symmetric monoidal (∞,1)-category of spectra
(also nonabelian homological algebra)
Let $\mathcal{C}$ be a stable (∞,1)-category. Then the (∞,1)-categories of non-negatively graded sequences in $C$ is equivalent to the (∞,1)-category of simplicial objects in an (∞,1)-category in $\mathcal{C}$
Under this equivalence, a simplicial object $X_\bullet$ is sent to the sequence of geometric realizations ((∞,1)-colimits) of its simplicial skeleta
This constitutes a filtering on the geometric realization of $X_\bullet$ itself
(Higher Algebra, theorem 1.2.4.1)
Given a simplicial object $X_\bullet$ in a stable (∞,1)-category $\mathcal{C}$, its image in the triangulated homotopy category $Ho(\mathcal{C})$ is identified by the ordinary Dold-Kan correspondence with a chain complex. On the other hand, by the discussion at spectral sequence of a filtered stable homotopy type in the section Filtered objects and their chain complexes, the skelton sequence $sk_\bullet X_\bullet$ also induces a chain complex. These are naturally isomorphic.
In particular therefore first page of the spectral sequence of a filtered stable homotopy type associated with the simplicial skeleton filtration consists of the Moore complexes of the simplicial objects $\pi_q(X_\bullet) \in \mathcal{A}^{\Delta^{op}}$
(Higher Algebra, remark 1.2.4.3, 1.2.4.4)
This infinity-Dold-Kan correspondence is theorem 12.8, p. 50 of
later absorbed in