Let $\Delta^{\bullet}$ be the cosimplicial scheme that at each level is given by the algebraic simplex over the field $k$ which contains $\mathbb{C}$. Let $\hat{\Delta}^{\bullet}$ be semi-localization of $\Delta^{\bullet}$ at its vertices (at each level). For every pre-sheaf $\mathcal{F}$ on smooth $k$-schemes one can consider the chain complex associated to $\mathcal{F}(\hat{\Delta}^{\bullet})$ and can calculate its homologies. Now let's assume that $\mathcal{F}$ is the pre-sheaf given by $n$-th algebraic De Rham cohomology of the schemes viewed as $\mathbb{C}$-schemes (Or equivalently the singular cohomology when considered with the analytical topology).
Note that the De Rham cohomology for a semi-local simplex over $k$, is defined as the colimit of De Rham cohomologies affine opens in a finite type $\mathbb{C}$-scheme.
Is it possible to calculate or gain any insight on the homologies of $\mathcal{F}(\hat{\Delta}^{\bullet})$ in this case? or is this some intractable object?
