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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 ( 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
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 (

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: )

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.

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