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nLab > Latest Changes: simplicial deRham complex

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeOct 20th 2009

    I started an entry simplicial deRham complex

    on differential forms on simplicial manifolds.

    In parts this is for me to collect some standard references and definitions (still very incomplete on that aspect, help is appreciated -- is there a good reference by Dupont that is online available?)

    and in parts this is to discuss the deeper abstract-nonsense origin of this concept.

    I am thinking that

    • with differential forms understood in the synthetic context as just the image under Dold-Kan of the cosimplicial algebra of functions on the simplicial object of infinitesimal simplices in some space

    • it follows that the simplicial deRham complex of a simplicial object is just the image under Dold-Kan of the cosimplicial algebra of functions on the realization of the bisimplicial object of infinitesimal simplices in the given simplicial space.

    This looks like it is prretty obvious, once one stares at the coend-formula, but precisely that makes me feel a bit nervous. Maybe i am being too sloppy here. Would appreciate you eyeballing this.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeOct 20th 2009

    Here is probably what I need to know:

    let F be a bisimplicial abelian group and \bar F := \int^{p,q} \mathbb{Z}(\Delta^p \times \Delta^q) \otimes F(k,l) the simplicial abelian group obtained as the F-weighted colimit over the simplicial abelian group \mathbb{Z}(\Delta^p \times \Delta^q) .

    Then is under Dold-Kan the image of $\bar F$ quasi-isomorphic to the total complex of F?

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeOct 20th 2009

    Oh, this IS just the Eilenberg-Zilber theorem, right?

    I just need something like exercise 1.6 in chapter 4 of Goerss-Jardine which should give me that \bar F = \int^n \mathbb{Z}\Delta^n \otimes F(\bullet,n) = d(F) and then Eilenberg-Zilber says d(F) \simeq Tot F .

    Am I on the right track here? Am a bit in a haste, unfortunately...

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeOct 20th 2009

    Sorry for all the self-replies.

    I have now typed the "supposed proposition" and its "supposed proof" into the page

    simplicial deRham complex --> reformulation in SDG.

    (scroll down a wee bit).

    Sanity checks are still very welcome.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeOct 20th 2009

    I am now pretty confident of the proof at

    simplicial deRham complex -> reformulation in SDG.

    I have also added on my personal web the page infinitesimal path oo-groupoid --> relation to simplicial deRham complex accordingly

1 to 5 of 5

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