The motivation to ask this question is some proposition of flasque sheaves.
Let's recall the definition of flasque sheaf:A sheaf $F$ on a topological space $X$ is flasque if for every inclusion $V\rightarrow U$ of open sets,the restriction map $F(U)\rightarrow F(V)$ is surjective.


It is known that the proposition of flasque sheaves is very similar to injective sheaves.For example,higher cohomology of flasque sheaves vanishes.On the noetherian topological space,direct limit of flasque sheaves is flasque.The category of quasi coherent sheaves on noetherian schemes has enough flasques.And we have localization of flasque sheaves is flasque.Notice that since any injective $O\_X$-module is flasque,then injective sheaves have all the propositions BUT the propositions for injectives can be proved independently without knowing they are flasque.


Notice that the injective sheaf is injective object of category of sheaves of $O\_X$-modules.So it's defined purely in terms of objects and morphism in this category without mentioning the section of this sheaf.But the definition of flasque sheaf is defined in the terms of certain properties of sections of this sheaf.So My question is 


1. Is there any definition of flasque sheaf in the category $O\_X-mod$ only in terms of objects and morphisms(section FREE)?
2. Notice that the definition of injective object in certain category actually depends on the class of morphisms we choose.For example,in the abelian category,the standard definition of injective objects is a special case of $E$-injectives where $E$ is class of all the monomorphisms in this category.We can of course formulate some other "non-standard" injectives by choosing other class of morphisms. For example,in the category of commutative affine schemes,we can choose $E$ to be the class of proper monomorphisms of affine schemes such that the correspondence surjective map between two commutative rings having square zero kernel.Then the $E$-injective objects in category of commutative affine schemes is precisely the smooth affine schemes in this category.Therefore,Is it possible that the flasque sheaves are certain "non standard" $E$-injectives in category $O\_X-mod$ with $E$ to be some class of morphisms in this category(We also notice that the class of morphisms E can be viewed as Grothendieck pretopology in this category,but maybe not finest).


If it is the case,then all the proposition of flasque sheaves can be proved purely general abstract just like injective sheaves.


I am not sure whether it is a stupid question,maybe it is known but I just did not get the right place to look at.Thanks



How to characterize flasque sheaves in more functorial way?
**Yes — by the Yoneda lemma, flasque sheaves can indeed be seen as $E$-injectives, where $E$ consists of the inclusion maps $\newcommand{\O}{\mathcal{O}}\newcommand{\restr}{\mathord{\upharpoonright}}\O\_X\restr\_U \to \O\_X\restr\_V$, for all pairs of opens $U \subset V$.**


Here $\O\_X \restr\_U$ is the sheaf of modules with $\O\_X\restr\_U(U')$ taken to be $\O\_X(U')$ if $U' \subseteq U$, and to be $0$ otherwise. This is the free $\O\_X$-module on the Yoneda sheaf of sets $\newcommand{\y}{\mathbf{y}}\y(U)$, defined by $\y(U)(U') = 1$ if $U' \subseteq U$ and $\emptyset$ otherwise. By the Yoneda lemma, for any sheaf of sets $F$, $F(U)$ is isomorphic to the set of sheaf maps $\y(U) \to F$. For $F$ a sheaf of modules, maps $\y(U) \to |F|$ (where $|F|$ is the underlying sheaf of sets of $F$) correspond to module maps $\O\_X\restr\_U \to F$; so elements of $F(U)$ correspond to such maps, and restriction $F(V) \to F(U)$ corresponds to composition with the inclusion map $\O\_X\restr\_U \to \O\_X \restr\_V$. So this restriction map is surjective exactly if $F$ is injective w.r.t. that inclusion map; and $F$ is flasque exactly if it’s injective w.r.t. the set of all such inclusions.


