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.
