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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeJan 28th 2010
   • (edited Jan 28th 2010)
   • PermaLink
   Author: Urs
   Format: Markdowncan anyone point me to some useful discussion of cosismplicial simplicial abelian groups <latex> \Delta \to [\Delta^{op}, Ab] </latex> and cosimplicial simplicial rings <latex> \Delta \to [\Delta^{op}, CRing] </latex> 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?

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

   

   and cosimplicial simplicial rings

   

   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?

2. • CommentRowNumber2.
   • CommentAuthorzskoda
   • CommentTimeJan 28th 2010
   • PermaLink
   Author: zskoda
   Format: MarkdownBut the totalization functor in either the second or in the third quadrant still give unboundedness in one direction only, isn't it ?

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

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeJan 28th 2010
   • (edited Jan 28th 2010)
   • PermaLink
   Author: Urs
   Format: MarkdownAm I mixed up? Let me see. We have a double complex <latex> C^{p,q} </latex> with one differential increasing <latex> p </latex> <latex> d : C^{p,q} \to C^{p+1,q}</latex> and the other decreasing <latex> q </latex> <latex> \delta : C^{p,q} \to C^{p,q-1}</latex> This gives me single differential <latex> D = d \pm \delta </latex> of definite degree (say +1) if the total degree is taken to be <latex> p - q </latex>. So if both <latex> p </latex> and <latex> q </latex> ranged in <latex> \mathbb{N} </latex> this gives degrees in all of <latex> \mathbb{Z} </latex>. And I'd think this is what I get from forming the chain complex of a cosimplicial simplicial abelian group: a chain complex (<latex> \delta </latex>) of cochain complexes (<latex> d </latex>). Do you agree?

   Am I mixed up? Let me see.

   We have a double complex with one differential increasing

   

   and the other decreasing

   

   This gives me single differential

   

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

   So if both and ranged in this gives degrees in all of .

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

   Do you agree?