Definition:Direct Image of Presheaf/Definition 1

Definition
Let $\mathbf C$ be a category.

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 $\mathcal F$ be a $\mathbf C$-valued presheaf on $T_1$.

The direct image of $\mathcal F$ via $f$ is the $\mathbf C$-valued presheaf $f_* \mathcal F$ on $T_2$ with:


 * $f_* \mathcal F \left({V}\right) = \mathcal F \left({f^{-1} \left({V}\right)}\right)$ for all open $V \subset T_2$


 * Restrictions $\operatorname{res}^U_V = \operatorname{res}^{f^{-1} \left({U}\right)}_{f^{-1} \left({V}\right)}$

Also see

 * Equivalence of Definitions of Direct Image of Presheaf