# Sheaf Theory/Sheaves

**Definition (sheaf)**:

Let be a site, a category, and . Then is called a **sheaf** if and only if the rightmost object and the rightmost morphism in the diagram

define an equalizer of its two leftmost arrows.

**Proposition (the morphism from sections on an open set of a sheaf to the corresponding stalk product is mono)**:

Let be a topological space, a category, and a sheaf. Then for each open , the morphism in

defined by the universal property of the stalk is a monomorphism.

**Proof:** Let be morphisms in such that .

**Proposition (morphism to a sheaf is uniquely determined by the induced morphisms on the stalks)**:

Let be a topological space, a category, a presheaf and a sheaf. Let

**Example (an epimorphism of sheaves need not be surjective on all open subsets)**:

Consider the topological space , whose open sets are given by , and ; it is easy to see that these three sets form a topology on . Then set

- , and .

The sole nontrivial restriction shall be given by the projection onto the second factor. This is a sheaf, because the sheaf condition is empty: There are no nontrivial open covers for any open set, and for trivial open covers the sheaf condition is always satisfied. Now, define another sheaf on exactly like , except that , and the restriction map is the identity. By the required compatibility with the restriction maps, there is a unique morphism that induces the identity . Hence, it is an epimorphism, since the compatibility with the restriction maps implies that a morphism emitting from is determined by its behaviour on . But it is certainly not surjective on .