Ordered Čech cochain complex of a cover

Let $X$ be a topological space, let $\mathcal{F}$ be a sheaf of abelian groups on $X$ (A sheaf on a topological space), and let $\mathcal{U}=(U_i{)}_{i\in I}$ be a cover of $X$ by open subsets, indexed by a set $I$ that is equipped with a linear order $<$.

For an integer $p\ge 0$ define the group of ordered $p$-cochains of $\mathcal{U}$ with values in $\mathcal{F}$ by $C^p(\mathcal{U},\mathcal{F}):={\prod }_{i_0<\cdots <i_p}\mathcal{F}(U_{i_0}\cap \cdots \cap U_{i_p}),$ the product of the section groups over all increasing $(p+1)$-tuples $i_0<\cdots <i_p$ in $I$; write an element as $s=(s_{i_0\cdots i_p})$. For $p<0$ put $C^p(\mathcal{U},\mathcal{F}):=0$. When $I$ has no increasing $(p+1)$-tuple the product is empty and $C^p(\mathcal{U},\mathcal{F})=0$; when an intersection $U_{i_0}\cap \cdots \cap U_{i_p}$ is empty, the corresponding factor is the one-element group $\mathcal{F}(\mathrm{\varnothing })=0$ (A set-valued sheaf has a unique section over the empty open set).

The Čech differential ${\delta }^p:C^p(\mathcal{U},\mathcal{F})\to C^{p+1}(\mathcal{U},\mathcal{F})$ is defined on an increasing $(p+2)$-tuple $i_0<\cdots <i_{p+1}$ by $({\delta }^{p}s{)}_{i_0\cdots i_{p+1}}={\sum }_{j=0}^{p+1}(-1{)}^j{s}_{i_0\cdots \widehat{i_j}\cdots i_{p+1}}{\mid }_{U_{i_0}\cap \cdots \cap U_{i_{p+1}}},$ where $\widehat{i_j}$ means that the index $i_j$ is omitted and the restriction is taken from $U_{i_0}\cap \cdots \widehat{U_{i_j}}\cdots \cap U_{i_{p+1}}$ to the full intersection (Sections, restrictions, and global sections of a presheaf). For $p<0$ set ${\delta }^p:=0$. The maps ${\delta }^p$ are group homomorphisms, componentwise sums of restrictions, so the data $(C^p(\mathcal{U},\mathcal{F}),{\delta }^p{)}_{p\in \mathbb{Z}}$ form a cochain complex of abelian groups; that ${\delta }^{p+1}\circ {\delta }^p=0$ is proved in The Čech differential squares to zero. The pair $(\mathcal{F},\mathcal{U})$ being fixed, the construction is functorial in $\mathcal{F}$: a morphism $\phi :\mathcal{F}\to \mathcal{G}$ of abelian sheaves induces componentwise maps $C^p(\mathcal{U},\phi )$ commuting with the differentials.