Filtration of cohomology via symmetric semisimplicial spaces

Abstract: Inspired by Deligne's use of the simplicial theory of
hypercoverings in defining mixed Hodge structures, we replace the indexing
category $\Delta$ by the \emph{symmetric simplicial category} $\Delta S$
and study (a class of) $\Delta S$-hypercoverings- which also happen to
appear in the avatar of modules over graded commutative monoids of the
form $\mathit{Sym} M$ for some space $M$. For $\Delta S$-hypercoverings we
construct a spectral sequence, somewhat like the
$\check{\mathrm{C}}$ech-to-derived category spectral sequence, obtaining
unified proof of old results and new-- like the computation of (in some
cases, stable) singular cohomology (with $\mathbb{Q}$ coefficients) and
\'etale cohomology (with $\mathbb{Q}_{\ell}$ coefficients) of the moduli
space of degree $n$ maps $C\to \mathbb{P}^r$, $C$ a smooth projective
curve of genus $g$, of unordered configuration spaces, of the moduli space
of smooth sections of a fixed $\mathfrak{g}^r_d$ that is $m$-very ample
for some $m$, some geoemetric Batyrev--Manin type conjectures over global
function fields for weighted projective spaces  etc. In the special case
when a $\Delta S$-object  $X_{\bullet}$ \emph{admits a symmetric
semisimplicial filtration by $M$}, the derived indecomposables of
$H^*(X_{\bullet})$ as a $H^*(\mathit{Sym} M)$-module (in the sense of
Galatius--Kupers--Randal--Williams) give the cohomology of the space of
\emph{$M$-indecomposables}.