Let $f:X\to Y$ be a continuous map between topological spaces. Let Shv(X), Shv(Y) be the category of sheaves of abelian groups over X, respectively, Y. They are abelian categories. There is a left exact functor $f_*: Shv(X)\to Shv(Y)$ by $f_*F(U):=F(f^{-1}U)$. A sheaf F over X is called flabby(in French,flasque) if for every $V\subset U$, $F(U)\to F(V)$ is surjective. Let Fl(X) be the full subcategory of Shv(X) whose objects are flabby sheaves. My question is why is Fl(X) an $f_*$-injective subcategory. To be specific, this contains 3 questions:
For every sheaf F, find a flabby sheaf I and an injective morphismes from F to I. This can be done by taking $I=\prod_{x\in X} i_{x*}F_x$, where $i_x$ is the injective map from {x} to X.
For a short exact sequence in Shv(X): $0\to F’\to F\to F’’\to 0$, if F’,F are flabby, then F’’ is flabby. This can be shown using snake’s lemma.
For a short exact sequence in Fl(X): $0\to F’\to F\to F’’\to 0$, the induced sequence : $0\to f_*F’\to f_*F\to f_*F’’\to 0$ is also exact. This is what I cannot solve.
A remark on 3, for the case when Y is a point, $f_*=\Gamma(X,-)$, in this case we can prove it using cohomology of Čech, or prove it directly. But I really have no idea on the general case about 3.
Any method and comment are appreciated!
Since $ F’$ is flasque,for any open subset $U$ of $X$,we can show there is a short exact sequence $0\rightarrow F’(U)\rightarrow F(U)\rightarrow F’’(U)\rightarrow 0$（this is well-known）.
We want to show the stalk of the given sequence is exact.$\forall p\in Y$,$(f_{*}F)_p=lim_{\rightarrow_{p\in V}}f_{*}F(V)=lim_{\rightarrow_{p\in V}}F(f^{-1}(V))$,since the functor $\lim_{\rightarrow}$ is exact,we get the conclunsion.