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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?