Let's consider an homotopy simplicial chain complex, that is a functor of $\infty$-categories $X_{\bullet} = \textrm{N}(\Delta^{op}) \to \mathcal{C}_{\ge 0}$, where $\mathcal{C}_{\ge 0}$ is the $\infty$-category of chain complexes in non-negative degrees.
There is a spectral sequence $E^{p,q}_r(X_{\bullet})$ associated to $X_{\bullet}$ constructed in Higher Algebra, Remark 1.2.4.4. It is obtained via the associated filtered object of (increasingly bigger) $p$-skeletons; this is the infinity version of Dold-Kan correspondence.
In case $X$ is a strict simplicial chain complex, I think this is the classical spectral sequence obtained by applying:
- The standard Dold-Kan correspondence, with alternating faces differential, obtaining a bicomplex;
- The filtered object associated to a bicomplex;
- The spectral sequence associated to a filtered object.
I think the passages (1)+(2) are the strict version of the infinity Dold-Kan correspondence, while (3) is the strict version of the Lurie spectral sequence (see Remark 1.2.2.1 in Higher Algebra).
My question. Consider the inclusion $\iota: \textrm{N}(\Delta_s^{op}) \to \textrm{N}(\Delta^{op})$, where $\Delta_s$ is the subcategory of $\Delta$ generated by the faces $d_i$. Given $X_{\bullet} : \textrm{N}(\Delta^{op}) \to \mathcal{C}_{\ge 0}$, one can first restrict it to $\textrm{N}(\Delta_s^{op}) $ and then left kan extend to $\textrm{N}(\Delta^{op})$. In this way we obtain an homotopy simplicial object $Y_{\bullet} = \textrm{Lan}_{\iota} \iota^* X_{\bullet}$ and a morphism $Y_{\bullet} \to X_{\bullet}$.
Is the induced morphism on spectral sequences $E^{p,q}_r(Y_{\bullet}) \to E^{p,q}_r(X_{\bullet})$ an isomorphism?
The following is a weaker version, which still convey the idea of depending only on faces (and it would actually suffice for my application):
Weaker Question. Suppose $X_{\bullet}, X'_{\bullet}: \textrm{N}(\Delta^{op}) \to \mathcal{C}_{\ge 0} $ are two homotopy simplicial chain complexes, and suppose we have an equivalence $i^* X_{\bullet} \simeq i^* X'_{\bullet}$. Is is true that $E^{p,q}_r(X_{\bullet}) \simeq E^{p,q}_r(X'_{\bullet})$?
Note. I think one can do the explicit calculations in the classical case and obtain a definitive answer, which I think is yes (at least in the weaker version), since we don't really need degeneracy maps to build the spectral sequence. In case the answer is already no in the classical case, I am sorry for the unnecessary generalization.
