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href="/nlab/show/equivariant+rational+homotopy+theory">equivariant</a>, <a class="existingWikiWord" href="/nlab/show/rational+stable+homotopy+theory">stable</a>, <a class="existingWikiWord" href="/nlab/show/parametrized+rational+homotopy+theory">parametrized</a>, <a class="existingWikiWord" href="/nlab/show/equivariant+rational+stable+homotopy+theory">equivariant & stable</a>, <a class="existingWikiWord" href="/nlab/show/parametrized+rational+stable+homotopy+theory">parametrized & stable</a>)</p> <h2 id="dgalgebra">dg-Algebra</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/graded+vector+space">graded vector space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+graded+vector+space">differential graded vector space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+graded+algebra">differential graded algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+dgc-algebras">model structure on dgc-algebras</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+equivariant+dgc-algebras">model structure on equivariant dgc-algebras</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+graded+coalgebra">differential graded coalgebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-coalgebras">model structure on dg-coalgebras</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+graded+Lie+algebra">differential graded Lie algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-Lie+algebras">model structure on dg-Lie algebras</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebra">L-∞ algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/model+structure+for+L-%E2%88%9E+algebras">model structure for L-∞ algebras</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+graded+Hopf+algebra">differential graded Hopf algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bar+and+cobar+construction">bar and cobar construction</a></p> </li> </ul> <h2 id="rational_spaces">Rational spaces</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/nilpotent+space">nilpotent space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/finite+type">finite type</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/rational+space">rational space</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/formal+space">formal space</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/rationalization">rationalization</a></p> </li> </ul> <h2 id="pl_de_rham_complex">PL de Rham complex</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+forms+on+simplices">differential forms on simplices</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/PL+de+Rham+complex">PL de Rham complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Sullivan+construction">Sullivan construction</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integration</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+theorem+of+dg-algebraic+rational+homotopy+theory">fundamental theorem of dg-algebraic rational homotopy theory</a></p> </li> </ul> <h2 id="sullivan_models">Sullivan models</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/minimal+Sullivan+model">minimal</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Sullivan+model">Sullivan model</a></li> </ul> <div> <p><strong>Examples of <a class="existingWikiWord" href="/nlab/show/Sullivan+models">Sullivan models</a></strong> in <a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rational homotopy theory</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Sullivan+model+of+an+n-sphere">Sullivan model of an n-sphere</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Sullivan+model+of+a+spherical+fibration">Sullivan model of a spherical fibration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Sullivan+model+of+complex+projective+space">Sullivan model of complex projective space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Sullivan+model+of+a+classifying+space">Sullivan model of a classifying space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Sullivan+model+of+mapping+space">Sullivan model of mapping space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Sullivan+model+of+free+loop+space">Sullivan model of free loop space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Sullivan+model+of+a+suspension">Sullivan model of a suspension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Sullivan+model+of+a+finite+G-quotient">Sullivan model of a finite G-quotient</a></p> </li> </ul> </div> <h2 id="related_topics">Related topics</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable homotopy theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homological+algebra">homological algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+theory">∞-Lie theory</a></p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#as_cofibrations'>As cofibrations</a></li> <li><a href='#rationalization'>Rationalization</a></li> <li><a href='#relation_to_nilpotent_algebras'>Relation to nilpotent <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</a></li> <li><a href='#relation_to_whitehead_products'>Relation to Whitehead products</a></li> </ul> <li><a href='#examples'>Examples</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>A <strong>Sullivan model</strong> of a <a class="existingWikiWord" href="/nlab/show/rational+space">rational space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a particularly well-behaved commutative <a class="existingWikiWord" href="/nlab/show/dg-algebra">dg-algebra</a> <a class="existingWikiWord" href="/nlab/show/quasi-isomorphism">quasi-isomorphic</a> to the dg-algebra of Sullivan forms on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>. These <em>Sullivan algebras</em> are precisely the cofibrant objects in the standard <a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-algebras">model structure on dg-algebras</a>.</p> <p>Sullivan models are a central tool in <a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rational homotopy theory</a>.</p> <h2 id="definition">Definition</h2> <p>Sullivan models are particularly well-behaved <a class="existingWikiWord" href="/nlab/show/differential+graded-commutative+algebras">differential graded-commutative algebras</a> that are equivalent to the dg-algebras of <a href="differential+forms+on+simplices#PiecewisePolynomialDifferentialForms">piecewise polynomial differential forms on topological spaces</a>. Conversely, every rational space can be obtained from a dg-algebra and the <em>minimal</em> Sullivan algebras provide convenient representatives that correspond bijectively to rational homotopy types under this correspondence.</p> <p>Abstractly, (relative) Sullivan models are the (relative) <a class="existingWikiWord" href="/nlab/show/cell+complexes">cell complexes</a> in the standard <a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-algebras">model structure on dg-algebras</a>.</p> <p>We now describe this in detail. First some notation and preliminaries:</p> <div class="num_defn" id="FiniteType"> <h6 id="definition_2">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/of+finite+type">of finite type</a>)</strong></p> <ul> <li> <p>A <a class="existingWikiWord" href="/nlab/show/graded+vector+space">graded vector space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> is <em><a class="existingWikiWord" href="/nlab/show/of+finite+type">of finite type</a></em> if in each degree it is finite dimensional. In this case we write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>V</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">V^*</annotation></semantics></math> for its degreewise dual.</p> </li> <li> <p>A <a class="existingWikiWord" href="/nlab/show/Grassmann+algebra">Grassmann algebra</a> is <em><a class="existingWikiWord" href="/nlab/show/of+finite+type">of finite type</a></em> if it is the Grassmann algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mo>∧</mo> <mo>•</mo></msup><msup><mi>V</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">\wedge^\bullet V^*</annotation></semantics></math> on a graded vector space of finite type</p> <p>(the dualization here is just convention, that will help make some of the following constructions come out nicely).</p> </li> <li> <p>A <a class="existingWikiWord" href="/nlab/show/CW-complex">CW-complex</a> is of finite type if it is built out of finitely many cells in each degree.</p> </li> </ul> </div> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> a <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>-<a class="existingWikiWord" href="/nlab/show/graded+vector+space">graded vector space</a> write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mo>∧</mo> <mo>•</mo></msup><mi>V</mi></mrow><annotation encoding="application/x-tex">\wedge^\bullet V</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/Grassmann+algebra">Grassmann algebra</a> over it. Equipped with the trivial differential <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">d = 0</annotation></semantics></math> this is a <a class="existingWikiWord" href="/nlab/show/semifree+dgc-algebra">semifree dgc-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><mi>V</mi><mo>,</mo><mi>d</mi><mo>=</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\wedge^\bullet V, d=0)</annotation></semantics></math>.</p> <p>With <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> our ground <a class="existingWikiWord" href="/nlab/show/field">field</a> we write <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>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(k,0)</annotation></semantics></math> for the corresponding <a class="existingWikiWord" href="/nlab/show/dg-algebra">dg-algebra</a>, the tensor unit for the standard <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal structure</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>dgAlg</mi></mrow><annotation encoding="application/x-tex">dgAlg</annotation></semantics></math>. This is the <a class="existingWikiWord" href="/nlab/show/Grassmann+algebra">Grassmann algebra</a> on the 0-vector space <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>0</mn><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><msup><mo>∧</mo> <mo>•</mo></msup><mn>0</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(k,0) = (\wedge^\bullet 0, 0)</annotation></semantics></math>.</p> <div class="num_defn" id="SullivanAlgebra"> <h6 id="definition_3">Definition</h6> <p><strong>(Sullivan algebras)</strong></p> <p>A <strong>relatived Sullivan algebra</strong> is a <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> of <a class="existingWikiWord" href="/nlab/show/differential+graded-commutative+algebras">differential graded-commutative algebras</a> that is an inclusion of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mo>↪</mo><mo stretchy="false">(</mo><mi>A</mi><msub><mo>⊗</mo> <mi>k</mi></msub><msup><mo>∧</mo> <mo>•</mo></msup><mi>V</mi><mo>,</mo><mi>d</mi><mo>′</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (A,d) \hookrightarrow (A \otimes_k \wedge^\bullet V, d') </annotation></semantics></math></div> <p>for <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>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A,d)</annotation></semantics></math> any <a class="existingWikiWord" href="/nlab/show/dgc-algebra">dgc-algebra</a> and for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> some <a class="existingWikiWord" href="/nlab/show/graded+vector+space">graded vector space</a>, such that</p> <ol> <li> <p>there is a <a class="existingWikiWord" href="/nlab/show/well+ordered+set">well ordered set</a> <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> indexing a <a class="existingWikiWord" href="/nlab/show/linear+basis">linear basis</a> <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>α</mi></msub><mo>∈</mo><mi>V</mi><mo stretchy="false">|</mo><mi>α</mi><mo>∈</mo><mi>J</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{v_\alpha \in V| \alpha \in J\}</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>;</p> </li> <li> <p>writing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mrow><mo><</mo><mi>β</mi></mrow></msub><mo>≔</mo><mi>span</mi><mo stretchy="false">(</mo><msub><mi>v</mi> <mi>α</mi></msub><mo stretchy="false">|</mo><mi>α</mi><mo><</mo><mi>β</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">V_{\lt \beta} \coloneqq span(v_\alpha | \alpha \lt \beta)</annotation></semantics></math> then for all basis elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mi>β</mi></msub></mrow><annotation encoding="application/x-tex">v_\beta</annotation></semantics></math> we have that</p> </li> </ol> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>′</mo><msub><mi>v</mi> <mi>β</mi></msub><mo>∈</mo><mi>A</mi><mo>⊗</mo><msup><mo>∧</mo> <mo>•</mo></msup><msub><mi>V</mi> <mrow><mo><</mo><mi>β</mi></mrow></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> d' v_\beta \in A \otimes \wedge^\bullet V_{\lt \beta} \,. </annotation></semantics></math></div> <p>This is called a <strong>minimal</strong> relative Sullivan algebra if in addition the condition</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>α</mi><mo><</mo><mi>β</mi><mo stretchy="false">)</mo><mo>⇒</mo><mo stretchy="false">(</mo><mi>deg</mi><msub><mi>v</mi> <mi>α</mi></msub><mo>≤</mo><mi>deg</mi><msub><mi>v</mi> <mi>β</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (\alpha \lt \beta) \Rightarrow (deg v_\alpha \leq deg v_\beta) </annotation></semantics></math></div> <p>holds. For a Sullivan algebra <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>0</mn><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><msup><mo>∧</mo> <mo>•</mo></msup><mi>V</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(k,0) \to (\wedge^\bullet V, d)</annotation></semantics></math> relative to the tensor unit we call the <a class="existingWikiWord" href="/nlab/show/semifree+dgc-algebra">semifree dgc-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><mi>V</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\wedge^\bullet V,d)</annotation></semantics></math> simply a <strong>Sullivan algebra</strong>, and we call it a <strong>minimal Sullivan algebra</strong> if <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>0</mn><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><msup><mo>∧</mo> <mo>•</mo></msup><mi>V</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(k,0) \to (\wedge^\bullet V, d)</annotation></semantics></math> is a minimal relative Sullivan algebra.</p> </div> <p>(e.g. <a href="#Hess06">Hess 06, def. 1.10, remark 1.11</a>)</p> <p>See also the section <a href="model+structure+on+dg-algebras#SullivanAlgebras">Sullivan algebras</a> at <a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-algebras">model structure on dg-algebras</a>.</p> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>The special condition on the ordering in the relative Sullivan algebra says that these morphisms are composites of <a class="existingWikiWord" href="/nlab/show/pushouts">pushouts</a> of the <a class="existingWikiWord" href="/nlab/show/cofibrantly+generated+model+category">generating cofibrations</a> for the <a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-algebras">model structure on dg-algebras</a>, which are the inclusions</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>↪</mo><mi>D</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> S(n) \hookrightarrow D(n) \,, </annotation></semantics></math></div> <p>where</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><msup><mo>∧</mo> <mo>•</mo></msup><mo stretchy="false">⟨</mo><mi>c</mi><mo stretchy="false">⟩</mo><mo>,</mo><mi>d</mi><mo>=</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> S(n) = (\wedge^\bullet \langle c \rangle, d = 0) </annotation></semantics></math></div> <p>is the dg-algebra on a single generator 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> with vanishing differential, and where</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><msup><mo>∧</mo> <mo>•</mo></msup><mo stretchy="false">(</mo><mo stretchy="false">⟨</mo><mi>b</mi><mo stretchy="false">⟩</mo><mo>⊕</mo><mo stretchy="false">⟨</mo><mi>c</mi><mo stretchy="false">⟩</mo><mo stretchy="false">)</mo><mo>,</mo><mi>d</mi><mi>b</mi><mo>=</mo><mi>c</mi><mo>,</mo><mi>d</mi><mi>c</mi><mo>=</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> D(n) = (\wedge^\bullet (\langle b \rangle \oplus \langle c \rangle), d b = c, d c = 0) </annotation></semantics></math></div> <p>with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math> an additional generator in degree <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>.</p> <p>Therefore for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><mi>dgcAlg</mi></mrow><annotation encoding="application/x-tex">A \in dgcAlg</annotation></semantics></math>, a pushout</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>S</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mi>ϕ</mi></mover></mtd> <mtd><mi>A</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>D</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>A</mi><mo>⊗</mo><msup><mo>∧</mo> <mo>•</mo></msup><mo stretchy="false">⟨</mo><mi>b</mi><mo stretchy="false">⟩</mo><mo>,</mo><mi>d</mi><mi>b</mi><mo>=</mo><mi>ϕ</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ S(n) &\stackrel{\phi}{\to}& A \\ \downarrow && \downarrow \\ D(n) &\to& (A \otimes \wedge^ \bullet \langle b \rangle, d b = \phi) } </annotation></semantics></math></div> <p>is precisely a choice <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>∈</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">\phi \in A</annotation></semantics></math> of a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">d_A</annotation></semantics></math>-closed element 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> and results in adjoining to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> the element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math> whose differential is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>b</mi><mo>=</mo><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">d b = \phi</annotation></semantics></math>. This gives the condition in the above definition: the differential of any new element has to be a sum of wedge products of the old elements.</p> <p>Notice that it follows in particular that the cofibrations in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>dgAlg</mi> <mi>proj</mi></msub></mrow><annotation encoding="application/x-tex">dgAlg_{proj}</annotation></semantics></math> are precisely all the <a class="existingWikiWord" href="/nlab/show/retracts">retracts</a> of relative Sullivan algebra inclusions.</p> </div> <div class="num_remark"> <h6 id="remark_2">Remark</h6> <p><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>-algebras)</strong></p> <p>Because they are <a class="existingWikiWord" href="/nlab/show/semifree+dga">semifree dga</a>s, Sullivan dg-algebras <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><mi>V</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\wedge^\bullet V,d)</annotation></semantics></math> are (at least for degreewise finite dimensional <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>) <a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebra">Chevalley-Eilenberg algebra</a>s of <a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E-algebra">L-∞-algebra</a>s.</p> <p>The co-commutative differential co-algebra encoding the corresponding <a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E-algebra">L-∞-algebra</a> is the free cocommutative algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mo>∨</mo> <mo>•</mo></msup><msup><mi>V</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">\vee^\bullet V^*</annotation></semantics></math> on the degreewise dual of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> with differential <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo>=</mo><msup><mi>d</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">D = d^*</annotation></semantics></math>, i.e. the one given by the formula</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 stretchy="false">(</mo><mi>D</mi><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>∨</mo><msub><mi>v</mi> <mn>2</mn></msub><mo>∨</mo><mi>⋯</mi><msub><mi>v</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">(</mo><mi>d</mi><mi>ω</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>v</mi> <mn>2</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>v</mi> <mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \omega(D(v_1 \vee v_2 \vee \cdots v_n)) = - (d \omega) (v_1, v_2, \cdots, v_n) </annotation></semantics></math></div> <p>for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo>∈</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">\omega \in V</annotation></semantics></math> and all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mi>i</mi></msub><mo>∈</mo><msup><mi>V</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">v_i \in V^*</annotation></semantics></math>.</p> </div> <div class="num_defn"> <h6 id="definition_4">Definition</h6> <p><strong>(Sullivan models)</strong></p> <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> a <a class="existingWikiWord" href="/nlab/show/simply+connected">simply connected</a> <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, a <strong>Sullivan (minimal) model</strong> 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> is a Sullivan (minimal) algebra <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>V</mi> <mo>*</mo></msup><mo>,</mo><msub><mi>d</mi> <mi>V</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\wedge^\bullet V^\ast, d_V)</annotation></semantics></math> equipped with a <a class="existingWikiWord" href="/nlab/show/quasi-isomorphism">quasi-isomorphism</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><msup><mo>∧</mo> <mo>•</mo></msup><msup><mi>V</mi> <mo>*</mo></msup><mo>,</mo><msub><mi>d</mi> <mi>V</mi></msub><mo stretchy="false">)</mo><mover><mo>⟶</mo><mo>≃</mo></mover><msubsup><mi>Ω</mi> <mi>pwpoly</mi> <mo>•</mo></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (\wedge^\bullet V^*, d_V) \stackrel{\simeq}{\longrightarrow} \Omega^\bullet_{pwpoly}(X) </annotation></semantics></math></div> <p>to the dg-algebra of <a class="existingWikiWord" href="/nlab/show/piecewise+polynomial+differential+forms">piecewise polynomial differential forms</a>.</p> </div> <h2 id="properties">Properties</h2> <h3 id="as_cofibrations">As cofibrations</h3> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p><strong>(cofibrations are relative Sullivan algebras)</strong></p> <p>The cofibrations in the <a class="existingWikiWord" href="/nlab/show/projective+model+structure+on+differential+graded-commutative+algebras">projective model structure on differential graded-commutative algebras</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>dgcAlg</mi> <mi>ℕ</mi></msub><msub><mo stretchy="false">)</mo> <mi>proj</mi></msub></mrow><annotation encoding="application/x-tex">(dgcAlg_{\mathbb{N}})_{proj}</annotation></semantics></math> are precisely the <a class="existingWikiWord" href="/nlab/show/retracts">retracts</a> of relative Sullivan algebra inclusions (def. <a class="maruku-ref" href="#SullivanAlgebra"></a>).</p> <p>Accordingly, the cofibrant objects in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>dgcAlg</mi> <mi>ℕ</mi></msub><msub><mo stretchy="false">)</mo> <mi>proj</mi></msub><mrow></mrow></mrow><annotation encoding="application/x-tex">(dgcAlg_{\mathbb{N}})_proj{}</annotation></semantics></math> are precisely the <a class="existingWikiWord" href="/nlab/show/retracts">retracts</a> of Sullivan algebras.</p> </div> <div class="num_prop"> <h6 id="proposition_2">Proposition</h6> <p>Minimal Sullivan models are unique up to <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>.</p> </div> <p>e.g <a href="#Hess06">Hess 06, prop 1.18</a>.</p> <h3 id="rationalization">Rationalization</h3> <div class="num_theorem" id="SullivanRationalizationEquivalence"> <h6 id="theorem">Theorem</h6> <p>Consider the <a class="existingWikiWord" href="/nlab/show/derived+adjunction">derived adjunction</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>Top</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>Ho</mi><mo stretchy="false">(</mo><mi>sSet</mi><mo stretchy="false">)</mo><munderover><mo>⊥</mo><munder><mo>⟶</mo><mrow><mi>ℝ</mi><msubsup><mi>Ω</mi> <mi>poly</mi> <mo>•</mo></msubsup></mrow></munder><mover><mo>⟵</mo><mrow><mi>𝕃</mi><msub><mi>K</mi> <mi>poly</mi></msub></mrow></mover></munderover><mi>Ho</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msub><mi>dgcAlg</mi> <mrow><mi>ℚ</mi><mo>,</mo><mo>≥</mo><mn>0</mn></mrow></msub><msup><mo stretchy="false">)</mo> <mi>op</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Ho(Top) \simeq Ho(sSet) \underoverset {\underset{\mathbb{R} \Omega^\bullet_{poly}}{\longrightarrow}} {\overset{\mathbb{L} K_{poly} }{\longleftarrow}} {\bot} Ho( (dgcAlg_{\mathbb{Q}, \geq 0})^{op} ) </annotation></semantics></math></div> <p>induced from the of the <a class="existingWikiWord" href="/nlab/show/PL+de+Rham+complex">PL de Rham complex</a>-<a class="existingWikiWord" href="/nlab/show/Quillen+adjunction">Quillen adjunction</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>dgcAlg</mi> <mrow><mi>ℚ</mi><mo>,</mo><mo>≥</mo><mn>0</mn></mrow></msub><msub><mo></mo><mi>proj</mi></msub><msup><mo stretchy="false">)</mo> <mi>op</mi></msup><munderover><mo>⊥</mo><munder><mo>⟶</mo><mrow><msub><mi>K</mi> <mi>poly</mi></msub></mrow></munder><mover><mo>⟵</mo><mrow><msubsup><mi>Ω</mi> <mi>poly</mi> <mo>•</mo></msubsup></mrow></mover></munderover><msub><mi>sSet</mi> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex"> (dgcAlg_{\mathbb{Q}, \geq 0}_{proj})^{op} \underoverset {\underset{K_{poly}}{\longrightarrow}} {\overset{\Omega^\bullet_{poly}}{\longleftarrow}} {\bot} sSet_{Quillen} </annotation></semantics></math></div> <p>(<a href="rational+homotopy+theory#SullivanRationalizationAdjunction">this theorem</a>).</p> <p>Then: On the <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><msubsup><mi>Top</mi> <mrow><mi>ℚ</mi><mo>,</mo><mo>≥</mo><mn>1</mn></mrow> <mi>fin</mi></msubsup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(Top_{\mathbb{Q}, \geq 1}^{fin})</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/nilpotent+topological+space">nilpotent</a> <a class="existingWikiWord" href="/nlab/show/rational+topological+spaces">rational topological spaces</a> of <a class="existingWikiWord" href="/nlab/show/finite+type">finite type</a> this adjunction restricts to an <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence of categories</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><msubsup><mi>Top</mi> <mrow><mi>ℚ</mi><mo>,</mo><mo>></mo><mn>1</mn></mrow> <mi>fin</mi></msubsup><mo stretchy="false">)</mo><munderover><mo>≃</mo><munder><mo>⟶</mo><mrow><mi>ℝ</mi><msubsup><mi>Ω</mi> <mi>poly</mi> <mo>•</mo></msubsup></mrow></munder><mover><mo>⟵</mo><mrow><mi>𝕃</mi><msub><mi>K</mi> <mi>poly</mi></msub></mrow></mover></munderover><mi>Ho</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msubsup><mi>dgcAlg</mi> <mrow><mi>ℚ</mi><mo>,</mo><mo>></mo><mn>1</mn></mrow> <mi>fin</mi></msubsup><msup><mo stretchy="false">)</mo> <mi>op</mi></msup><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Ho(Top_{\mathbb{Q}, \gt 1}^{fin}) \underoverset {\underset{\mathbb{R} \Omega^\bullet_{poly}}{\longrightarrow}} {\overset{\mathbb{L} K_{poly} }{\longleftarrow}} {\simeq} Ho( (dgcAlg_{\mathbb{Q}, \gt 1}^{fin})^{op} ) \,. </annotation></semantics></math></div> <p>In particular the <a class="existingWikiWord" href="/nlab/show/derived+adjunction+unit">derived adjunction unit</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>⟶</mo><msub><mi>K</mi> <mi>poly</mi></msub><mo stretchy="false">(</mo><msubsup><mi>Ω</mi> <mi>pwpoly</mi> <mo>•</mo></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> X \longrightarrow K_{poly}(\Omega^\bullet_{pwpoly}(X)) </annotation></semantics></math></div> <p>exhibits the <a class="existingWikiWord" href="/nlab/show/rationalization">rationalization</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> <p>This is the <a class="existingWikiWord" href="/nlab/show/fundamental+theorem+of+dgc-algebraic+rational+homotopy+theory">fundamental theorem of dgc-algebraic rational homotopy theory</a>, see there for more.</p> <p>It follows that the <a class="existingWikiWord" href="/nlab/show/cochain+cohomology">cochain cohomology</a> of the cochain complex of <a class="existingWikiWord" href="/nlab/show/piecewise+polynomial+differential+forms">piecewise polynomial differential forms</a> on any topological, hence equivalently that of any of its <a class="existingWikiWord" href="/nlab/show/Sullivan+models">Sullivan models</a>, coincides with its <a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a> with coefficients in the <a class="existingWikiWord" href="/nlab/show/rational+numbers">rational numbers</a>:</p> <div class="num_theorem"> <h6 id="theorem_2">Theorem</h6> <p>Let <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>V</mi> <mo>*</mo></msup><mo>,</mo><msub><mi>d</mi> <mi>V</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\wedge^\bullet V^*, d_V)</annotation></semantics></math> be a minimal Sullivan model of a simply connected rational topological 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>. Then there is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>V</mi></mrow><annotation encoding="application/x-tex"> \pi_\bullet(X) \simeq V </annotation></semantics></math></div> <p>between the <a class="existingWikiWord" href="/nlab/show/homotopy+groups">homotopy groups</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> and the generators of the minimal Sullivan model.</p> </div> <p>e.g. <a href="#Hess06">Hess 06, theorem 1.24</a>.</p> <h3 id="relation_to_nilpotent_algebras">Relation to nilpotent <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</h3> <p>Under the <a class="existingWikiWord" href="/nlab/show/formal+duality">formal duality</a> between <a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebras"><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</a> and their <a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebra">Chevalley-Eilenberg</a> <a class="existingWikiWord" href="/nlab/show/dgc-algebras">dgc-algebras</a>, the connected Sullivan models correspond <a class="existingWikiWord" href="/nlab/show/bijection">bijectively</a> to <a class="existingWikiWord" href="/nlab/show/connective">connective</a> <a class="existingWikiWord" href="/nlab/show/nilpotent+L-infinity+algebra">nilpotent <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</a> (<a href="#Berglund15">Berglund 2015, Thm. 2.3</a>).</p> <h3 id="relation_to_whitehead_products">Relation to Whitehead products</h3> <p>See at <em><a class="existingWikiWord" href="/nlab/show/the+co-binary+Sullivan+differential+is+the+dual+Whitehead+product">the co-binary Sullivan differential is the dual Whitehead product</a></em>.</p> <p><br /></p> <h2 id="examples">Examples</h2> <div> <p><strong>Examples of <a class="existingWikiWord" href="/nlab/show/Sullivan+models">Sullivan models</a></strong> in <a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rational homotopy theory</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Sullivan+model+of+an+n-sphere">Sullivan model of an n-sphere</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Sullivan+model+of+a+spherical+fibration">Sullivan model of a spherical fibration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Sullivan+model+of+complex+projective+space">Sullivan model of complex projective space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Sullivan+model+of+a+classifying+space">Sullivan model of a classifying space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Sullivan+model+of+mapping+space">Sullivan model of mapping space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Sullivan+model+of+free+loop+space">Sullivan model of free loop space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Sullivan+model+of+a+suspension">Sullivan model of a suspension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Sullivan+model+of+a+finite+G-quotient">Sullivan model of a finite G-quotient</a></p> </li> </ul> </div> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/formal+dg-algebra">formal dg-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/minimal+dg-module">minimal dg-module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/minimal+fibration">minimal fibration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+Sullivan+model">equivariant Sullivan model</a></p> </li> </ul> <h2 id="references">References</h2> <p>Original articles:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Dennis+Sullivan">Dennis Sullivan</a>, <em>Differential forms and the topology of manifolds</em>, in Proc. International Conf.: <em>Manifolds Tokyo (1973)</em>, Univ. Tokyo Press (1975) 37-49 [<a href="https://www.math.stonybrook.edu/~dennis/publications/PDF/DS-pub-0017.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Sullivan-DiffFormsManifolds.pdf" title="pdf">pdf</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Alan+J.+Deschner">Alan J. Deschner</a>, <em>Sullivan’s theory of minimal models</em>, MSc thesis, Univ. British Columbia (1976) [<a href="http://hdl.handle.net/2429/20052">doi:10.14288/1.0080132</a>, <a href="https://open.library.ubc.ca/media/stream/pdf/831/1.0080132/2">pdf</a>]</p> </li> <li id="Sullivan77"> <p><a class="existingWikiWord" href="/nlab/show/Dennis+Sullivan">Dennis Sullivan</a>, <em>Infinitesimal computations in topology</em>, Publications Mathématiques de l’IHÉS, 47 (1977), p. 269-331 (<a href="http://www.numdam.org/item/PMIHES_1977__47__269_0/">numdam:PMIHES_1977__47__269_0</a>)</p> </li> <li id="BousfieldGugenheim76"> <p><a class="existingWikiWord" href="/nlab/show/Aldridge+Bousfield">Aldridge Bousfield</a>, <a class="existingWikiWord" href="/nlab/show/Victor+Gugenheim">Victor Gugenheim</a>, <em><a class="existingWikiWord" href="/nlab/show/On+PL+deRham+theory+and+rational+homotopy+type">On PL deRham theory and rational homotopy type</a></em>, Memoirs of the AMS, vol. 179 (1976) (<a href="https://bookstore.ams.org/memo-8-179">ams:memo-8-179</a>)</p> </li> <li id="GriffithMorgan13"> <p><a class="existingWikiWord" href="/nlab/show/Phillip+Griffiths">Phillip Griffiths</a>, <a class="existingWikiWord" href="/nlab/show/John+Morgan">John Morgan</a>, <em>Rational Homotopy Theory and Differential Forms</em>, Progress in Mathematics Volume 16, Birkhauser (2013) (<a href="https://doi.org/10.1007/978-1-4614-8468-4">doi:10.1007/978-1-4614-8468-4</a>)</p> </li> </ul> <p>Review and application:</p> <ul> <li id="Halperin83"> <p><a class="existingWikiWord" href="/nlab/show/Steve+Halperin">Steve Halperin</a>, <em>Lectures on minimal models</em>, Mem. Soc. Math. Franc. no 9/10 (1983) (<a href="https://doi.org/10.24033/msmf.294">doi:10.24033/msmf.294</a>)</p> </li> <li id="FelixHalperinThomas00"> <p><a class="existingWikiWord" href="/nlab/show/Yves+F%C3%A9lix">Yves Félix</a>, <a class="existingWikiWord" href="/nlab/show/Stephen+Halperin">Stephen Halperin</a>, <a class="existingWikiWord" href="/nlab/show/Jean-Claude+Thomas">Jean-Claude Thomas</a>, Chapter II of: <em>Rational Homotopy Theory</em>, Graduate Texts in Mathematics, 205, Springer-Verlag, 2000 (<a href="https://link.springer.com/book/10.1007/978-1-4613-0105-9">doi:10.1007/978-1-4613-0105-9</a>)</p> </li> <li id="Hess06"> <p><a class="existingWikiWord" href="/nlab/show/Kathryn+Hess">Kathryn Hess</a>, around def 1.10 of <em>Rational homotopy theory: a brief introduction</em> (<a href="http://arxiv.org/abs/math.AT/0604626">arXiv:math.AT/0604626</a>)</p> </li> <li id="Menichi13"> <p><a class="existingWikiWord" href="/nlab/show/Luc+Menichi">Luc Menichi</a>, <em>Rational homotopy – Sullivan models</em>, in: <em>Free Loop Spaces in Geometry and Topology</em>, IRMA Lect. Math. Theor. Phys., EMS (2015) [<a href="https://arxiv.org/abs/1308.6685">arXiv:1308.6685</a>, <a href="https://doi.org/10.4171/153">doi:10.4171/153</a>]</p> </li> <li id="FelixHalperin17"> <p><a class="existingWikiWord" href="/nlab/show/Yves+F%C3%A9lix">Yves Félix</a>, <a class="existingWikiWord" href="/nlab/show/Steve+Halperin">Steve Halperin</a>, <em>Rational homotopy theory via Sullivan models: a survey</em>, Notices of the International Congress of Chinese Mathematicians Volume 5 (2017) Number 2 (<a href="https://arxiv.org/abs/1708.05245">arXiv:1708.05245</a>, <a href="https://dx.doi.org/10.4310/ICCM.2017.v5.n2.a3">doi:10.4310/ICCM.2017.v5.n2.a3</a>)</p> </li> </ul> <p>Dual interpretation as <a class="existingWikiWord" href="/nlab/show/nilpotent+L-infinity+algebra">nilpotent <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</a>:</p> <ul> <li id="Berglund15"><a class="existingWikiWord" href="/nlab/show/Alexander+Berglund">Alexander Berglund</a>, Def. 2.1 in: <em>Rational homotopy theory of mapping spaces via Lie theory 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> algebras</em>, Homology, Homotopy and Applications, Volume 17 (2015) Number 2 (<a href="https://arxiv.org/abs/1110.6145">arXiv:1110.6145</a>, <a href="http://dx.doi.org/10.4310/HHA.2015.v17.n2.a16">doi:10.4310/HHA.2015.v17.n2.a16</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on June 21, 2023 at 16:40:38. 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