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Atrium > Mathematics, Physics & Philosophy: total complexes from bicosimplicial groups

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- CommentRowNumber1.
- CommentAuthorUrs
- CommentTimeMay 12th 2010
- (edited May 12th 2010)
- PermaLink
Author: Urs
Format: MarkdownItexMaybe somebody can quickly help me out. I think I knew the answer to the following question, but now I must have gotten myself mixed up: let $F : \Delta \times \Delta \to Ab$ be a bi-cosimplicial abelian group and $$ diag F \simeq \int^{n} \Delta^n \cdot F_{n,\bullet} $$ the corresponding cosimplicial group. From the [[Eilenberg-Zilber theorem]] we have that its cochain complex is the total complex of the double complex obtained from $F$ $$ C diag F \simeq Tot C F \,. $$ But what I need is instead such a statement for the end expression $$ \int_n C(\mathbb{Z}(\Delta^n)) \otimes C(F_{n,\bullet}) $$ How is that related to $Tot C F_{\bullet,\bullet}$? Isn't that isomorphic?

Maybe somebody can quickly help me out. I think I knew the answer to the following question, but now I must have gotten myself mixed up:

let $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><mo>:</mo><mi>Δ</mi><mo>×</mo><mi>Δ</mi><mo>→</mo><mi>Ab</mi></mrow><annotation encoding="application/x-tex">F : \Delta \times \Delta \to Ab</annotation></semantics></math>$ be a bi-cosimplicial abelian group and
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>diag</mi><mi>F</mi><mo>≃</mo><msup><mo>∫</mo> <mi>n</mi></msup><msup><mi>Δ</mi> <mi>n</mi></msup><mo>⋅</mo><msub><mi>F</mi> <mrow><mi>n</mi><mo>,</mo><mo>•</mo></mrow></msub></mrow><annotation encoding="application/x-tex"> diag F \simeq \int^{n} \Delta^n \cdot F_{n,\bullet} </annotation></semantics></math>$
the corresponding cosimplicial group. From the Eilenberg-Zilber theorem we have that its cochain complex is the total complex of the double complex obtained from $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>$
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>C</mi><mi>diag</mi><mi>F</mi><mo>≃</mo><mi>Tot</mi><mi>C</mi><mi>F</mi><mspace width="0.16667em"/><mo>.</mo></mrow><annotation encoding="application/x-tex"> C diag F \simeq Tot C F \,. </annotation></semantics></math>$
But what I need is instead such a statement for the end expression
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mo>∫</mo> <mi>n</mi></msub><mi>C</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">(</mo><msup><mi>Δ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⊗</mo><mi>C</mi><mo stretchy="false">(</mo><msub><mi>F</mi> <mrow><mi>n</mi><mo>,</mo><mo>•</mo></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \int_n C(\mathbb{Z}(\Delta^n)) \otimes C(F_{n,\bullet}) </annotation></semantics></math>$
How is that related to $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Tot</mi><mi>C</mi><msub><mi>F</mi> <mrow><mo>•</mo><mo>,</mo><mo>•</mo></mrow></msub></mrow><annotation encoding="application/x-tex">Tot C F_{\bullet,\bullet}</annotation></semantics></math>$ ? Isn’t that isomorphic?

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Atrium > Mathematics, Physics & Philosophy: total complexes from bicosimplicial groups