^n N^\bullet \left( \int^{[k] \in \Delta} \Delta[k] \cdot \mathcal{O}(i(\mathfrak{a})_k) \right) </annotation></semantics></math></div> <p>with the <a class="existingWikiWord" href="/nlab/show/fat+simplex">fat simplex</a> replaced again by the ordinary simplex. But in brackets this is now by definition the image under the <a class="existingWikiWord" href="/nlab/show/monoidal+Dold-Kan+correspondence">monoidal Dold-Kan correspondence</a> of the <a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebra">Chevalley-Eilenberg algebra</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>⋯</mi><mo>≃</mo><msup><mi>H</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><msup><mi>N</mi> <mo>•</mo></msup><mi>Ξ</mi><mi>CE</mi><mo stretchy="false">(</mo><mi>𝔞</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \cdots \simeq H^n( N^\bullet \Xi CE(\mathfrak{a}) ) \,. </annotation></semantics></math></div> <p>By the <a class="existingWikiWord" href="/nlab/show/Dold-Kan+correspondence">Dold-Kan correspondence</a> we have hence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>⋯</mi><mo>≃</mo><msup><mi>H</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>CE</mi><mo stretchy="false">(</mo><mi>𝔞</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \cdots \simeq H^n(CE(\mathfrak{a})) \,. </annotation></semantics></math></div></div> <h2 id="examples">Examples</h2> <h3 id="classes_of_examples">Classes of examples</h3> <ul> <li> <p>a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Lie algebroid over the <a class="existingWikiWord" href="/nlab/show/point">point</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔞</mi><mo>=</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">\mathfrak{a} = *</annotation></semantics></math> is an <strong><a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E-algebra">L-∞-algebra</a></strong>;</p> </li> <li> <p>an <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>-<a class="existingWikiWord" href="/nlab/show/truncated">truncated</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Lie algebroid is a <strong>Lie <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>-algebroid</strong>;</p> </li> <li> <p>an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Lie algebroid the differential of whose <a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebra">Chevalley-Eilenberg algebra</a> is “co-binary”, i.e. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>:</mo><msup><mi>𝔞</mi> <mo>*</mo></msup><mo>→</mo><msup><mi>a</mi> <mo>*</mo></msup><mo>⊕</mo><msup><mi>a</mi> <mo>*</mo></msup><mo>∧</mo><msup><mi>g</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">d : \mathfrak{a}^* \to a^* \oplus a^* \wedge g^*</annotation></semantics></math>, is <strong>strict</strong>.</p> </li> </ul> <p>So in particular</p> <ul> <li> <p>a 1-Lie algebroid is a <strong><a class="existingWikiWord" href="/nlab/show/Lie+algebroid">Lie algebroid</a></strong>;</p> </li> <li> <p>a 1-Lie algebroid over the point is a <strong><a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a></strong>;</p> </li> <li> <p>a Lie <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>-algebroid over a point is a <strong><a class="existingWikiWord" href="/nlab/show/Lie+n-algebra">Lie n-algebra</a></strong>.</p> </li> <li> <p>a <em><a class="existingWikiWord" href="/nlab/show/BRST-complex">BRST-complex</a></em> is the <a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebra">Chevalley-Eilenberg algebra</a> of an action-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Lie algebroid of the <a class="existingWikiWord" href="/nlab/show/action">action</a> of an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebra, see <a class="existingWikiWord" href="/nlab/show/Lie+%E2%88%9E-algebroid+representation">Lie ∞-algebroid representation</a>;</p> <p>more generally, the complexes appearing in <a class="existingWikiWord" href="/nlab/show/BV-BRST+formalism">BV-BRST formalism</a> are derived <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Lie algebroids, whose Chevalley-Eilenberg algebra may have generators in negative degree.</p> </li> <li> <p>a <a class="existingWikiWord" href="/nlab/show/symplectic+Lie+n-algebroid">symplectic Lie n-algebroid</a> is a Lie <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>-algebroid equipped with a non-degrenerate bilinear <a class="existingWikiWord" href="/nlab/show/invariant+polynomial">invariant polynomial</a> of degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">n+2</annotation></semantics></math>. For low <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> this is</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">n = 0</annotation></semantics></math>: a <a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n = 1</annotation></semantics></math>: a <a class="existingWikiWord" href="/nlab/show/Poisson+Lie+algebroid">Poisson Lie algebroid</a>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">n = 2</annotation></semantics></math>: a <a class="existingWikiWord" href="/nlab/show/Courant+algebroid">Courant algebroid</a></p> </li> </ul> </li> </ul> <h3 id="LieAlgebroidsAsInfinLieAlgebroids">Lie algebroids regarded as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Lie algebroids</h3> <p>We discuss the traditional notion of <a class="existingWikiWord" href="/nlab/show/Lie+algebroid">Lie algebroid</a>s in view of their role as presentations for <a href="#GeneralAbstractDefinition">infinitesimal synthetic differential 1-groupoids</a>.</p> <h4 id="SmoothLociOfInfinitesimalSimplices">Smooth loci of infinitesimal simplices</h4> <p>In this section we characterize ordinary <a class="existingWikiWord" href="/nlab/show/Lie+algebroid">Lie algebroid</a>s <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 \to T X</annotation></semantics></math> as precisely those synthetic differential <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoids that under the <a href="#PresentationByCE">above presentation</a> are locally on any <a class="existingWikiWord" href="/nlab/show/chart">chart</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">U \to X</annotation></semantics></math> of their base space given by simplicial <a class="existingWikiWord" href="/nlab/show/smooth+loci">smooth loci</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>⋯</mi><mover><mover><mover><mo>→</mo><mo>→</mo></mover><mo>→</mo></mover><mo>→</mo></mover><mi>U</mi><mo>×</mo><mover><mi>D</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>rank</mi><mi>E</mi><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo><mover><mover><mo>→</mo><mo>→</mo></mover><mo>→</mo></mover><mi>U</mi><mo>×</mo><mover><mi>D</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>rank</mi><mi>E</mi><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mover><mo>→</mo><mo>→</mo></mover><mi>U</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \cdots \stackrel{\to}{\stackrel{\to}{\stackrel{\to}{\to}}} U \times \tilde D(rank E,2)\stackrel{\to}{\stackrel{\to}{\to}} U \times \tilde D(rank E,1) \stackrel{\to}{\to} U \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>D</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>k</mi><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\tilde D(k,n)</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/smooth+locus">smooth locus</a> of <a class="existingWikiWord" href="/nlab/show/infinitesimal+object">infinitesimal k-simplices</a> based at the origin in <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>. (These smooth loci have been considered in (<a href="#Kock">Kock, section 1.2</a>)).</p> <p>The following definition may be either taken as an informal but instructive definition – in which case the <a href="#FunctionsOnTwiddleD">next definition</a> is to be taken as the precise one – or in fact it may be already itself be taken as the fully formal and precise definition if one reads it in the <a class="existingWikiWord" href="/nlab/show/internal+logic">internal logic</a> of any <a class="existingWikiWord" href="/nlab/show/smooth+topos">smooth topos</a> with line object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> – which for the present purpose is the <a class="existingWikiWord" href="/nlab/show/Cahiers+topos">Cahiers topos</a> with line object <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>. (For an exposition of the latter perspective see (<a href="#Kock">Kock</a>)).</p> <div class="num_defn" id="FunctionsOnTwiddleD"> <h6 id="definition_6">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>,</mo><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">k,n \in \mathbb{N}</annotation></semantics></math>, an <strong>infinitesimal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/simplex">simplex</a></strong> in <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">R^n</annotation></semantics></math> based at the origin is a collection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mover><mi>ϵ</mi><mo stretchy="false">→</mo></mover> <mi>a</mi></msub><mo>∈</mo><msup><mi>R</mi> <mi>n</mi></msup><msubsup><mo stretchy="false">)</mo> <mrow><mi>a</mi><mo>=</mo><mn>1</mn></mrow> <mi>k</mi></msubsup></mrow><annotation encoding="application/x-tex">(\vec \epsilon_a \in R^n)_{a = 1}^k</annotation></semantics></math> of points in <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">R^n</annotation></semantics></math>, such that each is an infinitesimal neighbour of the origin</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>∀</mo><mi>a</mi><mo>:</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msub><mover><mi>ϵ</mi><mo stretchy="false">→</mo></mover> <mi>a</mi></msub><mo>∼</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> \forall a : \;\; \vec \epsilon_a \sim 0 </annotation></semantics></math></div> <p>and such that all are infinitesimal neighbours of each other</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>∀</mo><mi>a</mi><mo>,</mo><mi>a</mi><mo>′</mo><mo>:</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><msub><mover><mi>ϵ</mi><mo stretchy="false">→</mo></mover> <mi>a</mi></msub><mo>−</mo><msub><mover><mi>ϵ</mi><mo stretchy="false">→</mo></mover> <mrow><mi>a</mi><mo>′</mo></mrow></msub><mo stretchy="false">)</mo><mo>∼</mo><mn>0</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \forall a,a': \;\; (\vec \epsilon_a - \vec \epsilon_{a'}) \sim 0 \,. </annotation></semantics></math></div> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>D</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>k</mi><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo><mo>⊂</mo><msup><mi>R</mi> <mrow><mi>k</mi><mo>⋅</mo><mi>n</mi></mrow></msup></mrow><annotation encoding="application/x-tex">\tilde D(k,n) \subset R^{k \cdot n}</annotation></semantics></math> for the space of all such infinitesimal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-simplices in <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">R^n</annotation></semantics></math>.</p> </div> <p>Equivalently:</p> <div class="num_defn" id="FuntionsOnInfinitesimalSimplices"> <h6 id="definition_7">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>,</mo><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">k,n \in \mathbb{N}</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/smooth+algebra">smooth 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><mover><mi>D</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>k</mi><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>∈</mo><mi>SmoothAlg</mi></mrow><annotation encoding="application/x-tex"> C^\infty(\tilde D(k,n)) \in SmoothAlg </annotation></semantics></math></div> <p>is the unique lift through the forgetful functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>:</mo><mi>SmoothAlg</mi><mo>→</mo><msub><mi>CAlg</mi> <mi>ℝ</mi></msub></mrow><annotation encoding="application/x-tex">U : SmoothAlg \to CAlg_{\mathbb{R}}</annotation></semantics></math> of the 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>-algebra generated from <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 \times n</annotation></semantics></math> many generators</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><msubsup><mi>ϵ</mi> <mi>a</mi> <mi>j</mi></msubsup><msub><mo stretchy="false">)</mo> <mrow><mn>1</mn><mo>≤</mo><mi>j</mi><mo>≤</mo><mi>n</mi><mo>,</mo><mn>1</mn><mo>≤</mo><mi>a</mi><mo>≤</mo><mi>k</mi></mrow></msub></mrow><annotation encoding="application/x-tex"> (\epsilon_a^j)_{1 \leq j \leq n, 1 \leq a \leq k} </annotation></semantics></math></div> <p>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><mo>∀</mo><mi>a</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>j</mi><mo>′</mo><mo>:</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msubsup><mi>ϵ</mi> <mi>a</mi> <mi>j</mi></msubsup><msubsup><mi>ϵ</mi> <mi>a</mi> <mrow><mi>j</mi><mo>′</mo></mrow></msubsup><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> \forall a, j,j' : \;\; \epsilon_a^{j} \epsilon_a^{j'} = 0 </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>∀</mo><mi>a</mi><mo>,</mo><mi>a</mi><mo>′</mo><mo>,</mo><mi>j</mi><mo>,</mo><mi>j</mi><mo>′</mo><mo>:</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><msubsup><mi>ϵ</mi> <mi>a</mi> <mi>j</mi></msubsup><mo>−</mo><msubsup><mi>ϵ</mi> <mrow><mi>a</mi><mo>′</mo></mrow> <mi>j</mi></msubsup><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msubsup><mi>ϵ</mi> <mi>a</mi> <mrow><mi>j</mi><mo>′</mo></mrow></msubsup><mo>−</mo><msubsup><mi>ϵ</mi> <mrow><mi>a</mi><mo>′</mo></mrow> <mrow><mi>j</mi><mo>′</mo></mrow></msubsup><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \forall a,a',j,j' : \;\;\; (\epsilon_a^j - \epsilon_{a'}^j) (\epsilon_a^{j'} - \epsilon_{a'}^{j'}) = 0 \,. </annotation></semantics></math></div></div> <div class="num_remark" id="InterpretationOfRelations"> <h6 id="remark_2">Remark</h6> <p>In the above form these relations are the manifest analogs of the conditions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mover><mi>ϵ</mi><mo stretchy="false">→</mo></mover> <mi>a</mi></msub><mo>∼</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\vec \epsilon_a \sim 0</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mover><mi>ϵ</mi><mo stretchy="false">→</mo></mover> <mi>a</mi></msub><mo>−</mo><msub><mover><mi>ϵ</mi><mo stretchy="false">→</mo></mover> <mrow><mi>a</mi><mo>′</mo></mrow></msub><mo stretchy="false">)</mo><mo>∼</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">(\vec \epsilon_a - \vec \epsilon_{a'}) \sim 0</annotation></semantics></math>. But by multiplying out the latter set of relations and using the former, we find that jointly they are equivalent to the single set of relations</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>∀</mo><mi>a</mi><mo>,</mo><mi>a</mi><mo>′</mo><mo>,</mo><mi>j</mi><mo>,</mo><mi>j</mi><mo>′</mo><mo>:</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msubsup><mi>ϵ</mi> <mi>a</mi> <mi>j</mi></msubsup><msubsup><mi>ϵ</mi> <mrow><mi>a</mi><mo>′</mo></mrow> <mrow><mi>j</mi><mo>′</mo></mrow></msubsup><mo>+</mo><msubsup><mi>ϵ</mi> <mrow><mi>a</mi><mo>′</mo></mrow> <mi>j</mi></msubsup><msubsup><mi>ϵ</mi> <mi>a</mi> <mrow><mi>j</mi><mo>′</mo></mrow></msubsup><mo>=</mo><mn>0</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \forall a,a',j,j' : \;\;\; \epsilon_a^j \epsilon_{a'}^{j'} + \epsilon_{a'}^j \epsilon_{a}^{j'} = 0 \,. </annotation></semantics></math></div> <p>In this expression the roles of the two sets of indices is manifestly symmetric. Hence another equivalent way to state the relations is to say</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>∀</mo><mi>a</mi><mo>,</mo><mi>a</mi><mo>′</mo><mo>,</mo><mi>j</mi><mo>:</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msubsup><mi>ϵ</mi> <mi>a</mi> <mi>j</mi></msubsup><msubsup><mi>ϵ</mi> <mrow><mi>a</mi><mo>′</mo></mrow> <mi>j</mi></msubsup><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> \forall a,a', j: \;\;\; \epsilon_a^{j} \epsilon_{a'}^j = 0 </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>∀</mo><mi>a</mi><mo>,</mo><mi>a</mi><mo>′</mo><mo>,</mo><mi>j</mi><mo>,</mo><mi>j</mi><mo>:</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><msubsup><mi>ϵ</mi> <mi>a</mi> <mi>j</mi></msubsup><mo>−</mo><msubsup><mi>ϵ</mi> <mi>a</mi> <mrow><mi>j</mi><mo>′</mo></mrow></msubsup><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msubsup><mi>ϵ</mi> <mrow><mi>a</mi><mo>′</mo></mrow> <mi>j</mi></msubsup><mo>−</mo><msubsup><mi>ϵ</mi> <mrow><mi>a</mi><mo>′</mo></mrow> <mrow><mi>j</mi><mo>′</mo></mrow></msubsup><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> \forall a,a',j,j : \;\;\; (\epsilon_a^j - \epsilon_a^{j'})(\epsilon_{a'}^j - \epsilon_{a'}^{j'}) = 0 </annotation></semantics></math></div></div> <p>This appears as (<a href="#Kock">Kock, (1.2.1)</a>).</p> <p>Since <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><mover><mi>D</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>k</mi><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^\infty(\tilde D(k,n))</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/infinitesimally+thickened+point">Weil algebra</a> in the sense of <a class="existingWikiWord" href="/nlab/show/synthetic+differential+geometry">synthetic differential geometry</a>, its structure as an <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 extends uniquely to the structure of a <a class="existingWikiWord" href="/nlab/show/smooth+algebra">smooth algebra</a> (as discussed there) and we may think of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>D</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>k</mi><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\tilde D(k,n)</annotation></semantics></math> as an infinitesimal <a class="existingWikiWord" href="/nlab/show/smooth+locus">smooth locus</a>.</p> <div class="num_example" id="examplen2k2"> <h6 id="example">Example</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">n = 2</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">k = 2</annotation></semantics></math> we have that <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><mover><mi>D</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^\infty(\tilde D(2,2))</annotation></semantics></math> consists of elements 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 displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mi>f</mi><mo>+</mo><mi>a</mi><mo>⋅</mo><msub><mi>ϵ</mi> <mn>1</mn></msub><mo>+</mo><mi>b</mi><mo>⋅</mo><msub><mi>ϵ</mi> <mn>2</mn></msub><mo>+</mo><mo stretchy="false">(</mo><mi>ω</mi><mo>⋅</mo><msub><mi>ϵ</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>λ</mi><mo>⋅</mo><msub><mi>ϵ</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><mi>f</mi><mo>+</mo><msub><mi>a</mi> <mn>1</mn></msub><msubsup><mi>ϵ</mi> <mn>1</mn> <mn>1</mn></msubsup><mo>+</mo><msub><mi>a</mi> <mn>2</mn></msub><msubsup><mi>ϵ</mi> <mn>1</mn> <mn>2</mn></msubsup><mo>+</mo><msub><mi>b</mi> <mn>1</mn></msub><msubsup><mi>ϵ</mi> <mn>2</mn> <mn>1</mn></msubsup><mo>+</mo><msub><mi>b</mi> <mn>2</mn></msub><msubsup><mi>ϵ</mi> <mn>1</mn> <mn>2</mn></msubsup></mtd></mtr> <mtr><mtd></mtd> <mtd><mo lspace="verythinmathspace" rspace="0em">+</mo><mo stretchy="false">(</mo><msub><mi>ω</mi> <mn>1</mn></msub><msub><mi>λ</mi> <mn>2</mn></msub><mo>−</mo><msub><mi>ω</mi> <mn>2</mn></msub><msub><mi>λ</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo stretchy="false">(</mo><msubsup><mi>ϵ</mi> <mn>1</mn> <mn>1</mn></msubsup><msubsup><mi>ϵ</mi> <mn>2</mn> <mn>2</mn></msubsup><mo>−</mo><msubsup><mi>ϵ</mi> <mn>1</mn> <mn>2</mn></msubsup><msubsup><mi>ϵ</mi> <mn>2</mn> <mn>1</mn></msubsup><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} f + a \cdot \epsilon_1 + b \cdot \epsilon _2 + (\omega \cdot \epsilon_1) (\lambda \cdot \epsilon_2) &= f + a_1 \epsilon_1^1 + a_2 \epsilon_1^2 + b_1 \epsilon_2^1 + b_2 \epsilon_1^2 \\ & + (\omega_1 \lambda_2 - \omega_2 \lambda_1) \frac{1}{2}(\epsilon_1^1 \epsilon_2^2 - \epsilon_1^2 \epsilon_2^1) \end{aligned} </annotation></semantics></math></div> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>∈</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">f \in \mathbb{R}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>ω</mi><mo>,</mo><mi>λ</mi><mo>∈</mo><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><msup><mo stretchy="false">)</mo> <mo>*</mo></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(a, b, \omega, \lambda \in (\mathbb{R}^n)^*)</annotation></semantics></math> a collection of ordinary covectors and with “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⋅</mo></mrow><annotation encoding="application/x-tex">\cdot</annotation></semantics></math>” denoting the evident contraction, and where in the last step we used the above relations.</p> <p>It is noteworthy here that the coefficient of the term which is multilinear in each of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϵ</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\epsilon_i</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/wedge+product">wedge product</a> of two <a class="existingWikiWord" href="/nlab/show/covector">covector</a>s <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi></mrow><annotation encoding="application/x-tex">\lambda</annotation></semantics></math>: we may naturally identify the subspace 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><mover><mi>D</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^\infty(\tilde D(2,2))</annotation></semantics></math> on those elements that vanish if either <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϵ</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\epsilon_1</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϵ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\epsilon_2</annotation></semantics></math> are set to 0 as the space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mo>∧</mo> <mn>2</mn></msup><msubsup><mi>T</mi> <mn>0</mn> <mo>*</mo></msubsup><msup><mi>ℝ</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\wedge^2 T_0^* \mathbb{R}^2</annotation></semantics></math> of 2-forms at the origin of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^2</annotation></semantics></math>.</p> <p>Of course for this identification to be more than a coincidence we need that this is the beginning of a pattern that holds more generally. But this is indeed the case.</p> </div> <p>Let <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> be the set of <em>square</em> submatrices of the <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 \times n</annotation></semantics></math>-matrix <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msubsup><mi>ϵ</mi> <mi>i</mi> <mi>j</mi></msubsup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\epsilon_i^j)</annotation></semantics></math>. As a set this is isomorphic to the set of pairs of subsets of the same size of <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>⋯</mi><mo>,</mo><mi>k</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{1, \cdots, k\}</annotation></semantics></math> and <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>⋯</mi><mo>,</mo><mi>n</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{1, \cdots , n\}</annotation></semantics></math>, respectively. For instance the square submatrix labeled by <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><mn>3</mn><mo>,</mo><mn>4</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{2,3,4\}</annotation></semantics></math> and <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><mn>4</mn><mo>,</mo><mn>5</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{1,4,5\}</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>e</mi><mo>=</mo><mrow><mo>(</mo><mrow><mtable><mtr><mtd><msubsup><mi>ϵ</mi> <mn>1</mn> <mn>2</mn></msubsup></mtd> <mtd><msubsup><mi>ϵ</mi> <mn>4</mn> <mn>2</mn></msubsup></mtd> <mtd><msubsup><mi>ϵ</mi> <mn>5</mn> <mn>2</mn></msubsup></mtd></mtr> <mtr><mtd><msubsup><mi>ϵ</mi> <mn>1</mn> <mn>3</mn></msubsup></mtd> <mtd><msubsup><mi>ϵ</mi> <mn>4</mn> <mn>3</mn></msubsup></mtd> <mtd><msubsup><mi>ϵ</mi> <mn>5</mn> <mn>3</mn></msubsup></mtd></mtr> <mtr><mtd><msubsup><mi>ϵ</mi> <mn>1</mn> <mn>4</mn></msubsup></mtd> <mtd><msubsup><mi>ϵ</mi> <mn>4</mn> <mn>4</mn></msubsup></mtd> <mtd><msubsup><mi>ϵ</mi> <mn>5</mn> <mn>4</mn></msubsup></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> e = \left( \array{ \epsilon_1^2 & \epsilon_4^2 & \epsilon_5^2 \\ \epsilon_1^3 & \epsilon_4^3 & \epsilon_5^3 \\ \epsilon_1^4 & \epsilon_4^4 & \epsilon_5^4 } \right) \,. </annotation></semantics></math></div> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi><mo>∈</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">e \in E</annotation></semantics></math> an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>×</mo><mi>r</mi></mrow><annotation encoding="application/x-tex">r\times r</annotation></semantics></math> submatrix, we write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>det</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mi>σ</mi></munder><mi>sgn</mi><mo stretchy="false">(</mo><mi>σ</mi><mo stretchy="false">)</mo><msubsup><mi>ϵ</mi> <mn>1</mn> <mrow><mi>σ</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msubsup><msubsup><mi>ϵ</mi> <mn>2</mn> <mrow><mi>σ</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></msubsup><mi>⋯</mi><msubsup><mi>ϵ</mi> <mi>r</mi> <mrow><mi>σ</mi><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></mrow></msubsup><mo>∈</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mover><mi>D</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>k</mi><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> det(e) = \sum_{\sigma} sgn(\sigma) \epsilon_{1}^{\sigma(1)} \epsilon_2^{\sigma(2)} \cdots \epsilon_r^{\sigma(r)} \in C^\infty(\tilde D(k,n)) \,. </annotation></semantics></math></div> <p>for the corresponding <a class="existingWikiWord" href="/nlab/show/determinant">determinant</a>, given as a product of generators in <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><mover><mi>D</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>k</mi><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^\infty(\tilde D(k,n))</annotation></semantics></math>. Here the sum runs over all <a class="existingWikiWord" href="/nlab/show/permutation">permutation</a>s <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> of <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>⋯</mi><mo>,</mo><mi>r</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{1, \cdots, r\}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sgn</mi><mo stretchy="false">(</mo><mi>σ</mi><mo stretchy="false">)</mo><mo>∈</mo><mo stretchy="false">{</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>1</mn><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><mo stretchy="false">}</mo><mo>⊂</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">sgn(\sigma) \in \{+1, -1\} \subset \mathbb{R}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/signature">signature</a> of the permutation <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>.</p> <div class="num_prop" id="FunctionsOnSimplicesByMatrices"> <h6 id="proposition_6">Proposition</h6> <p>The elements <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><mover><mi>D</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>k</mi><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f \in C^\infty(\tilde D(k,n))</annotation></semantics></math> are precisely 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><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>e</mi><mo>∈</mo><mi>E</mi></mrow></munder><msub><mi>f</mi> <mi>e</mi></msub><mspace width="thickmathspace"></mspace><mi>det</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> f = \sum_{e \in E} f_e \; det(e) </annotation></semantics></math></div> <p>for <em>unique</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>f</mi> <mi>e</mi></msub><mo>∈</mo><mi>ℝ</mi><mo stretchy="false">|</mo><mi>e</mi><mo>∈</mo><mi>E</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{f_e \in \mathbb{R} | e \in E\}</annotation></semantics></math>. In other words, the map of <a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a>s</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><mo stretchy="false">|</mo><mi>E</mi><mo stretchy="false">|</mo></mrow></msup><mo>→</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mover><mi>D</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>k</mi><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathbb{R}^{|E|} \to C^\infty(\tilde D(k,n)) </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><mo stretchy="false">(</mo><msub><mi>f</mi> <mi>e</mi></msub><msub><mo stretchy="false">)</mo> <mrow><mi>e</mi><mo>∈</mo><mi>E</mi></mrow></msub><mo>↦</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>e</mi><mo>∈</mo><mi>E</mi></mrow></munder><msub><mi>f</mi> <mi>e</mi></msub><mi>det</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (f_e)_{e \in E} \mapsto \sum_{e \in E} f_e det(e) </annotation></semantics></math></div> <p>is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>.</p> </div> <div class="proof"> <h6 id="proof_6">Proof</h6> <p>This is a direct extension of the argument in the above example: a general product of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math> generators in <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><mover><mi>D</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>k</mi><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^\infty(\tilde D(k,n))</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><msubsup><mi>ϵ</mi> <mrow><msub><mi>i</mi> <mn>1</mn></msub></mrow> <mrow><msub><mi>j</mi> <mn>1</mn></msub></mrow></msubsup><msubsup><mi>ϵ</mi> <mrow><msub><mi>i</mi> <mn>2</mn></msub></mrow> <mrow><msub><mi>j</mi> <mn>2</mn></msub></mrow></msubsup><mi>⋯</mi><msubsup><mi>ϵ</mi> <mrow><msub><mi>i</mi> <mi>r</mi></msub></mrow> <mrow><msub><mi>j</mi> <mi>r</mi></msub></mrow></msubsup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \epsilon_{i_1}^{j_1} \epsilon_{i_2}^{j_2} \cdots \epsilon_{i_r}^{j_r} \,. </annotation></semantics></math></div> <p>By the relations in <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><mover><mi>D</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>k</mi><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^\infty(\tilde D(k,n))</annotation></semantics></math>, this is non-vanishing precisely if none of 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">i</annotation></semantics></math>-indices repeats and none of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math>-indices repeats. Furthermore by the relations, for any permutation <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> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math> elements, this is equal to</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>⋯</mi><mo>=</mo><mi>sgn</mi><mo stretchy="false">(</mo><mi>σ</mi><mo stretchy="false">)</mo><msubsup><mi>ϵ</mi> <mrow><msub><mi>i</mi> <mn>1</mn></msub></mrow> <mrow><msub><mi>j</mi> <mrow><mi>σ</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub></mrow></msubsup><msubsup><mi>ϵ</mi> <mrow><msub><mi>i</mi> <mn>1</mn></msub></mrow> <mrow><msub><mi>j</mi> <mrow><mi>σ</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></msub></mrow></msubsup><mi>⋯</mi><msubsup><mi>ϵ</mi> <mrow><msub><mi>i</mi> <mn>1</mn></msub></mrow> <mrow><msub><mi>j</mi> <mrow><mi>σ</mi><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></mrow></msub></mrow></msubsup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \cdots = sgn(\sigma) \epsilon_{i_1}^{j_{\sigma(1)}} \epsilon_{i_1}^{j_{\sigma(2)}} \cdots \epsilon_{i_1}^{j_{\sigma(r)}} \,. </annotation></semantics></math></div> <p>It follows that each such element 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>⋯</mi><mo>=</mo><mfrac><mn>1</mn><mrow><mi>r</mi><mo>!</mo></mrow></mfrac><mi>det</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \cdots = \frac{1}{r!} det(e) \,, </annotation></semantics></math></div> <p>where <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 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>×</mo><mi>r</mi></mrow><annotation encoding="application/x-tex">r \times r</annotation></semantics></math> sub-<a class="existingWikiWord" href="/nlab/show/determinant">determinant</a> given by the subset <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>i</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>i</mi> <mi>r</mi></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{i_1, \cdots, i_r\}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>j</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>j</mi> <mi>r</mi></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\{j_1, \cdots, j_r\})</annotation></semantics></math> as discussed above.</p> </div> <p>In (<a href="#Kock">Kock, section 1.3</a>) effectively this proposition appears as the “<a class="existingWikiWord" href="/nlab/show/Kock-Lawvere+axiom">Kock-Lawvere axiom</a> scheme for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>D</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>k</mi><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\tilde D(k,n)</annotation></semantics></math>” when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>D</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>k</mi><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\tilde D(k,n)</annotation></semantics></math> is regarded as an object of a suitable <a class="existingWikiWord" href="/nlab/show/smooth+topos">smooth topos</a>.</p> <div class="num_prop" id="TwiddleDsAsDOldKan"> <h6 id="proposition_7">Proposition</h6> <p>For any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>,</mo><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">k,n \in \mathbb{N}</annotation></semantics></math> we have a natural <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> of real commutative and hence of <a class="existingWikiWord" href="/nlab/show/smooth+algebra">smooth algebra</a>s</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>:</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mover><mi>D</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>k</mi><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>→</mo><mo>≃</mo></mover><msubsup><mo>⊕</mo> <mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow> <mi>n</mi></msubsup><mo stretchy="false">(</mo><msup><mo>∧</mo> <mi>i</mi></msup><msup><mi>ℝ</mi> <mi>k</mi></msup><mo stretchy="false">)</mo><mo>⊗</mo><mo stretchy="false">(</mo><msup><mo>∧</mo> <mi>i</mi></msup><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \phi : C^\infty(\tilde D(k,n)) \stackrel{\simeq}{\to} \oplus_{i = 0}^n (\wedge^i \mathbb{R}^k) \otimes (\wedge^i \mathbb{R}^n) \,, </annotation></semantics></math></div> <p>where on the right we have the algebras that appear degreewise in def. <a class="maruku-ref" href="#PresentationByMonoidalDoldKan"></a>, where the product is given on homogeneous elements by</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>ω</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><mi>λ</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>ω</mi><mo>∧</mo><mi>λ</mi><mo>,</mo><mi>x</mi><mo>∧</mo><mi>y</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (\omega, x) \cdot (\lambda, y) = (\omega \wedge \lambda , x \wedge y) \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_7">Proof</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>t</mi> <mi>a</mi></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{t_a\}</annotation></semantics></math> be the canonical basis for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>k</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^k</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msup><mi>e</mi> <mi>i</mi></msup><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{e^i\}</annotation></semantics></math> the canonical basis 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>. We claim that an isomorphism 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><mi>ϕ</mi><mo>:</mo><msubsup><mi>ϵ</mi> <mi>a</mi> <mi>i</mi></msubsup><mo>↦</mo><mo stretchy="false">(</mo><msub><mi>t</mi> <mi>a</mi></msub><mo>,</mo><msup><mi>e</mi> <mi>i</mi></msup><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \phi : \epsilon^i_a \mapsto (t_a , e^i) \,. </annotation></semantics></math></div> <p>To see that this defines indeed an algebra homomorphism we need to check that it respects the relations on the generators. For this compute:</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>ϕ</mi><mo stretchy="false">(</mo><msubsup><mi>ϵ</mi> <mi>a</mi> <mi>i</mi></msubsup><msubsup><mi>ϵ</mi> <mrow><mi>a</mi><mo>′</mo></mrow> <mrow><mi>i</mi><mo>′</mo></mrow></msubsup><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><mo stretchy="false">(</mo><msub><mi>t</mi> <mi>a</mi></msub><mo>∧</mo><msub><mi>t</mi> <mrow><mi>a</mi><mo>′</mo></mrow></msub><mo>,</mo><msup><mi>e</mi> <mi>i</mi></msup><mo>∧</mo><msup><mi>e</mi> <mrow><mi>i</mi><mo>′</mo></mrow></msup><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">(</mo><msub><mi>t</mi> <mrow><mi>a</mi><mo>′</mo></mrow></msub><mo>∧</mo><msub><mi>t</mi> <mi>a</mi></msub><mo>,</mo><msup><mi>e</mi> <mi>i</mi></msup><mo>∧</mo><msup><mi>e</mi> <mrow><mi>i</mi><mo>′</mo></mrow></msup><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>ϕ</mi><mo stretchy="false">(</mo><msubsup><mi>ϵ</mi> <mrow><mi>a</mi><mo>′</mo></mrow> <mi>i</mi></msubsup><msubsup><mi>ϵ</mi> <mi>a</mi> <mrow><mi>i</mi><mo>′</mo></mrow></msubsup><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} \phi(\epsilon_a^i \epsilon_{a'}^{i'}) & = (t_a \wedge t_{a'}, e^i \wedge e^{i'}) \\ & = -(t_{a'} \wedge t_{a}, e^i \wedge e^{i'}) \\ & = -\phi(\epsilon_{a'}^i \epsilon_{a}^{i'}) \end{aligned} \,. </annotation></semantics></math></div> <p>The inverse clearly exists, given on generators by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>ϕ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>:</mo><mo stretchy="false">(</mo><msub><mi>t</mi> <mi>a</mi></msub><mo>,</mo><msup><mi>e</mi> <mi>i</mi></msup><mo stretchy="false">)</mo><mo>↦</mo><msubsup><mi>ϵ</mi> <mi>a</mi> <mi>i</mi></msubsup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \phi^{-1} : (t_a, e^i) \mapsto \epsilon_a^i \,. </annotation></semantics></math></div></div> <div class="num_cor" id="SimplicialSmoothLocusOfLieAlgebroid"> <h6 id="corollary_2">Corollary</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔞</mi><mo>∈</mo><msub><mi>L</mi> <mn>∞</mn></msub><mi>Alg</mi></mrow><annotation encoding="application/x-tex">\mathfrak{a} \in L_\infty Alg</annotation></semantics></math> a 1-truncated object, hence an ordinary <a class="existingWikiWord" href="/nlab/show/Lie+algebroid">Lie algebroid</a> of rank <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> over a 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>, its image under the map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>:</mo><msub><mi>L</mi> <mn>∞</mn></msub><mi>Alg</mi><mo>→</mo><mo stretchy="false">(</mo><msup><mi>SmoothAlg</mi> <mi>Δ</mi></msup><msup><mo stretchy="false">)</mo> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">i : L_\infty Alg \to (SmoothAlg^\Delta)^{op}</annotation></semantics></math>, def. <a class="maruku-ref" href="#EmbeddingOfThePresentation"></a>, is such that its restriction to any <a class="existingWikiWord" href="/nlab/show/chart">chart</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">U \to X</annotation></semantics></math> is, up to <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</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>i</mi><mo stretchy="false">(</mo><mi>𝔞</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">|</mo> <mi>U</mi></msub><mo>:</mo><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>↦</mo><mi>U</mi><mo>×</mo><mover><mi>D</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>k</mi><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> i(\mathfrak{a})|_U : [n] \mapsto U \times \tilde D(k,n) \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_8">Proof</h6> <p>Apply prop. <a class="maruku-ref" href="#TwiddleDsAsDOldKan"></a> in def. <a class="maruku-ref" href="#PresentationByMonoidalDoldKan"></a>, using that by definition <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CE</mi><mo stretchy="false">(</mo><mi>𝔞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">CE(\mathfrak{a})</annotation></semantics></math> is given by the exterior algebra on locally free <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> modules, so that</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>CE</mi><mo stretchy="false">(</mo><mi>𝔞</mi><msub><mo stretchy="false">|</mo> <mi>U</mi></msub><mo stretchy="false">)</mo></mtd> <mtd><mo>≃</mo><mo stretchy="false">(</mo><msubsup><mo>∧</mo> <mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mrow> <mo>•</mo></msubsup><mi>Γ</mi><mo stretchy="false">(</mo><mi>U</mi><mo>×</mo><msup><mi>ℝ</mi> <mi>k</mi></msup><msup><mo stretchy="false">)</mo> <mo>*</mo></msup><mo>,</mo><msub><mi>d</mi> <mrow><mi>𝔞</mi><msub><mo stretchy="false">|</mo> <mi>U</mi></msub></mrow></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><mo stretchy="false">(</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo>⊗</mo><msup><mo>∧</mo> <mo>•</mo></msup><msup><mi>ℝ</mi> <mi>k</mi></msup><mo>,</mo><msub><mi>d</mi> <mrow><mi>𝔞</mi><msub><mo stretchy="false">|</mo> <mi>U</mi></msub></mrow></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} CE(\mathfrak{a}|_U) & \simeq (\wedge^\bullet_{C^\infty(U)} \Gamma(U\times \mathbb{R}^k)^*, d_{\mathfrak{a}|_U}) \\ & \simeq (C^\infty(U) \otimes \wedge^\bullet \mathbb{R}^k, d_{\mathfrak{a}|_U}) \end{aligned} \,. </annotation></semantics></math></div></div> <h4 id="TangentLieAlgebroid">Tangent Lie algebroid</h4> <p>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>, there is a standard notion of the <a class="existingWikiWord" href="/nlab/show/Lie+algebroid">Lie algebroid</a> which is the <a class="existingWikiWord" href="/nlab/show/tangent+Lie+algebroid">tangent Lie algebroid</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝔞</mi><mo>=</mo><mi>T</mi><mi>X</mi></mrow><annotation encoding="application/x-tex"> \mathfrak{a} = T X </annotation></semantics></math></div> <p>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>. We discuss this from the perspective of infinitesimal groupoids.</p> <div class="num_defn" id="InfinitesimalSingularSimplicialComplex"> <h6 id="definition_8">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>∈</mo><msub><mi>CartSp</mi> <mi>synthdiff</mi></msub></mrow><annotation encoding="application/x-tex">U \in CartSp_{synthdiff}</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/infinitesimal+singular+simplicial+complex">infinitesimal singular simplicial complex</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>X</mi> <mrow><mo stretchy="false">(</mo><msubsup><mi>Δ</mi> <mi>inf</mi> <mo>•</mo></msubsup><mo stretchy="false">)</mo></mrow></msup></mrow><annotation encoding="application/x-tex">X^{(\Delta^\bullet_{inf})}</annotation></semantics></math> is the simplicial <a class="existingWikiWord" href="/nlab/show/smooth+locus">smooth locus</a> which in terms 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> is the space 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><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(k+1)</annotation></semantics></math>-tuples of pairwise infinitesimal neighbour points in <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> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>U</mi> <mrow><mo stretchy="false">(</mo><msubsup><mi>Δ</mi> <mi>inf</mi> <mi>n</mi></msubsup><mo stretchy="false">)</mo></mrow></msup><mo>=</mo><mrow><mo>{</mo><msub><mi>x</mi> <mrow><msub><mi>i</mi> <mn>0</mn></msub></mrow></msub><mo>,</mo><mi>⋯</mi><msub><mi>x</mi> <mrow><msub><mi>i</mi> <mi>n</mi></msub></mrow></msub><mo>∈</mo><mi>U</mi><mo stretchy="false">|</mo><mo>∀</mo><mi>r</mi><mo>,</mo><mi>s</mi><mo>:</mo><msub><mi>x</mi> <mrow><msub><mi>i</mi> <mi>r</mi></msub></mrow></msub><mo>∼</mo><msub><mi>x</mi> <mrow><msub><mi>i</mi> <mi>s</mi></msub></mrow></msub><mo>}</mo></mrow></mrow><annotation encoding="application/x-tex"> U^{(\Delta^n_{inf})} = \left\{ x_{i_0}, \cdots x_{i_n} \in U | \forall r,s : x_{i_r} \sim x_{i_s} \right\} </annotation></semantics></math></div> <p>and whose face and degeneracy maps are as for the finite <a class="existingWikiWord" href="/nlab/show/singular+simplicial+complex">singular simplicial complex</a>.</p> <p>More explicitly, in terms of the spaces from def. <a class="maruku-ref" href="#FunctionsOnTwiddleD"></a> we may identify</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>U</mi> <mrow><mo stretchy="false">(</mo><msubsup><mi>Δ</mi> <mi>inf</mi> <mo>•</mo></msubsup><mo stretchy="false">)</mo></mrow></msup><mo>=</mo><mrow><mo>(</mo><mi>⋯</mi><mi>U</mi><mo>×</mo><mover><mi>D</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>dim</mi><mi>U</mi><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo><mover><mover><mo>→</mo><mo>→</mo></mover><mo>→</mo></mover><mi>U</mi><mo>×</mo><mover><mi>D</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>dim</mi><mi>U</mi><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mover><mo>→</mo><mo>→</mo></mover><mi>U</mi><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> U^{(\Delta^\bullet_{inf})} = \left( \cdots U \times \tilde D(dim U, 2) \stackrel{\to}{\stackrel{\to}{\to}}U \times \tilde D(dim U, 1)\stackrel{\to}{\to} U \right) \,, </annotation></semantics></math></div> <p>where 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> a <a class="existingWikiWord" href="/nlab/show/generalized+element">generalized element</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><mo stretchy="false">(</mo><msub><mover><mi>ϵ</mi><mo stretchy="false">→</mo></mover> <mi>a</mi></msub><msub><mo stretchy="false">)</mo> <mrow><mi>a</mi><mo>=</mo><mn>1</mn><mo>,</mo><mi>⋯</mi><mo>,</mo><mi>dim</mi><mi>U</mi></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x, (\vec \epsilon_a)_{a = 1, \cdots, dim U})</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>×</mo><mover><mi>D</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>dim</mi><mi>U</mi><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U \times \tilde D(dim U, n)</annotation></semantics></math> is thought of as a base point <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 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>dim</mi><mi>U</mi></mrow><annotation encoding="application/x-tex">dim U</annotation></semantics></math> infinitesimal paths starting at that basepoint</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>0</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>x</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>x</mi><mo>+</mo><msub><mi>ϵ</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><mi>x</mi><mo>+</mo><msub><mi>ϵ</mi> <mrow><mi>dim</mi><mi>U</mi></mrow></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (x_0, \cdots, x_n) = ( x, x + \epsilon_1, \cdots, x + \epsilon_{dim U} ) \,. </annotation></semantics></math></div> <p>The dual cosimplicial algebra is read off from this,</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>U</mi> <mrow><mo stretchy="false">(</mo><msubsup><mi>Δ</mi> <mi>inf</mi> <mo>•</mo></msubsup><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>(</mo><mi>⋯</mi><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>U</mi><mo>×</mo><mover><mi>D</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>dim</mi><mi>U</mi><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><munder><mo>←</mo><mrow><msubsup><mi>d</mi> <mn>1</mn> <mo>*</mo></msubsup></mrow></munder><mover><mo>←</mo><mrow><msubsup><mi>d</mi> <mn>0</mn> <mo>*</mo></msubsup></mrow></mover></mover><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> C^\infty(U^{(\Delta^\bullet_{inf})}) = \left( \cdots C^\infty(U \times \tilde D(dim U ,1)) \stackrel{\overset{d_0^*}{\leftarrow}}{\underset{d_1^*}{\leftarrow}} C^\infty(U) \right) \,. </annotation></semantics></math></div> <p>For instance for <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><mi>U</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f \in C^\infty(U)</annotation></semantics></math> we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>d</mi> <mn>1</mn> <mo>*</mo></msubsup><mi>f</mi><mo>=</mo><mi>f</mi></mrow><annotation encoding="application/x-tex">d_1^* f = f </annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msubsup><mi>d</mi> <mn>2</mn> <mo>*</mo></msubsup><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><msub><mi>ϵ</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><msub><mi>ϵ</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><mfrac><mrow><mo>∂</mo><mi>f</mi></mrow><mrow><mo>∂</mo><msup><mi>x</mi> <mi>i</mi></msup></mrow></mfrac><msubsup><mi>ϵ</mi> <mn>1</mn> <mi>i</mi></msubsup></mrow><annotation encoding="application/x-tex">(d_2^* f)(x,\epsilon_1) = f(x + \epsilon_1) = f(x) + \frac{\partial f}{\partial x^i} \epsilon^i_1</annotation></semantics></math>.</p> </div> <div class="num_not" id="InfinitesimalSingularNotAKanComplex"> <h6 id="note">Note</h6> <p>The object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>X</mi> <mrow><mo stretchy="false">(</mo><msubsup><mi>Δ</mi> <mi>inf</mi> <mo>•</mo></msubsup><mo stretchy="false">)</mo></mrow></msup></mrow><annotation encoding="application/x-tex">X^{(\Delta^\bullet_{inf})}</annotation></semantics></math> is not objectwise a <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a>: in general the composite of two first order neighbours produces a second order infinitesimal neighbour. Its <a class="existingWikiWord" href="/nlab/show/Kan+fibrant+replacement">Kan fibrant replacement</a> may be thought of as the infinitesikmal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoid whose morphisms are paths of a finite number of first order infinitesimal steps.</p> </div> <div class="num_prop" id="TangentLieAlgebroidAsSimplicialObject"> <h6 id="proposition_8">Proposition</h6> <p>The image of <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> under the embedding <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> from def. <a class="maruku-ref" href="#EmbeddingOfThePresentation"></a> is the <a class="existingWikiWord" href="/nlab/show/simplicial+object">simplicial</a> <a class="existingWikiWord" href="/nlab/show/smooth+locus">smooth locus</a> given by the <a class="existingWikiWord" href="/nlab/show/infinitesimal+singular+simplicial+complex">infinitesimal singular simplicial complex</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>T</mi><mi>X</mi><mo>=</mo><mrow><mo>(</mo><mi>⋯</mi><msup><mi>X</mi> <mrow><mo stretchy="false">(</mo><msubsup><mi>Δ</mi> <mi>inf</mi> <mn>2</mn></msubsup><mo stretchy="false">)</mo></mrow></msup><mover><mover><mo>→</mo><mo>→</mo></mover><mo>→</mo></mover><msup><mi>X</mi> <mrow><mo stretchy="false">(</mo><msubsup><mi>Δ</mi> <mi>inf</mi> <mn>1</mn></msubsup><mo stretchy="false">)</mo></mrow></msup><mover><mo>→</mo><mo>→</mo></mover><mi>X</mi><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> T X = \left( \cdots X^{(\Delta^2_{inf})} \stackrel{\to}{\stackrel{\to}{\to}} X^{(\Delta^1_{inf})} \stackrel{\to}{\to} X \right) </annotation></semantics></math></div> <p>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>.</p> <p>Moreover, the <a href="#Cohomology">intrinsic real cohomology</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo stretchy="false">(</mo><mi>T</mi><mi>X</mi><mo stretchy="false">)</mo><mo>∈</mo></mrow><annotation encoding="application/x-tex">i(T X) \in </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/SynthDiff%E2%88%9EGrpd">SynthDiff∞Grpd</a> is the <a class="existingWikiWord" href="/nlab/show/de+Rham+cohomology">de Rham cohomology</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></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>H</mi> <mi>SynthDiff</mi> <mi>n</mi></msubsup><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">(</mo><mi>T</mi><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><mi>ℝ</mi><mo stretchy="false">)</mo><mo>≃</mo><msubsup><mi>H</mi> <mi>dR</mi> <mi>n</mi></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> H^n_{SynthDiff}(i (T X), \mathbb{R}) \simeq H^n_{dR}(X) </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_9">Proof</h6> <p>The first statement may be checked locally on any <a class="existingWikiWord" href="/nlab/show/chart">chart</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">U \to X</annotation></semantics></math> where it follows from prop. <a class="maruku-ref" href="#SimplicialSmoothLocusOfLieAlgebroid"></a>. Since the <a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebra">Chevalley-Eilenberg algebra</a> of the <a class="existingWikiWord" href="/nlab/show/tangent+Lie+algebroid">tangent Lie algebroid</a> is the <a class="existingWikiWord" href="/nlab/show/de+Rham+complex">de Rham complex</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><mi>T</mi><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><msup><mi>Ω</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>d</mi> <mi>dR</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> CE(T X) = (\Omega^\bullet(X), d_{dR}) </annotation></semantics></math></div> <p>the second statement follows with prop. <a class="maruku-ref" href="#IntrinsicRealCohomologyByCECohomology"></a>.</p> </div> <h4 id="LieAlgebra">Lie algebra</h4> <p>Let <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">G</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a> with <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>. We describe how <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> looks when regarded as a special case of an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Lie algebroid.</p> <p>Write</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>B</mi></mstyle><mi>G</mi><mo>=</mo><mrow><mo>(</mo><mi>⋯</mi><mi>G</mi><mo>×</mo><mi>G</mi><mover><mover><mo>→</mo><mo>→</mo></mover><mo>→</mo></mover><mi>G</mi><mover><mo>→</mo><mo>→</mo></mover><mo>*</mo><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> \mathbf{B}G = \left( \cdots G \times G \stackrel{\to}{\stackrel{\to}{\to}} G \stackrel{\to}{\to} * \right) </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</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">G</annotation></semantics></math>, regarded as an an <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+groupoid">∞-Lie groupoid</a> modeled by a simplicial <a class="existingWikiWord" href="/nlab/show/smooth+space">smooth space</a>.</p> <p>We claim that a morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo>:</mo><mi>T</mi><mi>U</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex"> \omega : T U \to \mathbf{B}G </annotation></semantics></math></div> <p>from the tangent Lie algebroid of some <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>∈</mo></mrow><annotation encoding="application/x-tex">U \in </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/CartSp">CartSp</a> is <a class="existingWikiWord" href="/nlab/show/groupoid+of+Lie-algebra+valued+forms">flat Lie-algebra valued form</a> and how that can be used to find the <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> as the infinitesimal sub-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoid</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>b</mi></mstyle><mi>𝔤</mi><mo>↪</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex"> \mathbf{b}\mathfrak{g} \hookrightarrow \mathbf{B}G </annotation></semantics></math></div> <p>inside <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math>.</p> <p>Since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/simplicial+skeleton">2-coskeletal</a> (being the <a class="existingWikiWord" href="/nlab/show/nerve">nerve</a> of a <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a>) a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mi>U</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">T U \to \mathbf{B}G</annotation></semantics></math> is fixed already under its 2-<a class="existingWikiWord" href="/nlab/show/truncated">truncation</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>U</mi><mo>×</mo><mover><mi>D</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>ω</mi> <mn>2</mn></msub></mrow></mover></mtd> <mtd><mi>G</mi><mo>×</mo><mi>G</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo><mo stretchy="false">↓</mo><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mrow><msub><mi>p</mi> <mn>2</mn></msub></mrow></msup><msup><mo stretchy="false">↓</mo> <mo>⋅</mo></msup><msup><mo stretchy="false">↓</mo> <mrow><msub><mi>p</mi> <mn>1</mn></msub></mrow></msup></mtd></mtr> <mtr><mtd><mi>U</mi><mo>×</mo><mover><mi>D</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>ω</mi> <mn>1</mn></msub></mrow></mover></mtd> <mtd><mi>G</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>U</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>*</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ U \times \tilde D(2,n) &\stackrel{\omega_2}{\to}& G \times G \\ \downarrow \downarrow \downarrow && \downarrow^{p_2} \downarrow^{\cdot} \downarrow^{p_1} \\ U \times \tilde D(1,n) &\stackrel{\omega_1}{\to}& G \\ \downarrow \downarrow && \downarrow \downarrow \\ U &\to& * } \,. </annotation></semantics></math></div> <p>It is clear that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ω</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\omega_1</annotation></semantics></math> factors through the inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>D</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mi>dim</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>↪</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">\tilde D(1,dim(G)) \hookrightarrow G</annotation></semantics></math> that sends the unique point of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>D</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mi>dim</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\tilde D(1, dim(G))</annotation></semantics></math> to the neutral element (by respect for the degeneracy maps). Then from that one finds that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ω</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\omega_2</annotation></semantics></math> factors through the inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>D</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mi>dim</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>↪</mo><mi>G</mi><mo>×</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">\tilde D(2, dim(G)) \hookrightarrow G \times G</annotation></semantics></math> that sends the unique point of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>D</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mi>dim</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\tilde D(2,dim(G))</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>G</mi><mo>×</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">(e,e) \in G \times G</annotation></semantics></math>. And evidently these two factorizations are universal, in that every other factorization will uniquely factor through these</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>U</mi><mo>×</mo><mover><mi>D</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>ω</mi> <mn>2</mn></msub></mrow></mover></mtd> <mtd><mover><mi>D</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mi>dim</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mo>↪</mo></mtd> <mtd><mi>G</mi><mo>×</mo><mi>G</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo><mo stretchy="false">↓</mo><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mrow><msub><mi>p</mi> <mn>2</mn></msub></mrow></msup><msup><mo stretchy="false">↓</mo> <mo>⋅</mo></msup><msup><mo stretchy="false">↓</mo> <mrow><msub><mi>p</mi> <mn>1</mn></msub></mrow></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mrow><msub><mi>p</mi> <mn>2</mn></msub></mrow></msup><msup><mo stretchy="false">↓</mo> <mo>⋅</mo></msup><msup><mo stretchy="false">↓</mo> <mrow><msub><mi>p</mi> <mn>1</mn></msub></mrow></msup></mtd></mtr> <mtr><mtd><mi>U</mi><mo>×</mo><mover><mi>D</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>ω</mi> <mn>1</mn></msub></mrow></mover></mtd> <mtd><mover><mi>D</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mi>dim</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mo>↪</mo></mtd> <mtd><mi>G</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>U</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>*</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>*</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ U \times \tilde D(2,n) &\stackrel{\omega_2}{\to}& \tilde D(2,dim(G)) &\hookrightarrow& G \times G \\ \downarrow \downarrow \downarrow && \downarrow^{p_2} \downarrow^{\cdot} \downarrow^{p_1} && \downarrow^{p_2} \downarrow^{\cdot} \downarrow^{p_1} \\ U \times \tilde D(1,n) &\stackrel{\omega_1}{\to}& \tilde D(1, dim(G)) &\hookrightarrow& G \\ \downarrow \downarrow && \downarrow \downarrow && \downarrow \downarrow \\ U &\to& * &\to& * } \,. </annotation></semantics></math></div> <p>The universal object found this way we claim is the Lie algebra <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> in its incarnation as an infinitesimal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Lie groupoid</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>b</mi><mi>𝔤</mi></mtd> <mtd><mo>:</mo><mo>=</mo><mi>InitialObject</mi><mo stretchy="false">(</mo><mi>T</mi><mi>U</mi><mo stretchy="false">↓</mo><msup><mi>𝕃</mi> <mrow><msup><mi>Δ</mi> <mi>op</mi></msup></mrow></msup><mo stretchy="false">↓</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mrow><mo>(</mo><mi>⋯</mi><mover><mi>D</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mi>dim</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mover><mo>→</mo><mo>→</mo></mover><mo>→</mo></mover><mover><mi>D</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mi>dim</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>→</mo><mo>→</mo></mover><mo>*</mo><mo>)</mo></mrow></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} b \mathfrak{g} &:= InitialObject( T U\downarrow \mathbb{L}^{\Delta^{op}}\downarrow \mathbf{B}G) \\ & = \left( \cdots \tilde D(2,dim(G)) \stackrel{\to}{\stackrel{\to}{\to}} \tilde D(1,dim(G))\stackrel{\to}{\to} * \right) \end{aligned} \,. </annotation></semantics></math></div> <div class="num_prop" id="CEAlgebraFromNormalizedCochains"> <h6 id="proposition_9">Proposition</h6> <p>The normalized cochain complex of the cosimplicial alghebra of functions on this <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">b \mathfrak{g}</annotation></semantics></math> is isomorphic to the ordinary <a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebra">Chevalley-Eilenberg algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mo>∧</mo> <mo>•</mo></msup><msup><mi>𝔤</mi> <mo>*</mo></msup><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 stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\wedge^\bullet \mathfrak{g}^*, [-,-]^*)</annotation></semantics></math> 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>.</p> </div> <div class="proof"> <h6 id="proof_10">Proof</h6> <p>By the <a href="#spring">above discussion</a> we have that for <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><mover><mi>D</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>k</mi><mo>,</mo><mi>dim</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msub><mo stretchy="false">)</mo> <mi>top</mi></msub><mo>⊂</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mover><mi>D</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>k</mi><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^\infty(\tilde D(k,dim(G)))_{top} \subset C^\infty(\tilde D(k,n))</annotation></semantics></math> the subspace of those functions that are in the joint kernel of the co-degeneracy maps is naturally isomorpic to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mo>∧</mo> <mi>k</mi></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mi>dim</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow></msup><msup><mo stretchy="false">)</mo> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">\wedge^k (\mathbb{R}^{dim(G)})^*</annotation></semantics></math>, so that we have a natural isomorphism of vector spaces</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>N</mi><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>b</mi><mi>𝔤</mi><msub><mo stretchy="false">)</mo> <mi>k</mi></msub><mo>≃</mo><msup><mo>∧</mo> <mi>k</mi></msup><msup><mi>𝔤</mi> <mo>*</mo></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> N C^\infty(b \mathfrak{g})_k \simeq \wedge^k \mathfrak{g}^* \,. </annotation></semantics></math></div> <p>By the fact that everything is <a class="existingWikiWord" href="/nlab/show/simplicial+skeleton">2-coskeletal</a> it suffices to check that the differential in first degree</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>N</mi><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mover><mi>D</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mi>dim</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><msubsup><mi>p</mi> <mn>1</mn> <mo>*</mo></msubsup><mo lspace="verythinmathspace" rspace="0em">+</mo><msubsup><mi>p</mi> <mn>2</mn> <mo>*</mo></msubsup><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">(</mo><mo>⋅</mo><msup><mo stretchy="false">)</mo> <mo>*</mo></msup></mrow></mover><mi>N</mi><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mover><mi>D</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mi>dim</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> N C^\infty(\tilde D(1,dim(G))) \stackrel{p_1^* + p_2^* - (\cdot)^*}{\to} N C^\infty(\tilde D(2,dim(G))) </annotation></semantics></math></div> <p>is indeed the dual of the Lie bracket. But the product <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>⋅</mo> <mi>G</mi></msub><mo>:</mo><mi>G</mi><mo>×</mo><mi>G</mi><mo>→</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">\cdot_G : G \times G \to G</annotation></semantics></math> restricted along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>D</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mi>dim</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>↪</mo><mi>G</mi><mo>×</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">\tilde D(2,dim(G)) \hookrightarrow G \times G</annotation></semantics></math> to the <a class="existingWikiWord" href="/nlab/show/infinitesimal+space">infinitesimal space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>D</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mi>dim</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\tilde D(2, dim(G))</annotation></semantics></math> linearizes in each of its arguments: for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>,</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo>∈</mo><mover><mi>D</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mi>dim</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\vec x,\vec y) \in \tilde D(2,dim(G))</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><mover><mi>x</mi><mo stretchy="false">→</mo></mover><msub><mo>⋅</mo> <mi>G</mi></msub><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo>=</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>⋅</mo><msub><mo>∇</mo> <mi>x</mi></msub><msub><mo>⋅</mo> <mi>G</mi></msub><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo>+</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo>⋅</mo><msub><mo>∇</mo> <mi>y</mi></msub><msub><mo>⋅</mo> <mi>G</mi></msub><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo>+</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>⋅</mo><msub><mo>∇</mo> <mi>x</mi></msub><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo>⋅</mo><msub><mo>∇</mo> <mi>y</mi></msub><msub><mo>⋅</mo> <mi>G</mi></msub><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \vec x \cdot_G \vec y = \vec x \cdot \nabla_x \cdot_G (0,0) + \vec y \cdot \nabla_y \cdot_G (0,0) + \vec x \cdot \nabla_x \vec y \cdot \nabla_y \cdot_G(0,0) \,. </annotation></semantics></math></div> <p>Since the origin here corresponds to the neutral element 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">G</annotation></semantics></math> and since with one of its arguments the neutral element the operaton <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>⋅</mo> <mi>G</mi></msub></mrow><annotation encoding="application/x-tex">\cdot_G</annotation></semantics></math> is the identity, and since the double derivative produces the Lie bracket (keeping in mind that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>x</mi> <mi>i</mi></msup><msup><mi>y</mi> <mi>j</mi></msup><mo>+</mo><msup><mi>x</mi> <mi>j</mi></msup><msup><mi>y</mi> <mi>i</mi></msup><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">x^i y^j + x^j y^i = 0</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>D</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mi>dim</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\tilde D(2,dim(G))</annotation></semantics></math>), this is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>⋯</mi><mo>=</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>+</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo>+</mo><mo stretchy="false">[</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>,</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \cdots = \vec x + \vec y + [\vec x, \vec y] \,. </annotation></semantics></math></div> <p>Accordingly the alternating sum of co-face maps 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>d</mi></mtd> <mtd><mo>=</mo><msubsup><mi>p</mi> <mn>1</mn> <mo>*</mo></msubsup><mo lspace="verythinmathspace" rspace="0em">+</mo><msubsup><mi>p</mi> <mn>2</mn> <mo>*</mo></msubsup><mo lspace="verythinmathspace" rspace="0em">−</mo><msubsup><mo>⋅</mo> <mi>G</mi> <mo>*</mo></msubsup></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><msubsup><mi>p</mi> <mn>1</mn> <mo>*</mo></msubsup><mo lspace="verythinmathspace" rspace="0em">+</mo><msubsup><mi>p</mi> <mn>2</mn> <mo>*</mo></msubsup><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">(</mo><msubsup><mi>p</mi> <mn>1</mn> <mo>*</mo></msubsup><mo lspace="verythinmathspace" rspace="0em">+</mo><msubsup><mi>p</mi> <mn>2</mn> <mo>*</mo></msubsup><mo lspace="verythinmathspace" rspace="0em">+</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 stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</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></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} d &= p_1^* + p_2^* - \cdot_G^* \\ & = p_1^* + p_2^* - ( p_1^* + p_2^* + [-,-]^*) \\ & = - [-,-]^* \end{aligned} </annotation></semantics></math></div> <p>as it should be for the Chevalley-Eilenberg algebra of a Lie algebra.</p> </div> <p>The infinitesimal reasoning involved in this proof is discussed in (<a href="#Kock">Kock, section 6.8</a>).</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+infinity-algebroid+representation">Lie infinity-algebroid representation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sheaf+of+L-%E2%88%9E+algebras">sheaf of L-∞ algebras</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebroid+cohomology">∞-Lie algebroid cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebroid+valued+differential+forms">∞-Lie algebroid valued differential forms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Weil+theory">∞-Chern-Weil theory</a></p> </li> </ul> <h2 id="references">References</h2> <p>The term “Lie <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-algebroid” or “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebroid” as such is not as yet established in the literature, as most authors working with these objects think of them entirely in terms of <a class="existingWikiWord" href="/nlab/show/dg-algebra">dg-algebra</a>s or <a class="existingWikiWord" href="/nlab/show/NQ-supermanifolds">NQ-supermanifolds</a> and either ignore the relation to <a class="existingWikiWord" href="/nlab/show/Lie+theory">Lie theory</a> or take it more or less for granted.</p> <p>Possibly the first explicit appearance of the idea of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Lie algebroids recognized in their full <a class="existingWikiWord" href="/nlab/show/Lie+theory">Lie theoretic</a> meaning is</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Pavol+Severa">Pavol Severa</a>, <em>Some title containing the words “homotopy” and “symplectic”, e.g. this one</em> (<a href="http://arxiv.org/abs/math.SG/0105080">arXiv</a>)</li> </ul> <p>which uses “<a class="existingWikiWord" href="/nlab/show/NQ-supermanifolds">NQ-supermanifolds</a>”. Of course, as this article also points out, in hindsight one finds that much of this is already implicit in the much older theory of <a class="existingWikiWord" href="/nlab/show/Sullivan+model">Sullivan models</a> in <a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rational homotopy theory</a>, which is concerned with modelling <em><a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>s</em> by <a class="existingWikiWord" href="/nlab/show/dg-algebras">dg-algebras</a>. That these spaces can be regarded as <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoids">∞-groupoids</a> and as <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+groupoids">∞-Lie groupoids</a> in particular is clear in hindsight, but was possibly first explicitly realized in the above reference. See also <a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integration</a>, <a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory+in+an+%28%E2%88%9E%2C1%29-topos">rational homotopy theory in an (∞,1)-topos</a> and <a class="existingWikiWord" href="/nlab/show/function+algebras+on+%E2%88%9E-stacks">function algebras on ∞-stacks</a>.</p> <p>The explicit term <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Lie algebroid</em> / <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebroid</em> as such is due to</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>: <em>On <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Lie</em> (2008) [<a href="https://www.math.uni-hamburg.de/home/schreiber/action.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Schreiber-InfinityLie.pdf" title="">Schreiber-InfinityLie.pdf</a>]</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>, <a class="existingWikiWord" href="/nlab/show/Jim+Stasheff">Jim Stasheff</a>, Section A.1 of: <em><a class="existingWikiWord" href="/schreiber/show/Twisted+Differential+String+and+Fivebrane+Structures">Twisted Differential String and Fivebrane Structures</a></em>, Communications in Mathematical Physics, Volume 315, Issue 1 (2012) pp 169-213 (<a href="http://arxiv.org/abs/0910.4001">arXiv:0910.4001</a>, <a href="https://link.springer.com/article/10.1007/s00220-012-1510-3">doi:10.1007/s00220-012-1510-3</a>)</p> </li> </ul> <p>The term then appears in</p> <ul> <li>Andrew James Bruce, <em>From <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_{\infty}</annotation></semantics></math>-algebroids to higher Schouten/Poisson structures</em>, Rept. Math. Phys. 67:157-177, 2011 (<a href="https://arxiv.org/abs/1007.1389">arXiv:1007.1389</a>, <a href="https://doi.org/10.1016/S0034-4877(11)00010-3">doi:10.1016/S0034-4877(11)00010-3</a>)</li> </ul> <p>The dual monoidal Dold-Kan correspondence is discussed in</p> <ul> <li id="CastiglioniCortinas">J. L. Castiglioni, G. Cortiñas, <em>Cosimplicial versus DG-rings: a version of the Dold-Kan correspondence</em> , J. Pure Appl. Algebra 191 (2004), no. 1-2, 119–142, (<a href="http://arxiv.org/abs/math/0306289">arXiv:math.KT/0306289</a>) .</li> </ul> <p>The smooth spaces of infinitesimal simplices <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>D</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>k</mi><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\tilde D(k,n)</annotation></semantics></math> are considered in section 1.2 of</p> <ul id="Kock"> <li><a class="existingWikiWord" href="/nlab/show/Anders+Kock">Anders Kock</a>, <em>Synthetic differential geometry of manifolds</em> (<a href="http://home.imf.au.dk/kock/SGM-final.pdf">pdf</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on August 17, 2024 at 15:31:56. 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