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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJan 28th 2010
    • (edited Jan 28th 2010)

    can anyone point me to some useful discussion of cosismplicial simplicial abelian groups

     \Delta \to [\Delta^{op}, Ab]

    and cosimplicial simplicial rings

     \Delta \to [\Delta^{op}, CRing]

    I guess there should be a Dold-Kan correspondence relating these to unbounded (co)chain complexes (that may be nontrivial both in positive as well as in negative degree). I suppose it's kind of straightforward how this should work, but I'd still ike to know of any literature that might discuss this. Anything?

    • CommentRowNumber2.
    • CommentAuthorzskoda
    • CommentTimeJan 28th 2010

    But the totalization functor in either the second or in the third quadrant still give unboundedness in one direction only, isn't it ?

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJan 28th 2010
    • (edited Jan 28th 2010)

    Am I mixed up? Let me see.

    We have a double complex  C^{p,q} with one differential increasing  p

      d : C^{p,q} \to C^{p+1,q}

    and the other decreasing  q

      \delta : C^{p,q} \to C^{p,q-1}

    This gives me single differential

     D = d \pm \delta

    of definite degree (say +1) if the total degree is taken to be  p - q .

    So if both  p and  q ranged in  \mathbb{N} this gives degrees in all of  \mathbb{Z} .

    And I'd think this is what I get from forming the chain complex of a cosimplicial simplicial abelian group: a chain complex ( \delta ) of cochain complexes ( d ).

    Do you agree?