Morphism of sheaves ($\varphi :\mathcal{F}\to \mathcal{G}$)

A morphism of sheaves assigns a map $\mathcal{F}(U)\to \mathcal{G}(U)$ on every open set, commuting with all restriction maps. For sheaves of abelian groups, sheaves of rings or sheaves of modules, the maps must also preserve the relevant algebraic structure. It induces maps on every stalk, and equality of sheaf morphisms can be checked there.

For a morphism of sheaves $\varphi :\mathcal{F}\to \mathcal{G}$ of abelian groups or modules, the cokernel sheaf is the sheafification of the presheaf $U\mapsto \mathcal{G}(U)/\varphi (\mathcal{F}(U))$. Its stalk at $x$ is $\mathrm{coker}({\mathcal{F}}_x\to {\mathcal{G}}_x)$. The presheaf quotient itself need not satisfy the sheaf gluing axiom, which explains why a surjective morphism of sheaves need not be surjective on global sections.

For a continuous map $f:X\to Y$, an f-morphism from a sheaf on $Y$ to a sheaf on $X$ is a family of the displayed maps commuting with restrictions. It is equivalently a morphism of sheaves $\mathcal{G}\to f_{\ast }\mathcal{F}$. For each $x\in X$ it induces a map ${\mathcal{G}}_{f(x)}\to {\mathcal{F}}_x$ on stalks, by pulling back the neighbourhood representing a germ.

A morphism of sheaves of modules is surjective when the induced map on every stalk is onto. Equivalently every target section can locally be lifted. It need not be onto on global sections; the long exact sequence in sheaf cohomology describes that obstruction.

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.