Definition:Acyclic Sheaf
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Definition
Let $X$ be a topological space.
Let $\FF$ be an abelian sheaf on $X$.
Let $\map \Gamma {X, -} : \map {\mathbf {Ab} } X \to \mathbf {Ab}$ be the global sections functor on $X$.
Then $\FF$ is acyclic if and only if $\FF$ is $\map \Gamma {X, -}$-acyclic in the category of abelian sheaves $\map {\mathbf {Ab} } X$ on $X$.