Consider the category of presheaves of abelian groups over a topological space $X$ denoted by $\mathrm{Prshf}_X$. The category of abelian groups is denoted by $\mathrm{Ab}. $It's well known that the sheafification functor is exact. My question is whether it preserves injective objects.
EDIT: To my best knowledge of sheaves of abelian groups over a topological space, I view them through localization theory of abelian categories. Let me describe it now. The stalk at every point $x\in X$ can be seen as a (exact) functor $(-)_x:\mathrm{Prshf}_X\rightarrow \mathrm{Ab}$. Consider the intersection of the kernel of $(-)_x$ at every point $x\in X$: $$\mathcal N=\bigcap_{x\in X} \mathrm{ker}(-)_x$$ This is a localizing subcategory of $\mathrm{Prshf}_X$, i.e. it is closed under taking subobjects, quotient objects, and closed under extensions and arbitrary coproducts. Then the $\mathcal N$-closed objects(an object $A$of $\mathrm{Prshf}_X$ is $\mathcal N$-closed if $\mathrm{Hom}(\mathcal N,A)=0$ and $\mathrm{Ext}^1(\mathcal N, A)=0$ ) of the category $\mathrm{Prshf}_X$ are exactly sheaves. The full subcategory of $\mathcal N$-closed objects is denote by $\mathcal L$ (which is just the category of sheaves). Then we have the well-known adjoint pair $\mathrm (F,G):{Prshf}_X\rightleftharpoons \mathcal L$ where $F$ is the sheafification functor and $G$ is the inclusion functor. Since $\mathrm{Prshf}_X$ is a Grothendieck category, we may ask whether the sheafification functor preserves injectives. Two results are known (see Popescu, Abelian categories with applications,p182) :
If the injective presheaf $\mathcal I$ is an injective envelope of a presheaf $\mathcal F$ having no non-zero $\mathcal N$- subobjects, then we know the sheafification of $\mathcal I$ is injective.
If the topological space $X$ is good enough to make the category $\mathrm{Prshf}_X$ contain the injective envelope of each of its objects in $\mathcal N$, then the sheafification functor preserves injectives.