TITLE: What is the inverse image of a sheaf
QUESTION [8 upvotes]: Let $f : X \rightarrow Y$ be a continuous map of topological spaces and $\mathcal{G}$ a sheaf on $Y$.
What exactly is $f^{-1}\mathcal{G}$? It seems like we should be able to describe the sections $f^{-1}\mathcal{G}(U)$ over some open subset $U$ as equivalence classes of germs, but I get confused when I try and think what exactly the elements of $f^{-1}\mathcal{G}(U)$ are.

REPLY [4 votes]: @Fredrik
you ask for an example of a presheaf $\mathscr F = f_{pre}^{-1} (\mathscr G )$ which is not a sheaf:
Take a topological space $Y$, choose $X$ as two disjoint copies of $Y$ and set
$$f: X \longrightarrow Y$$
the canonical projection. Take $\mathscr G$ a sheaf on $Y$. For $V \subset Y$ open and $U := f^{-1}(V)$ we have $\mathscr F(U) = \mathscr G(V)$. On the other hand,
$$(f^{-1} \mathscr G) (U) := \mathscr F^{sh}(U)=\mathscr G(V) \times{} \mathscr G(V)$$
with the associated sheaf $\mathscr F^{sh}$.
Note. Due to comments changed coproduct to product.