Given an open cover $(U_i)_{\{i\in I\}}$ and $n\in\mathbb N$ and a sheaf $\mathcal F$, let us define $C(U, n)$ to be skew symmetric function of $$\prod\alpha : \Delta_n\to I,\mathcal F\left(U_{\alpha(0)}\cap\dots\cap U_{\alpha(n)}\right)$$