Definition:Direct Image of Presheaf

Definition
Let $\mathbf C$ be a category.

Let $X$ and $Y$ be topological spaces.

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

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

Definition 1
The direct image of $\mathcal F$ via $f$ is the $\mathbf C$-valued presheaf $f_*\mathcal F$ on $Y$ with:
 * $f_*\mathcal F(V) = \mathcal F(f^{-1}(V))$ for all open $V\subset Y$
 * Restrictions $\operatorname{res}^U_V = \operatorname{res}^{f^{-1}(U)}_{f^{-1}(V)}$

Definition 2
The direct image of $\mathcal F$ via $f$ is the $\mathbf C$-valued presheaf $f_*\mathcal F$ that is the composition $\mathcal F \circ \operatorname{Open}(f)$, where $\operatorname{Open}$ is the open subsets functor.

Also see

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