Consider the adjunction of $\infty$categories $\mathbb{Z}[]: \mathcal{Spaces} \rightleftarrows \text{Ch}_{\ge 0}(\mathbb{Z}): $ where the left adjoint takes a space to its singular chain complex over $\mathbb{Z}$. Is this adjunction monadic? By BarrBeckLurie, this would be equivalent to the right adjoint $$ preserving $$split geometric realizations. I've been unable to come up with a nice counterexample or proof, and no references of any sort, but I have to imagine someone else has given this some thought.

1$\begingroup$ Do you mean the 1category $\operatorname{Ch}_{\ge0}(\mathbb Z)$ of $\mathbb N$graded chain complexes, but not the connective derived category $\mathcal D_{\ge0}(\mathbb Z)$? Then this seems to be false: the right adjoint $\operatorname{Ch}_{\ge0}(\mathbb Z)\to\operatorname{An}$ does not seem to be conservative. $\endgroup$– Z. MCommented Jul 14 at 21:43

$\begingroup$ @Z.M I mean the $\infty$category of $\mathbb{N}$graded chain complexes up to quasiisomorphism  apologies if I used implicitly $1$categorical notation. $\endgroup$– K. StrongCommented Jul 15 at 19:06

$\begingroup$ Every 1category is an $\infty$category. There is a real issue: it makes sense to talk about chain complexes in an additive ($\infty$)category. In particular, there is a notion of chain complexes of spectra (which is rougly equivalent to completely filtered spectra), and Lurie has a version of Dold–Kan correspondence for stable $\infty$categories (which is further generalized to weakly idempotent additive $\infty$categories by Tashi Walde). Thus denoting the derived category by "chain complexes" is very misleading. $\endgroup$– Z. MCommented Jul 15 at 19:59
1 Answer
This answer assumes that $\def\Ch{{\sf Ch}}\def\Z{{\bf Z}}\Ch_{≥0}(\Z)$ refers to the derived ∞category of chain complexes, i.e., with quasiisomorphisms inverted up to a homotopy.
Recall that the right adjoint to $\def\sSet{{\sf sSet}}\def\N{{\sf N}}\N\Z[]\colon\sSet→\Ch_{≥0}$ can be computed as $UΓ$, where $\def\sAb{{\sf sAb}}Γ\colon\Ch_{≥0}(\Z)→\sAb$ is the Dold–Kan functor and $U\colon\sAb→\sSet$ is the forgetful functor.
The functor $Γ$ is an equivalence of ∞categories, so it suffices to establish that $U\colon\sAb→\sSet$ is a monadic functor.
The ∞category $\sAb$ of simplicial abelian groups is the ∞category of algebras over the algebraic theory of abelian groups. Every ∞category of algebras over an algebraic theory is monadic over the ∞category of spaces, which concludes the proof.

$\begingroup$ Thank you! For anyone else who is looking for a reference, arxiv.org/abs/1903.02991v1 (Example 2.5). $\endgroup$ Commented Jul 15 at 20:10