Cosheaf homology Global Sections

Question

Let X be a topological space and U={Ui} be some open covering of X. Let F be a cosheaf of abelian groups on X. Is the 0th cech homology group the same as the global sections on X?

Answer 1

I am not very familiar with this area, but wouldn't the following cosheaf give something distinct from a 0th Čech homology group? Let $X$ be a disconnected space with two components $A$ and $B$. Let the cosheaf $F$ be defined by $F(U)=\mathbb Z$ if $U\neq\emptyset$ and $U\subset A$. Let $F(U)=\mathbb Z_2$ if $U$ is nonempty and $U\subset B$. If $U$ meets both $A$ and $B$ then $F(U)=\mathbb Z\oplus\mathbb Z_2$ and if $U=\emptyset$ then $F(U)=\{0\}$. This seems to satisfy the cosheaf axioms.

The global sections are isomorphic to $\mathbb Z\oplus\mathbb Z_2$, which is a hybrid between the $0$th homology with coefficients in two different abelian groups.