For a topological space X, the category of sheaves on X with its values in (Ab) will form an Abelian Category.
Q1: Is it difficult to prove this?
Next, for the short exact sequence 0 ---> F ---> G ---> H ---> 0, its exactness is usually stated in terms of their stalks at each point x on X.
However, somebody told me that this is a ``theorem" rather than definition. Namely, once I know that the category of sheaves on X with values in (Ab) (we call this (Sh)_X makes an Abelian category, automatically the notion of exact sequence exists.
Hence, it only turns out that the short exact sequence defined via the characteristic of (Sh)_X being Abelian category is equivalent to the exactness of the given short exact sequence after taking stalk at an arbitrary point x on X.
Q2. I cannot see at all what this will mean. Please explain more plainly.
I heartily wish somebody's explanation. Sincerely, Pierre MATSUMI