category of sheaves of abelian groups properties

Question

I am working on some properties of categories. And I would like to know how could we characterize the category of sheaves of abelian groups on a point? It seems to be equivalent to the category of abelian groups but I do not know why?

Thanks.

Answer 1

It's even isomorphic to the category of abelian groups. The functor sends a sheaf $\mathcal{F}$ to its group of global sections $\Gamma(\mathcal{F})$. The inverse functor is that given an abelian group, you can build a sheaf on the one point space by assigning its sections on the point to be the abelian group (and its sections on the empty set to be 0).

(And given a morphism of sheaves, we get a morphism of the groups of global sections, and conversely.)

EDIT in response to comment: to build a presheaf of abelian groups on a topological space, you have to assign an abelian group $\mathcal{F}$ to each open subset $U$, and restriction maps $\mathcal{F}(V) \to \mathcal{F}(U)$ whenever $U \subseteq V$. The one-point space has only two open subsets. Moreover, for the presheaf to be a sheaf, we'll need $\mathcal{F}(\emptyset) = 0$ and the restriction map $\mathcal{F}(\{\text{pt}\}) \to 0$ has to be the zero map. So the only piece of data we actually get to pick is the abelian group $\mathcal{F}(\{\text{pt}\})$