# Stable cohomology of spaces via symmetric (semi)simplicial filtration

# Stable cohomology of spaces via symmetric (semi)simplicial filtration

We construct a spectral sequence quite like $\check{\mathrm{C}}$ech-to-derived category spectral sequence for hypercoverings on \Delta S$, a small category constructed from, and slightly larger than, $\Delta$. In effect this gives a unified proof of old results and new— e.g. $C$ 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 stacks, complete simplicial toric varieties etc. In the special case when we have a monoid over a graded commutative ring our spectral sequence corresponds to that of the derived indecomposables (in the sense of Galatius--Kupers--Randal—Williams) giving an alternative topological interpretation of the moduli spaces above.