Filtration of cohomology via symmetric semisimplicial spaces

In the simplicial theory of hypercoverings we replace the indexing category Δ Δ by the symmetric simplicial category Δ S Δ S and study (a class of) Δ ᵢₙⱼS Δ inj S -hypercoverings, which we call spaces admitting symmetric (semi)simplicial filtration —this special class happens to have a structure of a module over a graded commutative monoid of the form Sym\,M Sym M for some space M . For Δ S Δ S -hypercoverings we construct a spectral sequence, somewhat like the Čech-to-derived category spectral sequence. The advantage of working with Δ S Δ S over Δ Δ is that various combinatorial complexities that come with working on Δ Δ are bypassed, giving simpler, unified proof of results like the computation of (in some cases, stable) singular cohomology (with Q Q coefficients) and étale cohomology (with Q Q ℓ coefficients) of the moduli space of degree n maps C→ Pʳ C → P r with C a smooth projective curve of genus g , of unordered configuration spaces, of the moduli space of smooth sections of a fixed gʳd g d r that is m -very ample for some m etc. In the special case when a Δ ᵢₙⱼS Δ inj S -object X admits a symmetric semisimplicial filtration by M , we relate these moduli spaces to a certain derived tensor.