Let $\mathcal F$ be a sheaf on a topological space $X$ and $\mathcal F^\bullet$ a resolution of $\mathcal F$ by acyclic sheaves. Then $$ H^q(X,\mathcal F)\simeq H^q(\mathcal F^\bullet(X)), $$ where $H^q(X,\mathcal F)$ denotes the $q$ th sheaf cohomology group of $X$ with coefficients in $\mathcal F$