Let f:X\to Y be a
continuous mapping of
locally compact Hausdorff topological spaces, and let \mathrm{Sh}(-) denote the
category of sheaves of
abelian groups on a topological space. The
direct image with compact (or proper) support is the
functor :f_{!}:\mathrm{Sh}(X)\to \mathrm{Sh}(Y) that sends a sheaf \mathcal{F} on X to the sheaf f_{!}(\mathcal{F}) given by the formula :f_{!}(\mathcal{F})(U):=\{s\in\mathcal{F}(f^{-1}(U)) \mid {f\vert}_{\operatorname{supp}(s)}:\operatorname{supp}(s)\to U \text{ is proper}\} for every open subset U of Y. Here, the notion of a
proper map of spaces is unambiguous since the spaces in question are locally compact Hausdorff. This defines f_{!}(\mathcal{F}) as a subsheaf of the
direct image sheaf f_*(\mathcal{F}) and the functoriality of this construction then follows from basic properties of the support and the definition of sheaves. The assumption that the spaces be locally compact Hausdorff is imposed in most sources (e.g., Iversen or Kashiwara–Schapira). In slightly greater generality, Olaf Schnürer and
Wolfgang Soergel have introduced the notion of a "locally proper" map of spaces and shown that the functor of direct image with compact support remains well-behaved when defined for separated and locally proper continuous maps between arbitrary spaces. ==Properties==