Elementary example showing that $(f_*\mathcal{F})_{f(x)} \neq \mathcal{F}_x$

Question

Let $f: X \to Y$ be a continuous function and $\mathcal{F}$ be a sheaf (of sets, or abelian groups) over $X$. Is there an elementary example that shows $\mathcal{F}_x \neq (f_*\mathcal{F})_{f(x)}$. All the examples I've seen involve algebro-geometric concepts, but I was wondering if there was a simple one.

Answer 1

Let's take $X =\{a,b\}$, $Y=\{c\}$, and $f : X \to Y$ the unique function. We'll give each of $X$ and $Y$ the discrete topology.

A sheaf $\mathcal{F}$ on $X$ is essentially a pair of sets. More precisely, $\mathcal{F}$ is determined by $\mathcal{F}(\{a\}) =A$ and $\mathcal{F}(\{b\})=B$. Everything else is determined by the sheaf axiom. Note that we have $\mathcal{F}_a = A$.

A sheaf $\mathcal{G}$ on $Y$ is even easier: it's essentially just a set. More precisely, it is determined by its global sections. In the case of $\mathcal{G} = f_*\mathcal{F}$, we have $f_*\mathcal{F}(Y) = A \times B$, using notation from above. In particular, $f_* \mathcal{F}_{f(a)} = A \times B \neq A = \mathcal{F}_a$, assuming $B$ is not a singleton.