Definition:Direct Image of Presheaf
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Definition
Let $\mathbf C$ be a category.
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 $\FF$ be a $\mathbf C$-valued presheaf on $T_1$.
Definition 1
The direct image of $\FF$ via $f$ is the $\mathbf C$-valued presheaf $f_* \FF$ on $T_2$ with:
- $\map {f_* \FF} V = \map \FF {f^{-1} \sqbrk V}$ for all open set $V$ of $T_2$
- Restrictions $\operatorname{res}^U_V = \operatorname{res}^{f^{-1} \sqbrk U}_{f^{-1} \sqbrk V}$
Definition 2
The direct image of $\FF$ via $f$ is the $\mathbf C$-valued presheaf $f_*\FF$ that is the composition $\FF \circ \map {\operatorname{Open} } f$, where $\operatorname{Open}$ is the open subsets functor.