- Verify that the examples of §2.1 are indeed sheaves of differentiable function, or continuous functions, or smooth functions, or of functions on a manifold of \( \mathbb{R}^n \))
- Show that real-valued continuous functions on (open sets of) a topological space \(X\) form a sheaf.

- (sheaf of sections of a map) Suppose we are given a continuous map \( \mu : Y \to X \). Show that "sections of \(\mu\)" form a sheaf. More precisely, to each open set \( U \) of \(X\), associate the set of continuous maps \( s : U \to Y \) such that \( \mu \circ s = id\vert_U \). Show that this forms a sheaf. (For those who have heard of vector bundles, these are a good example.) This is motivation for the phrase "section of a sheaf".
- (This exercise is for those who know what a topological group is. If you don't know what a topological group is, you might be able to guess.) Suppose that \(Y\) is a topological group. Show that continuous maps to \(Y\) form a sheaf of \( groups.\)

- If \(\FF\) is a sheaf of sets on \(X\), then show that \( \Hom(\{p\},\FF) \cong \FF \), where \(\{p\}\) is the constant sheaf associated to the one element set \(\{p\}\).
- If \(\FF\) is a sheaf of abelian groups on \(X\), then show that \( \Hom_{Ab_X} (\underline{\Z}, \FF ) \cong \FF\) (an isomorphism of sheaves of abelian groups).
- If \(\FF\) is an \(\OO_X\)-module, then show that \(\Hom_{\Mod{\OO_X}} (\OO_X, f\FF) \cong \FF \) (an isomorphism of \(\OO_X\)-modules).

- Show that Exercise 2.4.A is false for general presheaves.
- Show that Exercise 2.4.D is false for general presheaves.
- Show that Exercise 2.4.E is false for general presheaves.

- \( \phi \) is a monomorphism in the category of sheaves.
- \( \phi \) is injective on the level of stalks: \( \phi_p : \FF_p \to \GG_p \) is injective for all \(p \in X \).
- \( \phi \) is injective on the level of open sets: \( \phi(U) : \FF(U) \to \GG(U) \) is injective for all open \( U \subset X \).

- \( \phi \) is an epimorphism in the category of sheaves
- \( \phi\) is surjective on the level of stalks: \( \phi_p : \FF_p \to \GG_p\) is surjective for all \( p \in X \)

- Verify that a morphism of sheaves is determined by the induced morphism of sheaves on the base.
- Show that a morphism of sheaves on the base gives a morphism of the induced sheaves. (Possible hint: compatible stalks.)

- Suppose \( \OO_X\) is a sheaf of rings on \(X\). Define (categorically) what we should mean by tensor product of two \( \OO_X \)-modules. Give an explicit construction, and show that it satisfies your categorical definition. Hint: take the "presheaf tensor product" — which needs to be defined — and sheafify. Note: \( \otimes_{\OO_X} \) is often written \( \otimes \) when the subscript is clear from the context. (An example showing sheafification is necessary will arise in Example 14.1.1.)
- Show that the tensor product of stalks is the stalk of the tensor product. (If you can show this, you may be able to make sense of the phrase "colimits commute with tensor products".)

Thanks for reading! 😁