Definition:Inverse Image Presheaf

Definition

Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ 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 $\FF$ be a $\mathbf C$-valued presheaf on $T_2$.

The inverse image presheaf of $\FF$ via $f$ is the presheaf $f^{-1}_{\operatorname {Psh} } \FF$ on $T_1$ with:

$\map {\paren {f^{-1}_{\operatorname{Psh} } \FF} } U = \ds \varinjlim_{V \mathop \supseteq \map f U} \map \FF V$ 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 $\set {V: V \supseteq \map f U} \subseteq \set {V : V \supseteq \map f W}$

Also see

• Definition:Inverse Image Sheaf
• Definition:Direct Image Presheaf