direct image (functor)

If $f:X\to Y$ is a continuous map of topological spaces, and if $\mathrm{𝐒𝐡𝐞𝐚𝐯𝐞𝐬}(X)$ is the category of sheaves of abelian groups on $X$ (and similarly for $\mathrm{𝐒𝐡𝐞𝐚𝐯𝐞𝐬}(Y)$), then the direct image functor $f_{*}:\mathrm{𝐒𝐡𝐞𝐚𝐯𝐞𝐬}(X)\to \mathrm{𝐒𝐡𝐞𝐚𝐯𝐞𝐬}(Y)$ sends a sheaf $ℱ$ on $X$ to its direct image $f_{*}ℱ$ on $Y$. A morphism of sheaves $g:ℱ\to 𝒢$ obviously gives rise to a morphism of sheaves $f_{*}g:f_{*}ℱ\to f_{*}𝒢$, and this determines a functor.

If $ℱ$ is a sheaf of abelian groups (or anything else), so is $f_{*}ℱ$, so likewise we get direct image functors $f_{*}:\mathrm{𝐀𝐛}(X)\to \mathrm{𝐀𝐛}(Y)$, where $\mathrm{𝐀𝐛}(X)$ is the category of sheaves of abelian groups on $X$.