Short Version
Given a cosimplicial space $X_\bullet$, what is the relationship between (co)chains on the totalisation of $X_\bullet$ and the totalisation of the cosimplicial chain complex obtained by first taking chains levelwise? $$Ch\left(\|X_\bullet\|\right) \qquad \text{vs.}\qquad \|[n] \mapsto Ch(X_n)\|$$

Simplicial Objects, Realisation, Chains
Given a simplicial set, I can get a topological space by realisation. To do this we model every [n] by a topological space (the cosimplicial space of simplices); the realisation functor is then the left Kan extension of this along the yoneda functor.
In fact, this is part of a general game giving us nerve and realisation (ncatlab.org/nlab/show/nerve+and+realization) functors in many contexts. For example I can realise a simplicial abelian group as a chain complex using as my model the cosimplicial chain complex that maps [n] to normalised chains on the n simplex. (see also: Dold-Kan correspondence ncatlab.org/nlab/show/Dold-Kan+correspondence)
The normalised chains functor from simplicial abelian groups to (bounded below) chain complexes of abelian groups is an example of such a realisation functor. $$ N: sAb \rightarrow Ch_+(Ab) $$ Alternatively we have a description in terms of the alternating face map complex modulo the degenerate subcomplex- see Moore Complex (ncatlab.org/nlab/show/Moore+complex).

Cosimplicial Object, Totalisation, Cochains
Dual to realisation is totalisation which turns a cosimplicial set into a space. More generally, give a cosimplicial object $X$ in (enriched) cateogry $C$ and given a cosimplicial object $D$ in $C$ as my model I can look at the totalisation of $X$ w.r.t. $D$ as cosimplicial morphisms $$ Tot_D(X) := Hom_\Delta(D,X)$$

Given a cosimplicial abelian group, I can form the totalisation using as my model the cosimplicial chain complex that maps [n] to normalised chains on the n simplex.
Alternatively, as before I can also get a (co)chain complex by using alternating sum of (co)face maps / Moore Complex ... from the dual Dold-Kan correspondence (for example here: http://arxiv.org/abs/math/0306289 )

1) My first question is basically this; is this last cochain complex the same as the one I obtain from totalisation? I assume yes, by duality?

2) My second question concerns a cosimplicial space (cosimplicial simplicial set).
I apply the free $R$-module functor to get a cosimplicial simplicial $R$-module and I want to end up with a (co)chain complex.
I could use normalised chains/Moore complex in each cosimplicial degree to get a cosimplicial chain complex $$(sM_R)^\Delta \stackrel{N^\Delta}{\longrightarrow} (Ch_+(M_R))^\Delta$$ and then perhaps get a cochain complex of chain complexes using the Moore complex construction again in the cosimplicial direction. An then...?
I could totalise the cosimplicial chain complex to get a chain complex directly? Or I could do things in reverse order?
Please excuse the confusion, I suppose I can't really see what is going on and I should ask, do I get something equivalent (quasi-isomorphic) in each case?

3) Finally, I could just totalise my cosimplicial space to begin with.
What is the relationship between chains on the totalisation of the cosimplicial space with the totalisation of the cosimplicial simplicial $R$-module of question 2 / or indeed the cosimplicial chain complex I get by taking chains in each degree?

thank you, a.


Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.