Definition:Inverse Image Presheaf

Definition
Let $T_1 = \left({S_1, \tau_1}\right)$ and $T_2 = \left({S_2, \tau_2}\right)$ be topological spaces.

Let $f: T_1 \to T_2$ be continuous.

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

Let $\mathcal F$ be a $\mathbf C$-valued presheaf on $T_2$.

The inverse image presheaf of $\mathcal F$ via $f$ is the presheaf $f^{-1}_{\operatorname{Psh}} \mathcal F$ on $T_1$ with:
 * $\left({f^{-1}_{\operatorname{Psh}} \mathcal F}\right) \left({U}\right) = \displaystyle \varinjlim_{V \supseteq f \left({U}\right)} \mathcal F \left({V}\right)$ 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 $\left\{ {V: V \supseteq f \left({U}\right)}\right\} \subseteq \left\{ {V : V \supseteq f \left({W}\right)}\right\}$

Also see

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