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.
Best Answer
I believe https://arxiv.org/abs/2109.01017 does what you want! The description of coherent chain complexes used there is a bit different than what you suggest, but they look equivalent at first glance.