The Stacks project

Exercise 111.32.11. Give an example of a topological space $X$ and a functor

\[ F : \mathop{\mathit{Sh}}\nolimits (X) \longrightarrow \textit{Sets} \]

which is exact (commutes with finite products and equalizers and commutes with finite coproducts and coequalizers, see Categories, Section 4.23), but there is no point $x \in X$ such that $F$ is isomorphic to the stalk functor $\mathcal{F} \mapsto \mathcal{F}_ x$.

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View Exercise 111.32.11 as pdf