Take the 2-minute tour ×  Let $\mathscr{F}$ be a presheaf and
$\mathscr{F}^+$ its sheafification, with the universal morphism
$\theta:\mathscr{F}\rightarrow\mathscr{F}^+$. Question is: is
$\theta$ always an inclusion? I'm pretty sure it isn't, but in
many cases it seems that it is. For example, if
$\phi:\mathscr{F}\rightarrow\mathscr{G}$ is a morphism of
sheaves, then $\operatorname{ker}\phi$, $\operatorname{im}\phi$,
$\operatorname{cok}\phi$ all seem to have this property. If it
isn't always the case, then is there any characterization of
such presheaves? A "subpresheaf" of a sheaf certainly has this
property (e.g., $\operatorname{im}\phi$), but what about
$\operatorname{cok}\phi$?  share|improve this question Just to
make sure, you are asking if the kernel of $\theta$ is always
trivial? –  M Turgeon Jun 3 '12 at 15:06 @M Turgeon: Yes. – 
ashpool Jun 3 '12 at 15:07 Dual question here. –  Zhen Lin Jun 3
'12 at 16:56  3 Answers 3  up vote 9 down vote accepted  No,
$\mathcal F \to \mathcal F^+$ is not injective