Practice problem

Sheaves and Algebraic Geometry

Let $\pi\colon X\to Y$ be a continuous map of topological spaces. Let $\mathcal{F}$ be a sheaf on $X$ and $\mathcal{G}$ a sheaf on $Y$ .

For convenience in this problem only, given open sets $U$ of $X$ and $V$ of $Y$ , call the pair $(U,V)$ comparable if any of the following equivalent conditions holds:

$\pi$ induces a restricted map $U\to V$ ;
$\pi(U)\subseteq V$ ; or equivalently,
$U\subseteq \pi^{-1}V$ .

What is the data of a morphism $\mathcal{G} \to \pi_\ast \mathcal{F}$ of sheaves over $Y$ equivalent to? Select all that apply.

Choose all correct answers

A morphism $\alpha_V\colon \mathcal{G}(V) \to \mathcal{F}(\pi^{-1}V)$ on each open set $V$ of $Y$ , such that the morphisms $\alpha_V$ are compatible under restriction of $V$ .

A morphism $\phi_{VU}\colon \mathcal{G}(V) \to \mathcal{F}(U)$ for every comparable pair $(U,V)$ , such that the morphisms $\phi_{VU}$ are compatible under restriction of comparable pairs $(U,V)$ .

A morphism $\beta_U\colon \varinjlim_{V\supseteq \pi(U)}\mathcal{G}(V) \to \mathcal{F}(U)$ on every open set $U$ of $X$ , such that the morphisms $\beta_U$ are compatible under restriction of $U$ .