Čech differential ($\delta :{\check{C}}^p\to {\check{C}}^{p+1}$)

The Čech differential sends a cochain on $(p+1)$-fold intersections to the alternating sum of its restrictions to $(p+2)$-fold intersections. Each pair of omitted indices occurs twice with opposite signs in the composite, so ${\delta }^2=0$. It defines the Čech cohomology of the cover.