Lurie's $\infty$-categorical Dold-Kan Correspondence relates simplicial objects and sequential diagrams in a stable $\infty$-category. Is there any reference for an equivalence to a category of homotopy chain complexes? That this equivalence holds is well-known, and I feel comfortable with the argument, but I'd like to save myself the trouble of writing it down and to give credit to whoever first did so. Actually I'm interested in cochain complexes, but that isn't important.

Specifically, I'm thinking of a homotopy coherent cochain complex as follows: let $\mathcal{P}_{fin}\mathbb{Z}_{>0}$ be the poset of finite subsets of the positive integers. Call $S\in\mathcal{P}_{fin}\mathbb{Z}_{>0}$ *orderly* if for all positive integers $n$, $n\leq \max(S)\implies n\in S$. A cochain complex in a pointed $\infty$-category $\mathcal{D}$ is a functor $$C:\mathcal{P}_{fin}\mathbb{Z}_{>0}\to \mathcal{D}$$ such that $C(S)=0$ whenever $S$ is not orderly.

dualDold-Kan correspondence? That's for cochain complexes andcosimplicial objects. I think it was first written down in 2004 by Castiglioni and Cortiñas: sciencedirect.com/science/article/pii/S0022404903003025 $\endgroup$