Here are some examples of morphisms f of topoi such that f^{-1}f_*F —> F is always surjective for any sheaf of *sets*:

- If f : X —> Y is a continuous map of topological spaces which induces a homeomorphism of X with a subset of Y.
- If f : Sets —> G-Sets is the morphism of topoi coming from mapping the point to the “classifying space” of the group G.
- If C is a site and f : Sh(C) —> PSh(C) is the morphism of topoi with f^{-1} equal to sheafification and f_* the forgetful functor.

In the first case the map f^{-1}f_*F —> F is actually always an isomorphism but in the second case it isn’t if G is nontrivial. Bhargav pointed out the last one and he also pointed out that you can similarly produce lots of examples for exampl X_{etale} —> X_{Zar} by comparing topologies.

By the way I think it is true that if f : X —> Y is a continuous map of Kolmogorov topological spaces with the property that f^{-1}f_*F —> F is always surjective for any sheaf of *sets*, then f induces a homeomorphism of X with a subset of Y. (I haven’t written out all the details however.)

Here is an example: f : X —> Y and X = {p, q} with discrete topology and Y= {*}. Then for any sheaf of *abelian* groups F the map f^{-1}f_*F —> F is surjective, but this does not hold for every sheaf of *sets*. Namely, a sheaf of sets (resp abelian groups) on X corresponds to a pair of sets (resp abelian groups) F_p, F_q (namely the stalks of F at p and at q). Then f_*F corresponds to F_p \times F_q. Thus we see that f^{-1}f_*F —> F is surjective if and only if the projections F_p \times F_q —> F_p and F_p \times F_q —> F_q are surjective. This is the case if and only if either both F_p and F_q are nonempty or both are empty. But for sheaves of sets F_p not empty and F_q empty can occur!

Of course this is somehow incredibly trivial. But since I’m used to thinking mostly about sheaves of abelian groups it is also very confusing. Namely, any sheaf of abelian groups on {p, q} is globally generated but as seen above it is not the case that every sheaf of sets on {p, q} is “globally generated” (i.e., the target of an epimorphism from a constant sheaf).