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1. • 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 $F : \Delta \times \Delta \to Ab$ 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?