Morphism of sheaves - Imported Knowlpedia Content

Definition

Let $\mathcal F$ and $\mathcal G$ be sheaves on a topological space $X$ . A morphism of sheaves $\varphi:\mathcal F\to\mathcal G$ consists of a map

\varphi_U:\mathcal F(U)\longrightarrow\mathcal G(U)

for every open subset $U\subseteq X$ , compatible with restriction: whenever $V\subseteq U$ , restricting after applying $\varphi_U$ gives the same result as applying $\varphi_V$ after restricting.

These maps induce maps on every stalk,

\varphi_x:\mathcal F_x\longrightarrow\mathcal G_x.

More generally, if $f:X\to Y$ , $\mathcal F$ is a sheaf on $Y$ , and $\mathcal G$ is a sheaf on $X$ , a morphism of sheaves along $f$ is a morphism into the direct image sheaf:

\mathcal F\longrightarrow f_*\mathcal G.

When the sheaves take values in rings, groups, or modules, every component map preserves the corresponding algebraic structure.