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nLab > Latest Changes: PL de Rham complex

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- CommentRowNumber1.
- CommentAuthorUrs
- CommentTimeAug 22nd 2020
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Author: Urs
Format: MarkdownItexstarting something <a href="https://ncatlab.org/nlab/revision/PL+de+Rham+complex/1">v1</a>, <a href="https://ncatlab.org/nlab/show/PL+de+Rham+complex">current</a>

starting something

v1, current
- CommentRowNumber2.
- CommentAuthorUrs
- CommentTimeAug 22nd 2020
- PermaLink
Author: Urs
Format: MarkdownItexAdded statement of the relation to the smooth de Rham complex ([here](https://ncatlab.org/nlab/show/PL+de+Rham+complex#RelationToSmoothdeRhamComplex)). <a href="https://ncatlab.org/nlab/revision/PL+de+Rham+complex/1">v1</a>, <a href="https://ncatlab.org/nlab/show/PL+de+Rham+complex">current</a>

Added statement of the relation to the smooth de Rham complex (here).

v1, current
- CommentRowNumber3.
- CommentAuthorUrs
- CommentTimeAug 23rd 2020
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Author: Urs
Format: MarkdownItexadded statement of the PL de Rham theorem <a href="https://ncatlab.org/nlab/revision/diff/PL+de+Rham+complex/3">diff</a>, <a href="https://ncatlab.org/nlab/revision/PL+de+Rham+complex/3">v3</a>, <a href="https://ncatlab.org/nlab/show/PL+de+Rham+complex">current</a>

added statement of the PL de Rham theorem

diff, v3, current
- CommentRowNumber4.
- CommentAuthorUrs
- CommentTimeAug 23rd 2020
- PermaLink
Author: Urs
Format: MarkdownItexadded statement of $\Omega^\bullet_{PLdR}$ being a Quillen left adjoint <a href="https://ncatlab.org/nlab/revision/diff/PL+de+Rham+complex/4">diff</a>, <a href="https://ncatlab.org/nlab/revision/PL+de+Rham+complex/4">v4</a>, <a href="https://ncatlab.org/nlab/show/PL+de+Rham+complex">current</a>

added statement of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>Ω</mi> <mi>PLdR</mi> <mo>•</mo></msubsup></mrow><annotation encoding="application/x-tex">\Omega^\bullet_{PLdR}</annotation></semantics></math>$ being a Quillen left adjoint

diff, v4, current
- CommentRowNumber5.
- CommentAuthorUrs
- CommentTimeAug 23rd 2020
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Author: Urs
Format: MarkdownItexadded statement of the [[fundamental theorem of dg-algebraic rational homotopy theory]]: $$ Ho \left( \big( DiffGradedCommAlgebras^{\geq 0}_{k} \big)^{op}_{proj} \right)^{\geq 1}_{fin} \underoverset { \underset {\;\;\; \mathbb{R} exp \;\;\;} {\longrightarrow} } { \overset {\;\;\; \mathbb{L} \Omega^\bullet_{PLdR}\;\;\;} {\longleftarrow} } {\simeq} Ho \big( HoSimplicialSets_{Qu} \big)^{\mathbb{Q}, fin_{\mathbb{Q}}}_{\geq 1, nil} $$ <a href="https://ncatlab.org/nlab/revision/diff/PL+de+Rham+complex/6">diff</a>, <a href="https://ncatlab.org/nlab/revision/PL+de+Rham+complex/6">v6</a>, <a href="https://ncatlab.org/nlab/show/PL+de+Rham+complex">current</a>

added statement of the fundamental theorem of dg-algebraic rational homotopy theory:
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>Ho</mi><msubsup><mrow><mo>(</mo><mo maxsize="1.2em" minsize="1.2em">(</mo><msubsup><mi>DiffGradedCommAlgebras</mi> <mi>k</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msubsup><msubsup><mo maxsize="1.2em" minsize="1.2em">)</mo> <mi>proj</mi> <mi>op</mi></msubsup><mo>)</mo></mrow> <mi>fin</mi> <mrow><mo>≥</mo><mn>1</mn></mrow></msubsup><munderover><mo>≃</mo><munder><mo>⟶</mo><mrow><mspace width="0.27778em"/><mspace width="0.27778em"/><mspace width="0.27778em"/><mi>ℝ</mi><mi>exp</mi><mspace width="0.27778em"/><mspace width="0.27778em"/><mspace width="0.27778em"/></mrow></munder><mover><mo>⟵</mo><mrow><mspace width="0.27778em"/><mspace width="0.27778em"/><mspace width="0.27778em"/><mi>𝕃</mi><msubsup><mi>Ω</mi> <mi>PLdR</mi> <mo>•</mo></msubsup><mspace width="0.27778em"/><mspace width="0.27778em"/><mspace width="0.27778em"/></mrow></mover></munderover><mi>Ho</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mi>HoSimplicialSets</mi> <mi>Qu</mi></msub><msubsup><mo maxsize="1.2em" minsize="1.2em">)</mo> <mrow><mo>≥</mo><mn>1</mn><mo>,</mo><mi>nil</mi></mrow> <mrow><mi>ℚ</mi><mo>,</mo><msub><mi>fin</mi> <mi>ℚ</mi></msub></mrow></msubsup></mrow><annotation encoding="application/x-tex"> Ho \left( \big( DiffGradedCommAlgebras^{\geq 0}_{k} \big)^{op}_{proj} \right)^{\geq 1}_{fin} \underoverset { \underset {\;\;\; \mathbb{R} exp \;\;\;} {\longrightarrow} } { \overset {\;\;\; \mathbb{L} \Omega^\bullet_{PLdR}\;\;\;} {\longleftarrow} } {\simeq} Ho \big( HoSimplicialSets_{Qu} \big)^{\mathbb{Q}, fin_{\mathbb{Q}}}_{\geq 1, nil} </annotation></semantics></math>$
diff, v6, current

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nLab > Latest Changes: PL de Rham complex