Let $M$ be a topological Hausdorff space. We use the following definitions (as they may vary):
A presheaf $\mathcal{F}$ is a collection of vector spaces $\mathcal{F}(U)$ for each open subset $U$ of $M$ equipped with restriction maps $R_{UV}:\mathcal{F}(U)\to\mathcal{F}(V)$ for any open subsets $U,V$ with $V\subset U$ for which the following holds:
- For any $U,V,W$ open subsets such that $W\subset V\subset U$, we have $R_{UW}=R_{VW}\circ R_{UV}$
- $R_{UU}=\text{Id}\vert_{\mathcal{F}(U)}$
An element of a presheaf is called a section of the presheaf.
A sheaf $\mathcal{F}$ is a presheaf such that the following holds for any open subset $U$ of $M$ and any open cover $\{U_{i}\}_{i\in I}$ of $U$:
- $f\in\mathcal{F}(U)$ such that $f\vert_{U_{i}}:=R_{UU_{i}}(f)=0$ implies $f=0$ (note that $f\vert_{\cup_{i\in I}U_{i}}=f\vert_{U}=f$).
- If $\{f_{i}\}_{i\in I}$ is a family of sections with $f_{i}\in\mathcal{F}(U_{i})$ for all $i\in I$ such that for all $i,j\in I$, $f_{i}\vert_{U_{i}\cap U_{j}}=f_{j}\vert_{U_{i}\cap U_{j}}$, then there exists $f\in\mathcal{F}(\cup_{i\in I}U_{i})=\mathcal{F}(U)$ such that $f\vert_{U_{i}}=R_{UU_{i}}(f)=f_{i}$ for all $i\in I$
A morphism $\psi:\mathcal{F}\to\mathcal{G}$ between two sheaves is a collection of homomorphisms $\psi_{U}:\mathcal{F}(U)\to\mathcal{G}(U)$ compatible with restrictions: for any open subsets $U,V$ with $U\subset V$ denote by $R^{\mathcal{H}}_{UV}$ the restriction map associated with the sheaf $\mathcal{H}=\mathcal{F}\text{ or }\mathcal{G}$; $\psi_{V}\circ R^{\mathcal{F}}_{UV}=R^{\mathcal{G}}_{UV}\circ\psi_{U}$
A stalk $\mathcal{F}_{x}$ of $\mathcal{F}$ in $x\in M$ can be seen as the set of equivalence classes for the following relation. Let $U$ and $U'$ be two open subsets containing $x$. We say that two functions $f\in\mathcal{F}(U)$ and $g\in\mathcal{F}(U')$ are equivalent if there exists an open subset $W\subset U\cap U'$ such that $R_{UW}(f)=R_{U'W}(g)$ (it is more properly defined with inductive limit). If $f\in\mathcal{F}(U)$, there canonically corresponds an equivalence class $f_{x}\in\mathcal{F}_{x}$ called the germ of $f$ in $x$.
There are two equivalent ways of defining an injective morphism between sheaves: either it has to be injective on each $\mathcal{F}(U)$ or it has to be injective on stalks.
My purpose is to prove that, for any sheaf $\mathcal{F}$ of vector spaces, there exists a monomorphism from $\mathcal{F}$ to some flasque sheaf $\mathcal{G}$. A natural candidate for $\mathcal{G}$ is the collection of $\mathcal{G}(U):=\prod_{x\in U}\mathcal{F}_{x}$ and a natural monomorphism is the following:
$$\psi_{U}:\mathcal{F}(U)\to\mathcal{G}(U):=\prod_{x\in U}\mathcal{F}_{x}:f^{U}\mapsto (f^{U}_{x})_{x\in U}$$
where $f^{U}$ denotes a section of $\mathcal{F}$ over $U$ (i.e. an element of $\mathcal{F}(U)$) and $f^{U}_{x}$ denotes the germ of $f^{U}$ in $x$. I can do it when I use the definition saying that $\psi$ is injective if every $\psi_{U}$ is injective but I don't manage to do it whith the other definition since I'm unsure of the stalk $\mathcal{G}_{x}$.
This is why I needed to prove the equivalence between the two definitions. Updated 03/02/16: I was able to prove this equivalence. Still, I would like to know if it is possible to determine the stalk of the particular $\mathcal{G}$ (defined previously) in some $x$.
EDIT: It is probably better if I show what I think. Restriction maps for $V\subset U$ send $(f_{x})_{x\in U}\to(f_{x})_{x\in V}$. Let $U,U'$ be two open subsets containing some point $y$. Let $\varphi\in\mathcal{G}(U)$ and $\gamma\in\mathcal{G}(U')$. By definition of $\mathcal{G}$, $\varphi=(f_{x})_{x\in U}$ and $\gamma=(g_{x})_{x\in U'}$. They are equivalent if there exists an open subset $W\subset U\cap U'$ such that $(f_{x})_{x\in W}=(g_{x})_{x\in W}$, i.e. $f_{x}=g_{x},\,\forall x\in W$. It leads me to think that $\mathcal{G}_{y}=\mathcal{F}_{y}$ for any $y\in M$. But is that possible?
Any help is appreciated.