# Reference for the equivalence between chain complexes and sequential diagrams in a stable $\infty$-category

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.

• Are you sure you don't want the dual Dold-Kan correspondence? That's for cochain complexes and cosimplicial objects. I think it was first written down in 2004 by Castiglioni and Cortiñas: sciencedirect.com/science/article/pii/S0022404903003025 Oct 6, 2022 at 11:01
• @DavidWhite yes, but because the definition of stable infinity-category is self-dual, the two are equivalent, and either will work for my purpose :) Thank you for the reference!! Oct 9, 2022 at 0:58