# Homotopy totalization and chains - reference

Simple case. Take $$X_{\bullet}$$ a cosimplicial space. Is it true that the chain complex of its homotopy totalization is quasi-isomorphic to the homotopy totalization of its chain complex? Because of the computations listed in "homotopy totalization" at nlab, this boils down to a quasi-isomorphism between $$C_* Hom( \mathbb{\Delta}^{\bullet}, X_{\bullet}) \ \ \ \textrm{ and }\ \ \ \textrm{Tot}(N(C_* X_{\bullet}))$$ where $$\textrm{Tot}$$ is the product totalization functor and $$N$$ is the normalized chain complex appearing in the dold-kan correspondence. Since this seems to be explicitly expressible, do you know a reference?

Hard case. Now consider $$X$$ to be a homotopy coherent cosimplicial space, that is a map $$N(\Delta) \to N(\textrm{Top})[W^{-1}]$$, where $$[W^{-1}]$$ is the $$\infty$$-localization at the homotopy equivalences. Consider also the $$\infty$$-category $$N(\textrm{Ch})[Q^{-1}]$$, where $$\textrm{Ch}$$ are chain complexes in abelian groups and $$Q$$ are quasi-isomorphisms. We can define a chain complex functor $$C_* : N(\textrm{Top})[W^{-1}] \to N(\textrm{Ch})[Q^{-1}]$$ by taking the nerve and localizing the classical chain complex functor $$C_* : \textrm{Top} \to \textrm{Ch}$$. At this point, the homotopy totalization is just a limit along $$\Delta$$, so is the following equivalence in chain complexes true? $$C_* \left (\lim^{\infty}_{\Delta} X_{\bullet} \right ) \simeq \lim^{\infty}_{\Delta} C_* X_{\bullet}$$ It is a sort of continuity for cosimplicial diagrams only. A reference would be highly appreciated.

• If I am interpreting correctly, the answer to both questions (which are the same as Dmitri says, there's no coherence issues) is "No, in general" and is a well studied problem having to do with the convergence of the Bousfield-Kan sseq (see, e.g. homepages.math.uic.edu/~bshipley/specseq.pdf ) Jun 1 at 18:59
• Thank you very much to both of you. Very insightful comments! I do even know the BK sseq, but I hadn't realized its convergence was essentially the failure or not of the qi above... Jun 1 at 20:16