Cosheaf homology Global Sections 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?
 A: 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.
A: The $0$th Čech homology group $ \check{\mathrm{H}}_0(X,F) $ of a topological space $ X $ with coefficients in an abelian cosheaf $ F $ is isomorphic to the global (co)sections of $ F $, i.e. $ F(X) $, in a natural way. This is actually an immediate consequence of Proposition 4.2. in Section V from the book "Bredon, Sheaf Theory". 
A sketch for this statement is as the following: 
$ \check{\mathrm{H}}_0(X,F) $ is the same as cokernel of $ \partial_0 $ (the 0th boundary operator). On the other hand, by cosheaf condition, the cokernel of  $ \partial_0 $ is $ F(X) $. Thus, the result is obtained.
