= Solution
For the usual additive <Čech cohomology> groups, $\mathcal F$ is understood to be a <sheaf of abelian groups>. Order the index set of the <open cover> $\mathfrak U=(U_i)$. The <Čech cochain group> is
$$
\check C^q(\mathfrak U,\mathcal F)=\prod_{i_0<\cdots<i_q}\mathcal F(U_{i_0}\cap\cdots\cap U_{i_q}),\qquad q\geq0.
$$
An empty intersection contributes the zero group. The <Čech coboundary> is the alternating sum of restriction maps:
$$
(\delta c)_{i_0\ldots i_{q+1}}
=\sum_{a=0}^{q+1}(-1)^a
c_{i_0\ldots\widehat{i_a}\ldots i_{q+1}}\big|_{U_{i_0}\cap\cdots\cap U_{i_{q+1}}}.
$$
Every double omission appears twice with opposite sign, so $\delta^2=0$. \b[The Čech cohomology is the cohomology of this cochain complex]:
$$
\boxed{\check H^q(\mathfrak U,\mathcal F)
=\frac{\ker(\delta:\check C^q\to\check C^{q+1})}{\operatorname{im}(\delta:\check C^{q-1}\to\check C^q)}.}
$$
Here $\check C^{-1}=0$. The <sheaf gluing axiom> identifies $\check H^0(\mathfrak U,\mathcal F)$ with $\Gamma(X,\mathcal F)$. This definition is for the fixed cover; no limit over refinements is part of the group requested here. Without the abelian-group structure, this additive <Čech cochain complex> is not defined.
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