Here’s a problem from the Sheaves section of Hartshorne. I’m having difficulty with the very last part…
Background. Consider the forgetful functor from the category of sheaves on to the category of presheaves on . Then if is a presheaf on , there is a universal morphism, say , from to this functor. is called the sheaf associated to the presheaf or alternatively its sheafification.
Let be an abelian group. Equip it with the discrete topology. The corresponding constant sheaf, denoted , is the one which maps each open set to the group of continuous functions , where is equipped with the discrete topology. The restriction maps are the obvious ones.
Problem. Let be an abelian group, and define the constant presheaf associated to on the topological space to be the presheaf for all , with restriction maps the identity. Show that the constant sheaf defined in the text is the sheaf associated to this presheaf.
Solution. Call this presheaf . We need to find a morphism with the property that for any morphism , there is a unique morphism such that .
Hence, let be an open set. We need a morphism of abelian groups . Recall that was the group of all continuous maps of into . The natural choice seems to be to send to the constant function . Let us verify that really is a morphism of presheaves: suppose we have an inclusion . Then the statement that the appropriate diagram commutes is the same as saying “the restriction to , of the constant function on , is the same as the constant function on ”, which is clear. So is a morphism of presheaves.
Before proceeding, we note that if is discrete then any continuous map is locally constant, in the sense that each admits some neighbourhood on which is constant. Locally constant maps are constant on each connected component.
Finally, suppose is a morphism of presheaves, i.e. for each open set we have a morphism of abelian groups . Define, for each open , a morphism as follows. If is connected, then since any continuous is constant (say for ), we may define . If is not connected, then we should have where are the connected components of , and then it should be clear where to go from here.
However, it seems like to finish, we instead want — we want to decompose any continuous as a finite sum of constant functions, and then the way to define will be clear. My question is: how do we know this is possible? What about, say, the function given by ?
EDIT: Maybe one should look at the maps induced on the stalks by , since to figure out what should be, it seems sufficient to only have local data about , which would certainly make our life easy since is locally constant…