Let $X$ be a topological space of dimension $0$. I learn from certain text claiming the functor $\Gamma(X,\;\cdot\;)$ gives rise to a categorical equivalence between the category of sheaf of abelian groups $\mathfrak{Ab}(X)$ and the category of abelian group $\mathfrak{Ab}$.
Explicitly why is that? What will be the inverse of the functor $\Gamma(X,\;\cdot\;)$?
Thanks in advance for your answer.
It suffices to work on connected components (as gluing is vacuous), and thus the problem reduces to $X=*$ the one-point space. The only sheaves on a one-point space are the constant sheaves. Thus, the inverse of $\Gamma(*,\;\cdot\;)$ is the assignment which takes an abelian group $A$ to the constant $A$-valued sheaf; i.e., $\mathcal{O}(U)=A$ for all $U\subseteq*$. Since the only nonempty inclusion in $*$ is $*$ itself, $\mathcal{O}(*)=A$ indeed determines the sheaf.