Let $F:\mathcal{A}\rightarrow\mathcal{B}$ be an exact functor between abelian categories, which are some category of sheaves of abelian groups or sheaves of modules...etc on certain topological spaces/schemes/etc.
Must $F$ send acyclic objects to acyclic objects (acyclic with respect to the global sections functor)?
If $\mathcal{A}$ has enough injectives, must $F$ send injective objects to acyclic objects?
- Are there reasonable criteria on $F,\mathcal{A},\mathcal{B}$ so that something like (1) or (2) is true?