Definition:Inverse Image Presheaf

Definition
Let $X$ an $Y$ be topological spaces.

Let $f : X \to Y$ be continuous.

Let $\mathbf C$ be a category which has all small inductive limits.

Let $\mathcal F$ be a $\mathbf C$-valued presheaf on $Y$.

The inverse image presheaf of $\mathcal F$ via $f$ is the presheaf $f^{-1}_{\operatorname{Psh}}\mathcal F$ on $X$ with:
 * $(f^{-1}_{\operatorname{Psh}}\mathcal F) (U) = \displaystyle\varinjlim_{V\supseteq f(U)}\mathcal F(V)$ where the inductive limit goes over open $V\subseteq Y$
 * $\operatorname{res}^U_W$ is the induced map on the inductive limit of the subset $\{V : V \supseteq f(U)\} \subseteq \{V : V \supseteq f(W)\}$

Also see

 * Definition:Inverse Image Sheaf
 * Definition:Direct Image Presheaf