Using the assumption that X is neither discrete nor indiscrete, we have in particular that X is nonempty. So, choose some point x in X. Let x_* : Ab -> Sh(X,Ab) denote the skyscraper sheaf functor, and let x^* : Sh(X,Ab) -> Ab denote the stalk functor. We note that x_* is exact, and that x^* x_* is (naturally isomorphic to) the identity.
Now note that there is a non-split exact sequence 0 -> Z -> Z -> Z/2 -> 0 in Ab. Applying x_* to this exact sequence gives an exact sequence in Sh(X,Ab) which cannot be split, since any splitting would give a splitting of the original sequence by applying x^*.
This issue has been created by Ben Spitz via the submission form on https://catdat.app/category/Sh(X,Ab)
Using the assumption that X is neither discrete nor indiscrete, we have in particular that X is nonempty. So, choose some point x in X. Let x_* : Ab -> Sh(X,Ab) denote the skyscraper sheaf functor, and let x^* : Sh(X,Ab) -> Ab denote the stalk functor. We note that x_* is exact, and that x^* x_* is (naturally isomorphic to) the identity.
Now note that there is a non-split exact sequence 0 -> Z -> Z -> Z/2 -> 0 in Ab. Applying x_* to this exact sequence gives an exact sequence in Sh(X,Ab) which cannot be split, since any splitting would give a splitting of the original sequence by applying x^*.
This issue has been created by Ben Spitz via the submission form on https://catdat.app/category/Sh(X,Ab)