Let $f:X\to Y$ be a morphism of locally compact topological spaces. Then there exist a unique subfunctor$f_!: Sh(X)\to Sh(Y)$ of the direct image functor $f_*$ such that for any abelian sheaf$F$ over $X$ the sections of $f_!(F)$ over $U^{open}\subset Y$ are those sections $s\in f_*(U)= \Gamma(f^{-1}(U),F)$ for which the restriction $f|_{supp(s)} : supp(s)\to U$ is a proper map.

This is called the direct image with compact support.

It follows that $f_!$ is left exact.

Let $p:X\to {*}$ be the map into the one point space. Then for any $F\in Sh(X)$ the abelian sheaf$p_!F$ is the abelian group consisting of sections $s\in \Gamma(X,F)$ such that $supp(s)$ is compact. One writes $\Gamma_c(X,F):= p_! F$ and calls this group a group of sections of $F$ with compact support. If $y\in Y$, then the fiber $(f_! F)_y$ is isomorphic to $\Gamma_c(f^{-1}y,F|_{f^{-1}(y)})$.