Kernel sheaf ($\ker \varphi$)

For a morphism of sheaves of abelian groups or modules, the kernel sheaf has sections $\ker (\mathcal{F}(U)\to \mathcal{G}(U))$. These already satisfy the sheaf gluing axiom. Its stalk at $x$ is the kernel of ${\mathcal{F}}_x\to {\mathcal{G}}_x$, since filtered colimits of modules preserve exactness. A sheaf morphism is injective exactly when its kernel sheaf is zero.