The Stacks project

Remark 6.31.13. Let $j : U \to X$ be an open immersion of topological spaces as above. Let $x \in X$, $x \not\in U$. Let $\mathcal{F}$ be a sheaf of sets on $U$. Then $j_!\mathcal{F}_ x = \emptyset $ by Lemma 6.31.4. Hence $j_!$ does not transform a final object of $\mathop{\mathit{Sh}}\nolimits (U)$ into a final object of $\mathop{\mathit{Sh}}\nolimits (X)$ unless $U = X$. According to our conventions in Categories, Section 4.23 this means that the functor $j_!$ is not left exact as a functor between the categories of sheaves of sets. It will be shown later that $j_!$ on abelian sheaves is exact, see Modules, Lemma 17.3.4.