Given a presehaf of abelian groups $\mathcal{F}$ and an open cover $\mathcal{U}:=\{U_i\}_{i=0}^n$, I can define the Cech cohomology groups $\check{H}^q(\mathcal{U},\mathcal{F})$.
It s well known that elements of $\check{H}^0(\mathcal{U},\mathcal{F})$ (i.e. 0-cocycles) satisfy the compatibility condition
$$ z_i\mid_{U_i\cap U_j}=z_j\mid_{U_i\cap U_j}~\forall i,j $$
and thus can be interpreted as compatibile families.
Can we interpret the elements of the first cohomology group $\check{H}^1(\mathcal{U},\mathcal{F})$ in a similiar way? I'm trying to achieve a concrete definition of this group.