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

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 $0 \to F \to G \to H \to 0$, its exactness is usually stated in terms of their stalks at each point $x$ in $X$.

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 $x$ in $X$.

Q2. I cannot see at all what this will mean. Please explain more plainly.

I heartily wish somebody's explanation.